More on Dynamic Nature of Number (2)

In yesterday's blog entry, I indicated how our experience of a number keeps switching as between both quantitative and qualitative aspects with respect to both base and dimensional expressions respectively in a two-way dynamic complementary manner.

In fact the situation in truth is even more intricate, with experience also switching as between both internal and external perceptions in each case.

So for example the quantitative (base) notion of an number such as "2" in an independent cardinal sense alternates as between both the external aspect (in the acknowledgement of the number "object") and also internal aspect (in the acknowledgement of the corresponding perception of the number "2").


The next key area is then to properly distinguish analytic from holistic type appreciation.

In brief the analytic approach - as I define it - attempts to separate the opposite key polarities of experience in an absolute type manner leading to a fixed unambiguous form of understanding.

Thus for example with respect to number, the external aspect  (as the number "object") is separated from the internal aspect (as number "perception) and with both in effect thereby reduced in terms of each other. More customarily the internal aspect is reduced in terms of the external so that we thereby attempt to understand the behaviour of number (i.e. as number "objects") in an unambiguous absolute type manner.

Likewise - and perhaps even more significantly - both quantitative and qualitative aspects are likewise separated in an unambiguous manner with the qualitative aspect then in effect reduced in terms of the quantitative. Thus for example - using again "2" to illustrate in conventional terms no clear distinction is made as between the quantitative  notion of "2" as an independent number and the corresponding qualitative notion of "2" (i.e. as twoness) whereby it is seen as interdependent with all other instances of "2".

And as I indicated in the last blog entry, this represents the precise reason why the crucial distinction as between the operations of addition and multiplication is not properly understood in Conventional Mathematics.

By contrast the basis of the holistic approach is that the opposite polarities (that govern all mathematical experience) are now considered in a dynamic relative interactive manner as complementary with each other.

So from a holistic perspective the notion of number necessarily represents a dynamic interaction as between its external and internal polarities (which are positive and negative with respect to each other).

Likewise in holistic terms, the notion of number necessarily represents a dynamic interaction as between its quantitative and qualitative aspects in  both quantitative and qualitative aspects (which are now real and imaginary with respect to each other),

And of course in this holistic context the very meaning of mathematical notions (such as positive and negative; real and imaginary, etc) themselves switch from their customarily understood analytic to a new distinctive holistic meaning.

So every mathematical notion, with a clearly defined meaning in analytic terms, can be given a coherent alternative meaning in a holistic manner.  

Ultimately the most comprehensive form of mathematical understanding entails the harmonious interaction of both analytic and holistic type meaning.


Therefore for a comprehensive mathematical worldview, I define 3 distinct types of Mathematics, that I term Type 1, Type and Type 3 respectively.

The first worldview relates to customary analytic type understanding of an absolute type nature. This is Type 1 Mathematics.
In formal terms, effectively all accepted Conventional Mathematics belongs to this one category.

The second worldview relates to the totally unrecognised (in formal terms) holistic type appreciation of mathematical symbols, which is of an approximate relative nature. This is Type 2 Mathematics.

Though completely unrecognised by the Mathematics profession, I have spent more than 50 years of my life in developing fundamental key notions of a holistic kind.

For example I have now long realised that all developmental processes (such as human transformation) are  of a holistic mathematical nature.
Thus in this context, Holistic Mathematics entails the elaborate mapping of all possible stages of development (physical and psychological) with their corresponding encoding in mathematical terms.

The third worldview, which is by far the most comprehensive entails the harmonious combination of both the analytic and holistic approaches. I generally refer to this as Radial Mathematics, which equally corresponds to Type 3 Mathematics.
In truth, as the analytic and holistic aspects are themselves complementary in nature, one cannot properly understand mathematical reality (with respect to any issue) without adopting this approach.

However it is important to appreciate that even though the analytic aspect (so heavily dominant in Type 1 Mathematics) is once again restored, it is done so in a relative - rather than absolute - manner.

So for example, if we assert the truth of the Pythagorean Theorem, for example, In Type 1 Mathematics, this will be understood in an absolute type manner. However in Type 3, though the proof still maintains an important validity, it is understood in a merely relative manner that necessarily is still strictly subject to uncertainty.

I will finish this entry by giving a simple illustration of the distinction between the three approaches.

In Type 1 terms the left and right turns at a crossroads are understood in an absolute type manner as unambiguously either left or right . This implies a linear (1-dimensional) approach where only one unambiguous direction (either N or S) in terms of approaching the crossroads is considered.

In Type 2 terms, the left and right turns are now understood in relative terms as paradoxically both left and right. This implies a circular (in this case 2-dimensional) approach where both possible directions (N and S) are simultaneously considered with respect to approaching the crossroads.

Thus what is left (approaching from a N direction) is likewise right (when approaching from the opposite S direction); and what is right (approaching from a N direction) is left (when approaching from the opposite S direction).

Thus Type 2 understanding is circular - rather than linear - in nature (though it must necessarily start with linear type appreciation). It is multi-dimensional in nature (with a minimum of 2 dimensions involved). 2-dimensional appreciation entails the simultaneous recognition of 2 opposite directions, serving as reference frames. Multi-dimensional in more general terms entails the simultaneous recognition of n distinct reference frames (that geometrically can be represented in holistic mathematical terms as the n roots of 1).

In Type 3 terms, left and right turns at a crossroads have a partial unambiguous linear interpretation as either left or right  (depending on relative context when N and S directions of approach are separated) while also having a holistic paradoxical circular interpretation as both left and right when the two reference frames (N and S) are simultaneously considered.

This type of understanding represents the changing frames in a movie.

At any given moment, just one frame will be in evidence; however because of the paradox created by opposite reference frames, these keep switching in complementary fashion. Thus a limited partial validity applies to the analytic interpretation associated with each frame, while the overall holistic appreciation of the complementary nature of these frames ensures that these partial interpretations continually change.

More on Dynamic Nature of Number (1)

I am continuing here my most recent refinements relating to the truly dynamic interactive nature of number.

To keep this at its  very simplest, we will illustrate here with respect to the number 2.

Now as will become quickly apparent, every number in this context is defined with respect to both a base and dimensional number aspect that are complementary in quantitative and qualitative terms.
Thus once again in the expression a^b, a is the base and b the dimensional number accordingly!

So it is important to appreciate how the base number (representing a quantity) is complementary to its (default) dimensional number, which relatively - has a qualitative meaning.

1) Thus when I refer to "2" as a number quantity, this is necessarily defined with respect to a (default) dimensional number of 1.

Therefore it is more properly written as 2¹.

Thus 2 is thereby a specific actual number that is defined with respect to an overall linear dimension that potentially relates to all real numbers.

Thus when we use 1 to represent this dimension it strictly carries a qualitative - as opposed to quantitative - meaning.

The very notion of quantitative implies independence (from all other numbers).
However to enable such a number to be then related with other numbers, we require the corresponding notion of general number interdependence which is - relatively - of a qualitative nature
And this number interdependence is provided by the dimensional notion of number (which in this default case represents the 1st dimension i.e. the number line).

So I represent 2 as a specific number quantity, by highlighting this number (which is emphasised here in an explicit manner) in black, while the default dimensional number of 1 is shown in light grey (to show that it remains merely implicit in this instance).


2) We next look at the reverse notion of "2", now written as 2¹.
The number "1" which now represents the dimensional aspect of 1, carries an actual quantitative meaning i.e. as applying to all actual numbers (in this context natural numbers) on the number line.

The number now "2" represents the qualitative aspect of 2, where this number is now understood as in common with all other classes of 2 objects. For example if we have 3 columns with 2 items in each row, then we can strike a one-to-one correspondence as between the three columns (where each contains "2" items).

What is vital however to appreciate here is "2" is not now being used in a quantitative sense (where it is viewed as independent of other numbers) but rather qualitatively, whereby "2" is now seen to be interdependent with the members of each column (i.e. each column is similar in containing 2 members)

This in fact represents the fundamental difference as between addition and multiplication.

With addition two number quantities are combined (without change of qualitative dimension).

So 2¹ + 3¹ = 5¹.

However when we multiply thee numbers we strictly combine both quantitative and qualitative meaning.

So  2¹ * 3¹ = 6¹ * 1².

Thus both a quantitative change in units, as well as a qualitative change in the dimensional nature of the units takes place. Thus is we have a small table with length 3 ft. and width 2 ft. respectively we can immediately recognise that its are is 6 square feet. However in the conventional treatment of number multiplication the qualitative dimensional aspect of transformation is simply ignored.

Therefore referring again to the rows and columns, we must recognise initially the independence of the rows and columns (as separate).
However equally in then multiplying 3 by 2, we recognise the common quality of twoness with respect to each of the 3 columns.

So once again 2¹ represents the situation where the base number "2" is now of a qualitative nature and "1" as dimensional number assumes a quantitative identity as applying to any actual (natural) number on the (1-dimensional) line. In this way we can multiply 2 by 1, 2, 3, 4,.......

However with multiplication a change takes place to the qualitative aspect.
So in this context "2" refers to the two-dimensional plane which now potentially stretches in two directions in an infinite manner

Thus in moving from 1) to 2), the meaning of both base and dimensional numbers switch.

In 1), the base number "2" is quantitative and the dimensional number "1" is qualitative; however in 2), the base number "2" is now qualitative, and the dimensional number "1" is quantitative.   

3) We now look at the interpretation of 1².  

"2" as dimensional number is now used in a qualitative sense, representing the simple multiplication operation 1 * 1. Once again the length and width of the unit square are not independent of each other but related to each other in a specific manner. Therefore "2" is here qualitative. 

However  "1" as base number has a quantitative meaning representing the one (2-dimensional) object that results. So just as in 1) we defined an actual number i.e. "2" with respect to the number line (as potentially infinite), here we are defining an actual object i.e. 1 object with respect to the 2-dimensional plane that is potentially infinite. 

Thus in this context, the base number is quantitative and the dimensional number qualitative respectively.   

4) Finally we look at the interpretation of  1².  

Here "2" representing dimension carries an actual meaning, whereby it can be applied to classes of 1 object. So for example if we were comparing areas of different fields, these would all be of a 2-dimensional nature, that would apply in the case of each field. In this sense each field (as a unit) would be in common with each other field so that "1" would now have a qualitative meaning. 

And "2" now representing the dimensional number (with an actual finite significance) would be thereby quantitative in nature.

Thus again in 3) the base number "1" is quantitative and the dimensional number "2" is qualitative; however in 4) the base number "1" is now qualitative and the dimensional number "2" is quantitative.


Thus in the dynamics of experience, both base and dimensional numbers (which psychologically are represented through corresponding perceptions and concepts) keep switching as between both their quantitative and qualitative meanings respectively in a dynamic complementary manner.  

Therefore in the simplest possible case the number "1" can have a base meaning (corresponding to the actual rational perception of "1" that is of an independent quantitative nature.
However equally "1" can have a base meaning (corresponding to the potential intuitive perception of "1") that is of a common qualitative nature as applying to all classes of 1 object).

Then "1" can have a dimensional meaning (corresponding to the potential intuitive concept of "1" that is of an common qualitative nature (i.e. the number line as potentially applying to all numbers).

Finally "1" can have a dimensional meaning (corresponding to the actual rational concept of "1" that is of a common quantitative nature (i.e. the number line as actually applying to specific numbers).

Connection Between Zeta 1 and Zeta 2 Functions

As I have outlined on many occasions, there are really two zeta functions (of equal importance) that can only be properly understood in a dynamic interactive manner.

What has long interested me however is a certain unexpected property of the Zeta 2 Function, which then becomes enshrined in the Zeta 1 Function as its central feature.

Once again the Zeta 2 Function starts from the consideration of each prime as comprising a unique grouping of natural numbered objects (in ordinal terms).

So once again to illustrate, 3 is a prime which - by definition - is thereby compromised of  1st, 2nd and 3rd members.

Now when 3 is defined in the standard cardinal manner, it is comprised of homogeneous units in quantitative terms, i.e. 3 = 1 + 1 + 1.

This notion of 3 in Type 1 terms is then represented as 3¹. So 3 is here explicitly defined as a number quantity with respect to 1 (representing the 1-dimensional number line) which remains implicit.

However because quantitative and qualitative notions interact in a complementary manner, therefore to represent the corresponding qualitative notion of 3, it now explicitly is represented as a dimensional number that is defined with respect to an implicit base number that is 1.

What this entails in effect is that we cannot explicitly recognise qualitative ordinal distinctions (i.e. relationships between numbers) without implicitly recognising each number as an independent unit in quantitative terms.

So in the first case we are concentrating on the independent nature of the number 3 (in quantitative terms).

In the second case we are now concentrating on the interdependent  nature of 3 (in qualitative terms).
But both notions in dynamic interactive terms are of a merely approximate nature.

In this way we can keep switching as between cardinal (quantitative) and ordinal (qualitative) notions of 3 in a relative manner.

The big issue then arises as to how one can convert - as it were - the qualitative notion of 3 (expressing the relationship between its individual members) in a quantitative manner.

And because the qualitative notion is literally in this context 3-dimensional, this entails taking the three roots of 1.

Put more accurately in requires taking the cube root of 1¹, 1², 1³  respectively i.e. 1^1/3, 1^2/3 and 1^3/3
respectively.

These 3 roots of 1 then uniquely express in a quantitative manner the notions of 1st, 2nd and 3rd (in the context of 3 members) .

So expressed as an equation, 1 = s³. 

Now one solution (i.e. 1 = s) is trivial, as it is always one of the roots of unity. Thus dividing by
1 –  s,  we get 1 + s¹  + s² = 0.

The two solutions to this equation could thereby be referred to as the non-trivial zeros. In this case these would represent the 1st and 2nd members of a group of 3.  The third member (of a group of 3) would be represented by the trivial solution . In other words when we already know the 1st and 2nd members, the 3rd member therefore can be unambiguously identified in an absolute type manner.


So generalising for all primes the Zeta 2 Function can be expressed as the solutions to the finite equation 1 + s¹  + s²  + s³  +….. + s^{t – 1} = 0 (where t is prime).

Then because of the unique relationships as between the primes and the natural numbers (where every natural number is composed of a unique combination of prime factors),

 1 + s¹  + s²  + s³  +….. + s^{t – 1}  = 0 (where t is a natural number).

Thus for any natural number t, we have t – 1 non trivial solutions. These represent in a quantitative manner the 1st, 2nd, 3rd,..... (t – 1) ordinal rankings with respect to t members of a group.


The issue the arises as to to what happens when this finite equation is extended in an infinite manner.

In other words what can we say about the value of the infinite Zeta 2 equation

 1 + s¹  + s²  + s³  +….  or alternatively  s⁰ + s¹  + s²  + s³  +…  ?

This seems very interesting as its terms bear an inverse form to the Zeta 1 Function,

i.e. 1 ^{– s}  +  2 ^{– s}  + 3 ^{– s}  + 4 ^{– s}  + .....

Now returning to the infinite Zeta 2 expression, what we are now attempting to do is to estimate its value for each set of zeros that arise when this is finite.

Now the simplest case is for t = 2, where just one trivial zeros arises i.e.  –  1.

Then substituting this value in infinite expression we get the alternating series

1 –  1 + 1 –  1 +........


Now clearly if the infinite series has an even number of terms its value = 0.

If however it has an odd number of terms, its value = 1.

Now because the probability of the series having an even or odd number of terms is equal i.e. 1/2 then we can say that its expected value = (0 + 1)/2 = .5.

And this in fact is the value that is customarily given for this series.

However initially, it might not seem at all obvious what the expected value of the series might be when a multiple set of non-trivial zeros arise (as in every case where t > 2 in the finite series).


So we will now explore the case where t = 3. This means that the two relevant zeros are –  .5 + .866i and –  .5 + .866i respectively  (expressed correct to 3 decimal places).

Now when the first value is substituted in the infinite equation i.e.  1 + s¹  + s²  + s³  +…., we get

1 –  .5 + .866i  –  .5 – .866i, which then keeps recurring with every 3 terms.

Therefore if we assume that the infinite series is made up of multiples of 3 i.e. 3n terms, then the value of the infinite series = 0.

If however the infinite series comprises 3n + 1 terms, its value = 1.

Finally if the infinite series comprises 3n + 2 terms, then its value = .5 + .866i

Thus we have 3 possible values here!


However we must equally consider the value of the infinite series for the other zero!

So the first 3 terms (which will then continually repeat) are

1  –  .5 – .866i  –  .5 + .866i .

Once more if the infinite series comprises 3n terms its value = 1.

If the series comprises 3n + 1 terms its value = 0.

Finally if the series has 3n + 2 terms its value = .5 –.866i.


Thus considering both possible zeros in this case (where t = 3) we have 3 * 2 ( = 6) possible values for the series.

Now on the assumption that all these outcomes are equally likely, the expected value of the infinite series = (1 + 0  + .5 + .866i + 1 + 0 + + .5 –. .866i )/2 = 3/6 = .5.

Now in more general terms the series for any finite value t will yield t * (t  – 1) possible values for the infinite series.

And it is postulated here that the expected value in all cases =   .5!


When looked on in the appropriate manner, this result is immensely revealing.

It is no accident that with the Riemann Hypothesis (in relation to the Zeta 1 function) that all non-trivial zeros are postulated to lie on the imaginary line through .5.

In fact the unexpected link with the much simpler Zeta 2 infinite function provides a remarkably simple way of expressing the true nature of the Riemann Hypothesis which is intimately tied up with the probabilistic nature of reality.

The very essence of probability is that it tries to bridge both finite and infinite realms (which are quite distinct from each other).

We could equally say that both finite and infinite realms relate to the quantitative and qualitative aspects respectively of the number system.

Now if we take the simple case of tossing an unbiased coin, we may well maintain that the probability of getting a H or a T is equally likely i.e. = .5.

However there is a subtle problem here. The postulate that both outcomes are equally likely strictly relates to a potential infinite order.

However when we empirically carry out trials where we repeatedly toss the coin, we are now dealing with the actual finite realm.

Now of course with a finite number of trials the number of H's and T''s recorded is unlikely to be equal. However the assumption that is made - which is strictly an act of faith - is that somehow the (actual) finite can eventually be successfully bridged with the (potential) infinite case.

So therefore if we were to keep increasing the number of tosses, we would be confident that the actual behaviour (of recorded tosses) would approximate ever more closely to the assumed potential behaviour in the infinite case (i.e. that both outcomes = .5).


Therefore we can look at the Riemann Hypothesis as the very condition that is required to justify the very assumptions that are made with respect to all probabilistic behaviour (which properly transcends rational behaviour).

Put even more simply the Riemann Hypothesis is required to properly underpin our  assumption that repeated tosses of an unbiased coin will approximate ever more closely to an equal outcome of H's and T's .

Now as this assumption - which is properly an act of faith - entails a relationship as between both finite and infinite (or alternatively quantitative and qualitative) aspects of understanding, it cannot thereby be rationally proved (using the accepted axioms of Mathematics). These are based on merely reduced quantitative notions, which allow for no distinct holistic - as opposed to analytic - appreciation of mathematical symbols.  

Thus in the end the true significance of the Riemann Hypothesis is that the very nature of the number system (and all Mathematics) is inherently dynamic, entailing a two-way interaction as between finite and infinite notions (i.e. quantitative and qualitative meaning).

In short the very nature of mathematical reality - when appropriately understood - is that it is inherently probabilistic!

The True Nature of the Zeta Zeros (4)

So far we have looked at the nature of the two sets of zeta zeros, from just one (relatively) fixed perspective.

Thus starting with the standard quantitative cardinal notion of prime numbers (in analytic terms), we demonstrated how the Zeta 1 (Riemann) zeros, represent the complementary qualitative notion of the primes (in a holistic manner), where they are intimately related with the successive factors of the (composite) natural numbers.

Likewise starting with the standard qualitative ordinal notion of natural numbers (in analytic terms), we demonstrated how the Zeta 2 zeros, represent the complementary quantitative notion of these numbers (again in a holistic manner), where they are intimately related with the successive prime roots of 1!

So in both cases, the zeta zeros represent the holistic complements of what we customarily interpret in a merely analytic (i.e. linear rational) manner.
From a psycho spiritual perspective, the zeta zeros represent the (hidden) unconscious shadow, therefore, of what we customarily seek to understand in a merely conscious manner.

Thus the enormous consequence of properly appreciating the nature of these two sets of zeros (in their bi-directional relationship with the primes and natural numbers) is that the very nature of Mathematics itself must radically change to explicitly incorporate, as equal partners, both conscious and unconscious aspects of understanding.


However true appreciation of this dynamic relationship (linking the primes and natural numbers to both sets of zeros) can be shown to be of an even more refined subtle nature, in that through the very dynamics of experience, reference frames continually switch.

Therefore we have to consider this relationship from the - equally valid - opposite perspective.

So here we start with the holistic qualitative notion of prime numbers (in holistic terms), to demonstrate how the Zeta 1 (Riemann) zeros, now represent the complementary quantitative notion of these primes (in an analytic type manner).

So what does this precisely mean?

We we are of course familiar with the quantitative notion of a prime e.g. 3, as an independent "building block" of the natural number system, which would be represented on the number line.

The corresponding qualitative notion of "3" could now be represented as "threeness" relating to its unique 1st, 2nd and 3rd members, that can be represented as equidistant points on the unit circle. So each prime here represents in effect a unique manner of configuring group interdependence!

However if we then attempt to give meaningful expression to the multiplication of such primes, they must likewise assume a quantitative identity.

So when we start from the quantitative perspective, we are led to the realisation that the multiplication of primes must likewise entail a corresponding qualitative identity!

However equally in complementary fashion, when we start from the qualitative perspective, we are led to the realisation that the multiplication of primes must likewise entail a corresponding quantitative identity to be meaningful.

This entails that the corresponding generation of Zeta 1 (Riemann) zeros equally has a quantitative interpretation (now as the analytic shadow of the holistic nature of primes).

There is a close parallel here with respect to conventional understanding where it is recognised that the primes and the zeros are dual to each other.

Unfortunately, however because of the rigid assumptions underlying conventional interpretation, the true dynamic implications of this duality are not appreciated. In other words the recognition that the primes and the (Riemann) zeros are dual to each other should properly suggest that they are complementary to each other in a dynamic interactive manner!    .

However from the conventional perspective, it is indeed recognised that we can from one perspective use the Riemann zeros to eliminate remaining deviations with respect to the precise calculation of the the number of primes (to a given number on a real scale).

Equally it is recognised that we can use the primes to eliminate remaining deviations with respect to the precise calculation of zeros (to a given number on an imaginary scale).

Thus when we appreciate this relationship properly in a dynamic interactive manner, we come to the realisation that both the primes and Riemann zeros both possess analytic and holistic aspects, which continually interchange with each other.

Then the refined appreciation of this dynamic interaction (which requires the marriage of both pure reason and pure contemplative insight) brings one to that original intersection of both the quantitative and qualitative aspects of meaning (that lies as the final partition between phenomenal and ineffable reality).


Equally we can switch the frames of reference with respect to appreciation of the corresponding relationship as between the Zeta 2 zeros and the number system.

So here we start with the quantitative notion of numbers representing dimensions. So just as we can use the primes to represent number objects (as base numbers), equally we can use the primes to represents objects (representing dimensions e.g. as in 3 dimensions. So here the corresponding Zeta 2 zeros (obtained through the prime roots of 1) would be interpreted in complementary qualitative manner (through the collective combination of all the natural numbered roots)..

Thus once more a full realisation requires the ability to appreciate the zeros in both an analytic and holistic manner and also each natural number in both an analytic and holistic manner with the relationship between both complementary.


Thus when we combine bi-directional appreciation of the number system with respect to primes and natural numbers and both sets of zeros from the two interchanging perspectives as outlined then the true synchronistic relativity of the number system can be appreciated (from both a cardinal and ordinal perspective) whereby experience can approximate as close as is possible in the phenomenal realm to absolute union with reality.

So the refined rational appreciation of the number system (that fundamentally underlies all reality), ultimately cannot be separated from contemplative union with this reality.

The True Nature of the Zeta Zeros (3)

In the last two blog entries, I have outlined how the the two sets of zeros enable conversion of both cardinal and ordinal notions (as analytically understood) in a complementary holistic manner.

So again with the Zeta 1 (Riemann) zeros, we start with the quantitative notion of primes (i.e. 2, 3, 5, ...), which serve as the building blocks of the natural number system, except 1, (again in a quantitative manner).

However through the process of multiplication of primes, a qualitative dimension is also crucially involved, whereby the primes (and by extension other natural numbers) now serve uniquely as factors of other composite numbers.

Therefore the absolute independent identity of primes is thereby lost, when they are combined as factors of subsequent natural numbers!

So in truth, a new unique qualitative identity (of a merely relative nature) is thereby established.

For example if we take the composite number 12 to illustrate, we perhaps readily appreciate that it is composed from two initial prime building blocks i.e. 2 and 3.

However this means that in the context of 12, both 2 and 3 as constituent factors obtain a new qualitative identity. In other words through their relationship to 12 - and the qualitative aspect inherently relates to number relationships - 2 and 3 are thereby uniquely reflected in a new light.
However other natural numbers (already attained from prime building blocks) are now also uniquely reflected in a new light i.e. 4, 6 and 12.

Thus, from this perspective associated with 12, are 5 natural number factors (including primes and other composite natural numbers derived from primes).

The factors - where each is uniquely reflected in the light of the composite natural number concerned - thereby express the qualitative nature of the natural number system.


And there is a direct link as between the accumulated frequency of such factors and the corresponding frequency of the famed Zeta 1 (Riemann) zeros.

So a stated before the frequency of these non-trivial zeros to t (on the imaginary line) matches very closely the corresponding (accumulated) frequency of factors to n (on the real line) where n = t/2π.

For example, the frequency of zeros to t = 628 (on the imaginary line) = 91.
The corresponding (accumulated) frequency of factors to n = 100, i.e. n = t/2π (on the real line) = 98, which already in relative terms is fairly close.

Indeed there is equally a close relationship as between the accumulated sums of factors and corresponding sum of zeros.

(In attaining the sums of factors we multiply each composite number - the primes are excluded - by the number of its factors. So for 12 (as illustrated), we would add 60 i.e. 12 * 5.

Thus in this manner, the accumulated sum of factors to 100 (on the real line) = 20367 (according to my estimate).
The corresponding sum of zeros to 628 (on the imaginary line) with the result then divided by 2π = 20133. So the relative accuracy here is already close to 99%!

This thereby reveals the very nature of the Zeta 1 (Riemann) zeros as an imaginary (i.e. holistic) expression of the qualitative nature of the primes. And because of  complementarity as between analytic and holistic notions, the primes and zeros thereby both necessarily lie on lines (real and imaginary respectively) which mutually mirror each each other in a perfect manner!

However once more, this can only be properly understood from a dynamic interactive perspective, where the zeros, as representative of holistic qualitative appreciation, are clearly recognised as directly complementary with the standard analytic appreciation of the primes in quantitative terms.

So we start with the (conscious) analytic appreciation of the primes as absolute in a merely quantitative manner.

Then we are ultimately led, through the zeros, to this (hidden) shadow recognition of the corresponding (unconscious) holistic recognition of the primes, as purely relative in a qualitative manner.

Thus the true integration in understanding of both aspects (i.e. primes and zeros) ultimately requires the full incorporation of both analytic and holistic modes of mathematical appreciation.

This in turn requires from a psycho spiritual perspective the consequent full integration of both conscious and unconscious aspects of all mathematical understanding.

Thus the poverty of present conventional understanding is starkly revealed, where no formal recognition whatsoever of the holistic aspect yet exists!

And quite simply the true role of the zeros cannot be grasped in the absence of this holistic aspect.


So the Zeta 1 (Riemann) zeros express, in a holistic numerical manner, the hidden qualitative aspect of the cardinal primes (through their collective relationship to the natural number system).

Then the Zeta 2 zeros likewise express, in a complementary holistic mathematical manner, the hidden quantitative aspect of the ordinal natural numbers (in their relationship to individual primes representing groups).

So once again the ordinal notions of 1st, 2nd, 3rd,...are inherently of a qualitative nature (expressing in any given finite context, a relationship between a group of numbers).
Then, through obtaining the successive prime roots of 1, these ordinal members, except 1, can be uniquely expressed in an (indirect) circular quantitative manner.
And because of the fundamental importance of primes as building blocks (now as unique "circles of interdependence"), the ordinal nature of all numbers can likewise be expressed in a quantitative manner (through the corresponding roots of that number).

Now, the Zeta 2 zeros  represent all these roots, except 1, thereby expressing the qualitative nature of ordinal numbers in an (indirect) quantitative manner.
And once again the interpretation of such roots is properly of a holistic - rather than analytic - nature.

Thus, again for example, the 3 roots of 1, i.e. 1, .– 5 + .866i, and  – .5 – .866i express in an indirect quantitative manner the ordinal notions of 1st, 2nd and 3rd (in the context of 3 members) .

However the interpretation is properly of a circular (holistic) nature, where the relative independence of each individual root as quantitative (representing each distinct ordinal identity) can only be properly understood in the context of the collective interdependent identity (through addition) of all 3 members. 

And in every case, this collective sum = 0, representing the corresponding qualitative nature of these members. In this way, through appropriate holistic understanding, both the (individual) quantitative and (collective) qualitative nature of the roots (expressing their ordinal identity) can be fully balanced in a dynamic relative manner!


And again it is similar in a complementary manner with respect to the Zeta 1 (Riemann) zeros.

Here, through multiplication, each individual zero has a qualitative identity. This expresses, at each such point on the imaginary scale, a location where opposites (from an analytic perspective) are reconciled in a dynamic interactive manner. So randomness and order are opposites from an analytic perspective! However these two notions are dynamically reconciled for each point representing a non-trivial zero. Likewise the notions of prime and (composite) natural numbers are opposite in analytic terms. However once again these two opposing notions are reconciled in a holistic manner with respect to each zero!

However each individual zero can only be properly understood in the context of the collective nature of all zeros! And it is from this latter collective perspective that the quantitative nature of these zeros can be appreciated, in their ability to smooth out discrepancies as between the general behaviour of primes (as quantities) and their unique unpredictable behaviour in local regions of the number system.

The True Nature of the Zeta Zeros (2)

In yesterday's blog entry, I attempted to outline the true nature of the two sets of zeta zeros (Zeta 1 and Zeta 2).

As both of these sets are ultimately fully complementary with each other, they can only be properly understood therefore in a dynamic interactive manner, where they are seen to play a truly key role ensuring consistency with respect to all subsequent number operations.

So properly understood number - and indeed all mathematical relationships - entail both quantitative and qualitative aspects.
Whereas the quantitative relates to the notion of independence, the qualitative - by contrast - relates to the complementary notion of interdependence (i.e. where numbers are defined in relation to each other).

Thus number is necessarily of a dynamic relative nature entailing the interaction of both quantitative and qualitative aspects.

The relationship between quantitative and qualitative in turn leads to the corresponding interaction as between the cardinal and ordinal aspects of number, which are mutually distinct and cannot be successfully reduced in terms of each other.

So the key overriding issue with respect to the number system - and again by extension all mathematical relationships - is the prior need to ensure complete consistency in the relationship of both cardinal and ordinal meaning (reflecting its quantitative and qualitative aspects).

Here the relationship between the primes and the natural numbers is of the utmost importance, for it is through this relationship, entailing cardinals and ordinals, that both aspects (quantitative and qualitative) are mediated in a bi-directional manner.

And at the heart of this relationship between the primes and natural numbers lies a direct paradox!.
Whereas from the cardinal perspective, the primes appear as the independent building blocks of the natural number system (in quantitative terms), from the corresponding ordinal perspective, each prime is already necessarily defined by its natural number members (in a qualitative manner).

Thus, when one allows for recognition of both quantitative and qualitative aspects (which in truth are equally important) the primes are seen in dynamic terms to inherently combine two extreme complementary tendencies (with respect to independence and interdependence respectively).

Thus from one perspective, the primes appear as the most independent of numbers (in an absolute unchanging manner).
Then from an equally valid alternative perspective, the primes are understood as the most interdependent of all numbers (in a purely relative fashion approaching complete ineffability).


Therefore we must properly view the relationship between the primes and natural numbers (and natural numbers and primes) in a bi-directional interactive manner, whereby their mutual identity continually changes as between quantitative and qualitative (and qualitative and quantitative) aspects.

And remarkably this is what happens in actual experience, where appreciation of the cardinal and ordinal aspects of number keep interchanging!

However the consistent interplay of both aspects implicitly requires another two key sets of numbers, which very much mirror both cardinal and ordinal aspects i.e. Zeta 1 and Zeta 2 zeros.

Indeed in a very important psycho spiritual sense, these two sets represent the unconscious shadow counterparts to the consciously recognised cardinal and ordinal aspects of number.

Thus the role of these two sets of zeros is to enable conversion, in a fully consistent manner, as between both quantitative and qualitative (and qualitative and quantitative) aspects.

So, as we saw yesterday, the Zeta 1 (Riemann) zeros provide an (indirect) means of conversion (via the primes) from the recognised quantitative notion of cardinal numbers to their (unrecognised) qualitative counterpart notion.

Likewise from the opposite perspective, the Zeta 2 zeros again provide an (indirect) means of conversion (via the natural numbers) from the inherent qualitative nature of the ordinal numbers to their (unrecognised) quantitative counterpart notion. Now, as we have seen, this entails the simple task of obtaining roots of 1, which of course is well known in conventional mathematical terms. However the deeper appreciation, that these can in fact represent the important quantitative conversion of ordinal type notions, is completely missing!

Thus once again the key role of the two sets of zeros (Zeta 1 and Zeta 2) is to enable the perfect conversion in two-way fashion from the quantitative to the qualitative (and qualitative to quantitative) aspects of number.

Without belief in the guarantee of such consistency we would have no reason to belief in the subsequent consistency of  any mathematical operation and the whole edifice would be thereby built on sand.

However this consistency, attained through the mediation of the zeta zeros, cannot be proved or disproved in a conventional mathematical manner, as its very acceptance is already implicit in the use of standard mathematical axioms.

So ultimately a massive act of faith underlies the whole mathematical enterprise.

However in moving towards a true knowledge of the number system, one must be prepared for a radical change in customary beliefs.

Indeed the most profound fact about our number system is that ultimately it operates in a totally synchronous manner, which can only be properly approached through holistic - rather than analytic - interpretation.

Associated with this is an appreciation of the extraordinary status of the dynamic interactive nature of primes as representing the purest relative form of knowledge in the phenomenal universe.

G.H. Hardy must now be surely turning in his grave!

The True Nature of the Zeta Zeros (1)

Once more, we return to this key issue with the attempt to provide a simple intuitively accessible explanation of the nature of the two sets of zeta zeros i.e. Zeta 1 and Zeta 2.

As perhaps, the latter set is easier to appreciate, we will start with the Zeta 2 zeros. Then through a complementary form of interpretation, the true nature of the better known Zeta 1 (i.e. Riemann) zeros can then be revealed in a coherent manner.


The Zeta 2 are intrinsically related to the ordinal nature of number. So corresponding for example to the cardinal notions of 1, 2, 3 we have the corresponding ordinal notions of 1st, 2nd and 3rd respectively..

However, whereas the cardinal in this context relate directly to the quantitative aspect of number, the ordinal notions, by contrast are directly associated with the corresponding qualitative aspect.

So once again, the cardinal relates the individual notion of each number as independent (in a quantitative manner).
By contrast, the ordinal relates to the collective notion of a group of numbers as interdependent with each other (in a qualitative manner).

However, crucially, conventional mathematical interpretation (in formal terms) is of a grossly reduced nature (i.e. where in every context, qualitative notions are necessarily reduced in quantitative terms).

Thus in the conventional treatment of ordinal number notions, their inherent qualitative nature is not explicitly recognised, but rather referred to in more neutral terms as representative of number rankings that correspond directly with cardinal notions.

Now from one legitimate perspective, this may indeed appear to be true.

Thus when we express ordinal notions with respect to the infinite number system (in linear fashion), no problem seemingly arises with 1st, 2nd and 3rd (as in our example) seeming to directly correspond in an unambiguous fashion with the absolute cardinal notions of 1, 2, and 3.

However this all subtly changes when we switch to the expression of ordinal notions within a finite group context, where they are now revealed to be of a strictly relative nature.

So for example within the (default) group of 3, 1st, 2nd and 3rd have a relative identity (that can be expressed in circular fashion). So we could represent the 3 ordinal positions as 3 equidistant points on the unit circle. However, depending on context, 1st, 2nd and 3rd could be equally identified with each point.

If, for example, we then consider the (default) group of 5, 1st, 2nd, 3rd, 4th and 5th now equally have a relative identity, that can again be expressed by 5 equidistant points on the unit circle.

However on reflection, it becomes clear that the very meaning of 1st, 2nd and 3rd in this latter group has now changed.

And there is no finite limit to such change, as we can keep increasing the size of the group with the relative meaning of the rankings likewise changing!

In fact this can readily be appreciated in conventional situations (though the enormous mathematical significance of this is not properly grasped).

For example if I told you that I entered a competition and came 2nd (out of 1000 entrants), you might indeed be impressed.

If however if in fact there had only been 2 entrants, finishing 2nd would not however have constituted much of an achievement.

So in forming judgement as regards ordinal number rankings, implicitly we acknowledge that the relative significance of such rankings changes (depending on the overall number of the group).

Thus the ordinal notion of 2nd (to give just one example) can be given an unlimited set of relative interpretations (depending on the number of members in the finite group to which it belongs).

And this of course equally applies to every ordinal notion which likewise can be given an unlimited set of relative interpretations.

Now the significance of the primes in this context is that the ordinal interpretation of its natural number members is always unique for such groups.

However, we still have the big problem of finding a satisfactory quantitative manner of uniquely expressing these relative ordinal notions!

This is where the Zeta 2 zeros come in!

By defining the unit circle in the complex plane, the various ordinal notions (unique for each prime group) can be simply obtained through obtaining the corresponding prime roots of 1 (easily obtained though the Euler Identity).

Thus is we take the simplest case, the prime group will be 2 which enables in this context consideration of both a 1st and 2nd member.

So the 1st and 2nd roots correspond to 1^1/2 and 1^2/2 respectively (in the Type 2 number system)
=  – 1 and + 1 respectively.

Now one of these roots i.e. + 1, is not unique and this is true in every case. Thus, here, the 2nd (in the case of 2) has an absolute rather than relative meaning (which is denoted by + 1).

What this means is that once the position of the 1st member is chosen, the position of the 2nd (in the case of 2) is then automatically known in an absolute manner.

So the various prime roots of 1 correspond to the simple equation, 1 – x^t  = 0 (where t is initially prime).

However one of these roots, i.e. 1 – x, = 0 is of an absolute nature and not unique!

Therefore to find the unique (i.e. truly relative) solutions we divide by 1 – x^t  by 1 – x and solve thereby for 

1 + x¹  + x² + ... + x^{t – 1} = 0 (where t is initially prime).

Now this can ultimately be extended for all natural numbers from 2 to t (due to the unique relationship between the primes and natural numbers).

These solutions constitute the Zeta 2 zeros.

So the important role of the Zeta 2 zeros is that they enable an (indirect) quantitative means of uniquely expressing the purely relative nature of ordinal rankings, except in the default case i.e. the 
t^th member of a group of t).

So remarkably they provide the means of consistently converting (in this context of ordinal numbers) from qualitative to corresponding quantitative expression.

Furthermore this connection is of a holistic - rather than strict - analytic nature.

Thus each root - and we must always include the non-unique root of 1 here as relative considerations can only be made with respect to what is initially understood as independent - has a certain (relative) independent identity (as separate) while the addition of roots displays (relative) interdependence (as combined).

Thus in the case of the two roots of 1, – 1 and + 1 are relatively independent of each other (in quantitative terms).

However the combined sum i.e. – 1  + 1 = 0 displays corresponding (relative) interdependence, which is strictly of a qualitative nature.  And this is exemplified by the fact that universally, the sum of roots of 1 (representing the natural number ordinal members of a group) = 0.


By a complementary form of understanding - and ultimately both approaches are completely complementary in a dynamic interactive manner -  we can now perhaps explain simply the key significance of the corresponding Zeta (i.e. Riemann) zeros.

As we know, from the cardinal perspective, the primes are the building blocks in quantitative terms of the natural number system.

Now once again we start off with recognition of the individual primes as fully independent numbers (in an absolute manner).

However on deeper reflection - just as we have already seen with the ordinals - the primes can be shown to possess a merely relative identity.

This appreciation comes through recognition of the - formally unrecognised - qualitative aspect of  multiplication.

For example we may initially understand the first  two primes i.e. 2 and 3 as number quantities in an absolute manner.

However when we multiply these two numbers (i.e. 2 * 3) a qualitative - as well as quantitative - transformation is involved.

The first clue to this comes from looking at the operation in a geometrical manner. So the representation of 2 * 3  would entail a rectangle (measured in square i.e. 2-dimensional units).

However, this qualitative transformation of a dimensional nature, is simply ignored in the conventional treatment of number multiplication. So from this reduced perspective 2 * 3 = 6 (i.e. 6¹).

In this way, we then form the utterly misleading impression that all the composite natural numbers can likewise be defined in absolute terms (in a merely quantitative manner).

However the more profound appreciation of the true nature of multiplication, entails both the notions of number independence (as quantitative) and number interdependence (as qualitative).

So for example if one lays out two rows of similar coins (with 3 in each row) clearly one must recognise the (quantitative) independence of each coin. However, for multiplication to validly take place, equally one must recognise that the coins in each row can be placed in mutual correspondence with each other (as possessing a shared common identity) So this latter recognition relates to the qualitative aspect. Therefore in the multiplication operation i.e. 2 * 3, 2 here defines the mutual correspondence (i.e. interdependent identity)  of the 3 coins in each row!

So all multiplication implicitly always entails this vital qualitative aspect (which in conventional mathematical terms remains completely unrecognised.

Thus quite simply we cannot reconcile the two operations of addition and multiplication without proper recognition of both the quantitative and qualitative aspects of number.


Now just as earlier in the case of  ordinal numbers we saw - how for example, the notion of 2nd could acquire an unlimited number of relative interpretations depending on the size of the number group to which it is related - likewise the cardinal notion of 2 can acquire an unlimited number of relative interpretations, through the process of multiplication.

Thus in the case of 2 * 2  = 4, 2 acquires a new relative identity as a unique factor of this composite number.

Then in the case of 2 * 3 = 6, 2 and indeed - a different context 3, likewise attain  new identities as unique factors of 6.

Thus as all prime numbers can in principle be combined without limit as factors of resulting natural numbers (entailing the qualitative notion of interdependence), therefore all primes possess unique relative identities that are ultimately without finite limit.

So just as the Zeta 2 zeros provide a unique means of indirectly expressing the unique inherent qualitative identity of ordinal numbers in an (indirect) quantitative manner, the Zeta 1 (Riemann) zeros, provide a corresponding unique means of expressing the inherent quantitative identity of cardinal numbers in an (indirect) qualitative manner.

Now you might remember how I defined the two aspects of the number system as Type 1 and Type 2 respectively.

Whereas the Type 1 related to a base quantitative notion (with respect to the default dimensional value of 1), the Type 2 related to a dimensional qualitative notion (with respect to the default base value of 1).

So the Zeta 1 zeros relate directly to dimensional values that express the hidden qualitative aspect of the primes (which arises through multiplication with other primes).

In this context, I  have mentioned before how a remarkably close relationship connects the frequency of the Zeta 1 (Riemann) non-trivial zeros with the corresponding frequency of the natural factors of the composite numbers.

Thus we can use knowledge of the frequency of the Zeta 1 zeros (up to t on the imaginary scale) to accurately predict the corresponding accumulated frequency of natural factors of the composites up to n (on the real scale) where n = /t/2π.

So the Zeta 1 (Riemann) zeros provide a remarkable way of expressing (indirectly) in precise numerical format, the hidden qualitative transformations that are involved through the multiplication of primes.

In this way, they express the purely relative nature of the primes (ultimately as pure energy states) which represents the opposite extreme to conventional quantitative notions where they appear as rigid and absolute.

And, as before, the nature of these zeros is strictly of a holistic nature.

In the natural number system we get a considerable amount of discontinuity in the movement as between the primes (with no factors other than themselves and 1) and the composite natural numbers (with 2 or more factors), which increases as we ascend the number scale.

The Zeta 1 zeros therefore represent a continual smoothing out of this discontinuity with locattions taken, so that in the identity of each zero, opposites are reconciled such as between notions of randomness and order and primes (without factors) with composite natural numbers (with factors).

So this reconciliation of opposites (in the identity of each zero) represents the qualitative aspect of number behaviour from this complementary perspective. Then the combination of all such zeros represents their quantitative aspect (which can be used to eliminate deviations in the general calculation of the frequency of primes).

I cannot stress however how far we have moved here from conventional mathematical understanding (based on mere analytic interpretation where the quantitative aspect is clearly divorced in absolute terms from its corresponding qualitative aspect).

Here we have arrived at the other extreme of pure holistic interpretation (where both quantitative and qualitative aspects are fully harmonised with each other in a truly relative manner).

And as it stands, there is no way whatsoever of grasping the true significance of the zeros (Zeta 1 and Zeta 2) from the conventional mathematical perspective (as it gives no formal recognition  to the holistic aspect of mathematical symbols).

In fact in their most profound sense they represent (both sets) a mysterious alchemy whereby we can convert consistently in two-way fashion from both qualitative to quantitative format (and quantitative to qualitative) with respect to the interpretation of number.