More on Dynamic Nature of Number (4)

We have seen how number - when properly understood - entails a dynamic interaction between its quantitative and qualitative aspects (which are complementary with each other).

This raises the key issue of consistency with regard to the two aspects, when used in relation to one another.  

So again in Type 3 (radial) terms, it is now clearly understood that both the quantitative aspect of number (as independent) and the corresponding qualitative aspect (as interdependence) necessarily are of a relative nature.
 

Type 3 understanding represents both Type 1 (cardinal) and Type 2 (ordinal) aspects of number (where both are understood in dynamic terms as complementary).

So  strictly the cardinal notion of number (as explicitly understood) requires implicit understanding of the corresponding ordinal notion.

For example In Type 1 (base) terms, the cardinal notion of number is understood in a quantitative manner.

Therefore, using the number "3" to illustrate, this is explicitly understood as the sum of independent homogeneous units (that lack qualitative distinction).  Therefore 3 = 1 + 1 + 1.

However implicitly, a qualitative relationship must be recognised as existing between these individual units. Otherwise there would be no means of arriving at their composite total = 3.

Therefore explicitly the adding of 1 + 1 + 1 (in an independent quantitative manner) implicitly requires the qualitative ordinal recognition of these units (as 1st, 2nd and 3rd respectively).

Then in a reverse manner in explicit Type 2 (dimensional) terms, the ordinal notion of number as comprised of related ordinal components, implicitly requires the cardinal notion of number (as comprised of independent units).

Therefore we can only explicitly recognise the 1st, 2nd and 3rd ordinal members of a group of 3 members, if we already implicitly recognise 3 in cardinal terms (as composed of single independent units).

So in dynamic interactive terms cardinal and ordinal notions of number, ultimately, mutually imply each other in a holistic synchronous manner.


The huge question then arises as to the consistency of quantitative (cardinal) and qualitative (ordinal) aspects with respect to each other.

This implies for example that an indirect quantitative means is required to convert - as it were - qualitative to quantitative notions (that then can be seen to be related in a consistent manner).

Now again, in conventional mathematical terms, this key issue is avoided through an (unrecognised) form of reductionism whereby qualitative (ordinal) notions are viewed in one limited manner.

If we start with just one item which in cardinal terms is 1, we can unambiguously view this as the 1st (of a group of 1).

If we then move on to two items, which again in cardinal terms is 2, we can once again view this additional member as the 2nd (of a group of 2).

Then if we move on to three items, which now in cardinal terms is 3, we can now view the additional member as the 3rd (of a group of 3).

In this way we can continue to unambiguously identify additional ordinal members, by always considering its as the nth member of a group of n.

We can in fact mathematically show how each additional ordinal member, thereby becomes indistinguishable from corresponding cardinal members


Now quantitative meaning relates directly to linear (i.e. 1-dimensional) rational interpretation (which accurately characterises the conventional mathematical approach).

Dimensional meaning - which relative to base cardinal interpretation is of an ordinal nature - is represented as 1ⁿ.(where n is the dimesnion defined with respect to the default fixed base of 1).

Therefore to reduce from n-dimensional to 1-dimensional format we obtain the solution of
 xⁿ = 1ⁿ

Therefore x¹ = 1^n/n  = 1¹ = 1 (in cardinal terms).

Thus for the 1st unit x¹ = 1; then for the 2nd unit x¹ = 1; likewise for the 3rd unit x¹ = 1.

So the qualitative (ordinal) notion of 1st + 2nd + 3rd  has now been successfully reduced in mere quantitative (cardinal) terms as 1 + 1 + 1.

Therefore, the linear interpretation of ordinal notions is only possible through treating each additional ordinal member as the nth member of the nth group.

In this way ordinal notions seem compatible with absolute cardinal notions of number (in a Type 1 manner).

However when we treat ordinal notions in a truly relative manner, within each group, this comforting position breaks down irretrievably (in effect requiring a greatly enlarged mathematical framework).

More on Dynamic Nature of Number (3)

Yesterday, we looked at the distinction as between the analytic and holistic approaches to number interpretation.

Once again with the analytic, an absolute type interpretation results. At a deeper level this represents a linear (i.e. 1-dimensional) approach, implying in any context single unambiguous polar reference frames. So the external is clearly separated from the internal pole; likewise the quantitative is clearly separated from the qualitative aspect.
Therefore the customary analytic interpretation of number is with respect to its independent (external) objective identity in a merely quantitative sense!

However with the holistic, a relative type interpretation by contrast results, which always entails the dynamic interaction of opposite polarities.
So number from this perspective, necessarily entails a dynamic interaction as between internal (mental) and external (objective) aspects; equally it entails a dynamic interaction as between quantitative (independent) and qualitative (interdependent i.e. relational) aspects.

In extremes, the holistic aspect in itself represents the complementary opposite of the analytic aspect.
Whereas the analytic extreme leads to the absolute interpretation of number as unchanging phenomena of form, the holistic extreme leads to the purely relative interpretation of numbers as energy states (that ultimately are ineffable).

Insofar as the natural numbers are concerned, the analytic interpretation corresponds directly with standard cardinal notions.

However a major (unrecognised) problem relates to the corresponding conventional treatment of ordinal numbers, which properly requires holistic - rather than analytic - appreciation.

When one reflects carefully on it, the ordinal notion of number implies qualitative as well as quantitative considerations.

Say we are trying to rank the exam results of pupils in a class of 20!
Now this requires the quantitative notion of 20 and the recognition of each pupil as independent.
So standard cardinal notions here apply with 2 = 1 + 1, and 3 = 1 + 1 + 1 and so on.

However the attempt to rank each student as 1st, 2nd, 3rd, etc strictly implies the qualitative notion of interdependence, whereby we can place the student positions in relationship to each other.

Therefore in the standard analytic interpretation of number, ordinal notions are therefore necessarily reduced in a merely quantitative type manner.
Indeed, in this context it is interesting how the qualitative notion is never even mentioned in connection with the ordinal treatment of numbers, where the seemingly safer more neutral term of "rankings" is employed.
However, once again, rankings necessarily entail the qualitative notion of the relationship between numbers.
And if we view natural numbers as absolutely independent (in a cardinal manner), well then this begs the significant question of how these numbers can be related with each other!

So clearly there is an enormous question, regarding the very nature of ordinal numbers, which is completely overlooked in conventional mathematical terms.

Now if we confine ourselves for the moment to a finite group of numbers, we can begin to appreciate how ordinal rankings are merely of a relative nature (that keep changing depending on context).

So if for example we envisage a class of 2, obtaining 2nd place might not appear an achievement.
as in this context it represents last place!

However as the size of the class increases, the relative significance of 2nd changes. So 2nd in the context of 20, has very different connotations from 2nd in the context of 2. Again 2nd in the context of 200 would appear even more impressive!

So the meaning of 2nd - as indeed the meaning of every ordinal number - continually changes as the corresponding cardinal magnitude of the finite group to which it relates, itself is increased.

It is only when we view the cardinal number set as infinite, that the ordinal nature of number appears unambiguous.

So therefore if we consider 1st, 2nd, 3rd, 4th etc as ordinal members of an infinite class, then their meaning appears as invariant, whereby they can be represented in identical terms with the corresponding cardinal point of 1, 2, 3,  4 etc. on the number line.

In this (1-dimensional) linear manner - which again underlines all standard analytic interpretation -
ordinal notions can seemingly be successfully reduced to their cardinal counterparts in similar fashion.


However there is n an important unappreciated paradox about what is involved here.

When for example, we refer to the cardinal numbers 1, 2, 3 and 4 for example, these - by definition - are given a limited finite identity.

However, when we refer to the corresponding ordinal numbers 1st, 2nd, 3rd and 4th, these, by contrast necessarily pertain - with respect to their unambiguous identity - to number relationships which entail an infinite class (in cardinal terms).

So ultimately, the seeming correspondence as between cardinal and ordinal numbers in conventional mathematical terms, is based on the fundamental reduction of infinite to finite notions. And this strictly, is the same problem that underlines the attempted reduction of all qualitative notions in merely quantitative terms.

Thus the remarkable conclusion that we have reached - which has profound implications for conventional mathematical understanding - is that the ordinal notion of number is inherently associated with the Type 3 mathematical worldview (where both quantitative and qualitative notions are related).

Ordinal notions thereby represent both the quantitative notion of the independence (of each individual natural number) with the qualitative notion of the interdependence (with respect to the relationship between these numbers).

Thus in Type 3 terms, we at last realise clearly that the very notions of independence and interdependence (with respect to number) are themselves necessarily of a relative nature.

What this means is that both cardinal and ordinal notions mutually imply each other in a relative -rather than absolute - manner.

Using Jungian notions, thus there is a hidden shadow side to the interpretation of the qualitative aspect of number (in a quantitative fashion).

Also, there is also a hidden shadow side to the quantitative aspect of number (in a qualitative manner).

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.