The Holistic Nature of the Number System (5)

Yesterday, I introduced again the two Zeta Functions (Zeta 1 and Zeta 2) which correspond in turn to the quantitative and qualitative aspects of the number system (i.e. Type 1 and Type 2) respectively.

Now properly understood - in dynamic interactive terms – these two Functions are complementary with each other.

Once again the Zeta 1 Function (i.e. the standard Riemann Function) is given in conventional terms as the infinite series,

1^–s  + 2^–s  + 3^–s  + 4^–s  +……..,

where each of the natural numbers is raised to the negative of the same dimensional value (i.e. s).

By contrast the Zeta 2 Function is given as the finite series,

1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime),

where in reverse, s as the quantitative value is raised to each of the natural values (up to t – 1).

The value of s will of course be distinct for each Function!

So through each Function we have the expression of the natural number system (both in finite and infinite terms) alternating in each case as between dimensional values (that are qualitative) and - relatively - base number values (of a quantitative nature).

Now the zeta zeros (Zeta 1 and Zeta 2 respectively) represent the solutions of each Function, where the sum of terms in the series expression = 0.

 

Thus when we look at these solutions in a dynamic interactive manner, the crucial key as to their nature is revealed.

Putting it simply, the zeta zeros represent the solutions for which both the quantitative and qualitative aspects of the number system are perfectly reconciled (from two complementary perspectives).

In fact it is easier to initially appreciate the precise nature of such holistic solutions from the Zeta 2 perspective (which we will return to shortly).

So the Zeta 2 solutions show how the internal nature of each prime number is comprised of a series of individual natural numbers in ordinal terms, with the (indirect) quantitative nature of each of these natural numbers (in isolation) perfectly reconciled in a holistic manner with the overall qualitative interdependence of this prime number grouping.

 

Now the uniqueness of each prime number in this context is that all its natural number ordinal members (except 1) are uniquely defined in an (indirect) quantitative manner. In other words these prime number roots are unique for each prime number group.

Thus for example 5 is a prime number. Now by definition 5 as a group is composed of its 5 individual members which are 1^st, 2^nd 3^rd, 4^th and 5^th  in natural number ordinal  terms. Each of these ordinal members can then be indirectly defined in a quantitative (relatively independent) manner through the corresponding 5 roots of 1.

Then when we combine these roots to demonstrate their corresponding relative interdependence (in a qualitative manner) their sum = 0.

So in this way we can demonstrate a unique holistic balance for each prime number as between its individual members (in quantitative terms) and their overall collective interdependence (in a qualitative manner).

We are accustomed to interpreting prime numbers in an absolute analytic fashion as the independent quantitative building blocks of the natural number system.

So from this perspective, the qualitative relational aspect of the primes become fully separated and then fully reduced to the quantitative aspect.

And once again the analytic aspect of the number system is defined by such absolute reductionism (whereby it is formally interpreted in a merely quantitative manner)!

However the holistic aspect is altogether distinctive in nature (and lying at the other extreme) where quantitative elements, while maintaining their independent identity in a relative manner, are yet seen - relatively - as fully interdependent with each other in qualitative collective terms.

And it is vital to grasp that zeta zeros directly represent the latter holistic interpretation of number!

 

And so once again this is exactly why the very nature of these zeros is - necessarily - completely missed through the conventional mathematical approach.

We cannot therefore hope to appreciate the holistic nature of number (entailing the balanced interaction of its quantitative and qualitative aspects) in an analytic manner (that is formally based on mere quantitative interpretation).

Now I have spelt out in some detail here the holistic nature of the Zeta 2 zeros.

Though we will return to this in future entries in this “holistic” series on my blog, the general nature of the - greater recognised - Zeta 1 zeros can easily be expressed.

Just as the Zeta 2 zeros are defined for the internal ordinal natural members of each individual prime, the Zeta 1 zeros are defined externally for the number system as a whole with respect to the collective nature of the primes with respect to the (composite) cardinal natural numbers.

So the non-trivial zeros of the Zeta 1 function ensure a complementary type of holistic harmony as between the qualitative interaction of primes and the quantitative natural numbers resulting.
 

 

Thus the Zeta 1 and Zeta 2 serve as complementary mirrors to each other.

In the case of the Zeta 2 each cardinal prime is defined in an ordinal manner through a finite set of natural number members (1, 2, 3…p).

In the case of Zeta 2, the cardinal natural numbers are defined in terms of the ordinal activity of the prime number groupings that uniquely define their nature.  

 

When one can dynamically interpret these relationships simultaneously in both Zeta 1 and Zeta 2 terms, the primes and natural numbers are revealed as perfect mirrors of each other (in an ultimately ineffable manner). And this is because the Zeta zeros likewise serve as perfect holistic mirrors of each other.

Another interesting feature worth emphasising here is that the true holistic nature of the zeta zeros (Zeta 1 and Zeta 2) implies that they in fact represent energy states.

Once again the conventional analytic interpretation of number is based on the apparent existence of numbers absolutely existing as rigid immutable forms.

So in analytic quantitative terms “2” for example is interpreted as a fixed number symbol.

 

However just as matter can be transformed into energy (and energy into matter), it is exactly similar in mathematical terms.

So the understanding of numbers existing as absolute forms represents the extreme analytic interpretation (where the qualitative aspect is reduced to the quantitative).

However when, at the other extreme, we understand numbers in pure holistic terms, they are understood as representing energy states.

Of course both of these aspects necessarily interact to a degree in experience, representing the inherent dynamic nature of number.

So here number forms in rational (analytic) terms interact with number energy states in a (holistic) intuitive manner leading to a continual dynamic transformation in the experience of number.

Of course in formal terms such experience is greatly misrepresented in a mere rational (analytic) manner.

 

Recent strong correlations as between the Zeta 1 zeros and energy states in quantum physics suggest that in some sense the zeros do indeed represent energy states!

However none of this can have intuitive resonance within the restricted conventional manner in which number is presently interpreted.

However it makes perfect sense when one interprets the zeros in an appropriate holistic manner.

However what is still being completely missed is that from a dynamic perspective, physical and psychological aspects are complementary.

So not alone do the zeta zeros represent physical energy states, they equally represent psychological - or more correctly - psycho-spiritual energy states.

Therefore, correctly understand the zeta zeros (in appropriate terms)  represent psycho-spiritual energy states that have complementary correlates in nature (in a psycho-physical manner).

And of course in dynamic interactive terms, physical nature is necessarily understood in a psycho-physical manner!

The spiritual traditions often make the distinction as between dual and nondual reality.

Dual reality - literally - relating to “twoness” refers to analytic type understanding, where the poles of understanding (such as internal/external and quantitative/qualitative are clearly separated.

Nondual reality - literally not-two - refers to holistic type understanding, where the opposite poles of understanding (that necessarily condition all experience) are understood as complementary and ultimately identical in a spiritual manner.

So in these terms, the zeta zeros relate directly to a pure nondual intuitive type appreciation at the deepest possible level of the ultimate interdependence of both the qualitative and quantitative (and indeed internal and external) aspects of the number system.

Now as with all phenomenal experience, holistic appreciation is necessarily of a relative approximate nature (which entails a degree of supporting analytic activity).

So the zeta zeros can indeed be encoded in a phenomenal numerical manner.

However it is important to remember that their inherent nature is holistic and thereby they relate to the most dynamically interactive experience that is humanly possible, while remaining in the world of phenomena. In this sense they are like the most fleeting of virtual particles.

We are a long, long way in our present evolution from being able to appreciate their inherent nature truly in the appropriate interactive manner (which will entail the most advanced states of spiritual contemplative development).

However we have come far enough to begin to understand their true nature, which is as far away as is possible from the rigid analytic notion of number that dominates conventional mathematical interpretation.        

The Holistic Nature of the Number System (4)

As we have seen the ordinal interpretation of number relates to number interdependence, as opposed to the corresponding cardinal notion of number as independent.

Whereas analytic interpretation is indeed appropriate with respect to the cardinal notion of independence, by contrast holistic appreciation is required for the ordinal interpretation of number as interdependent.

Once again analytic interpretation corresponds with rational understanding (of a linear nature).

However holistic interpretation corresponds directly with intuitive understanding (that is indirectly rationally expressed in a circular i.e. paradoxical manner).

 

Again the essence of analytic interpretation is that it is 1-dimensional i.e. based in any context on just one independent pole of reference.

The essence of holistic interpretation is that any number but 1 serves as the dimension. In the simplest case, which in many ways serves as the blueprint for all holistic interpretation, two interdependent poles of reference are used.

I have explained on many occasions the relationship as between these two approaches in the context of a crossroads.

Once again if we adopt independent polar frames of reference, we can give unambiguous interpretation to left and right turns at a crossroads.

So if one approaches the crossroads (travelling North), one can give an unambiguous definition of left and right to each turn; then when one approaches the crossroads travelling South, again both turns are unambiguous.

The reason for this lack of ambiguity is that the reference frame is independent, referring to just one pole (which again is the essence of 1-dimensional interpretation).

However, if one now simultaneously attempts to include both reference frames North and South, then definition of left and right is paradoxical. So each turn at the crossroads can be both left and right depending on context.

The reason for the paradox is now due to the fact that the reference frames are interdependent (defined in a 2-dimensional manner).

So the very essence of paradox is that it is nondual (and thereby cannot be grasped in an analytic fashion).

And the appreciation of such paradox relates to a distinctive type of holistic understanding (where the two reference frames are united in a directly intuitive manner).

Therefore, though we indirectly can express such nondual appreciation in a circular (paradoxical) rational manner, in direct terms it is of a holistic intuitive nature.

 

So the holistic appreciation of 2, is the recognition of  + 1 and – 1 as complementary opposite poles of recognition.

Therefore, the very basis of intuitive insight in experience is the recognition of the complementary - and ultimately identical - nature of external (i) (objective) and internal (subjective) polarities and (ii) part (quantitative) and whole (qualitative)  polarities.

Indeed these four polarities, representing the 4-dimensional qualitative nature of dynamic interaction in experience, can be conveniently represented - like the directions on a compass - by four equidistant points on the circle of unit radius (in the complex plane).

In like manner the n-directional qualitative nature of dynamic interaction between polarities can be represented by n equidistant points on the circle!

As I have said, in many ways 2-dimensional interpretation serves as the simple blueprint for all holistic recognition.

The important point here is that with holistic recognition we must always start with (1-dimensional) analytic interpretation.

So before we can recognise the two turns at a crossroads in holistic interdependent terms as both left and right, we must initially recognise them in independent terms as unambiguously either left or right. So we must apply two independent reference frames separately (as independent) before combining both (as interdependent).

 

Now in qualitative terms "+" simply implies positing in a conscious manner and + 1 implies such positing with respect to one independent reference frame.

Thus once again, Conventional Mathematics in this qualitative sense, is 1-dimensional with the one root of 1 = + 1.

In qualitative terms,  "– " implies (dynamic) negation in an unconscious manner. Thus the very means by which the unconscious is activated in experience is through the dynamic negation of consciously posited reference frames.

Without the unconscious it would be thereby impossible to switch reference frames. Thus the all important process (in any context) of relating part to whole (and whole to part) implicitly implies unconscious recognition. And as we have seen, intuitive appreciation in experience relates directly to unconscious type recognition!

So the holistic appreciation of interdependence is directly of an unconscious nature.

Once again Conventional Mathematics cannot deal with this notion (except in a grossly reduced sense) as it formally defines relationships in a merely conscious manner.

Thus the Type 1 aspect of the number system relates directly to the analytic (conscious) aspect of mathematical experience.

The Type 2 aspect relates - by contrast - to its holistic (unconscious) aspect.

 

So again from a Type 1 perspective, 2 is defined as 2¹. From a Type 2 perspective, 2 is defined as 1².

Then associated with the Type 1 system is the standard Riemann Zeta Function (which I refer to as Zeta 1),

i.e. 1^–s  + 2^–s  + 3^–s  + 4^–s  +……..

Here each of the natural numbers is raised to the negative of the same dimensional value (i.e. s).

However equally associated with the Type 2 aspect is an alternative Zeta Function (which I refer to as Zeta 2),

i.e. 1 + s¹  + s²  + s³  +….. + s^{t – 1}
 

Here, in reverse s as the quantitative value is raised to each of the natural values (up to t – 1).

So whereas Zeta 1 is conventionally defined as an infinite series, Zeta 2 is of a finite nature (though it can be given a qualified meaning in infinite terms).

Now the derivation of the non-trivial zeros for Zeta 1 relates to the equation

1^–s  + 2^–s  + 3^–s  + 4^–s  +……..  = 0

The corresponding derivation of non-trivial zeros (whose role is as yet unrecognised) relates to the equation,

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

The simplest example is where t = 2, and

1 + s¹  = 0,

so that s =  – 1.

 

As we know  + 1 will always constitute one of the roots of 1.

In this sense it is a trivial root (as it is repeated in every case).

The deeper qualitative significance of this root is that it implies that we must always start with analytic interpretation (within independent reference frames) before we can achieve their holistic interdependence.

 

However the fascinating feature of the Zeta 2 zeros is that they already have eliminated the common root of 1).

Thus where t is prime the solutions of the Zeta 2 equation relate to the t – 1 non trivial roots (which are directly of a holistic nature geared to the interpretation of interdependence).

 

Now the unconscious holistic nature of these solutions is indicated by the fact that their sum = – 1.

Thus in one sense, the higher dimensional interpretations (where t is prime > 2), relate to ever more refined unconscious (i.e. holistic) type appreciation.

The uniqueness of the prime numbers in this regard is that there non-trivial roots are unique (i.e. never duplicated in other prime root solutions).

Thus in Type 1 terms a prime number is unique because of its independence in cardinal terms i.e. has no natural number factors.

In Type 2 terms a prime number is unique because of its interdependence in ordinal terms i.e. all members (apart from the 1^st) are defined in a non-repetitive manner.  

The Holistic Nature of the Number System (3)

The quantitative and qualitative aspects of the number system are intimately related to the (pure) nature of addition and multiplication respectively.

As we have seen number can be defined in two ways (which dynamically interact in mathematical experience).

Now to see this clearly, numbers must be defined with respect to base and dimensional numbers (which properly in a dynamic manner, are quantitative and qualitative with respect to each other).

So again to take the simplest (non-trivial) case of the number 2, this can be defined in two distinct ways.

In the conventional quantitative approach - related to (pure) addition

2 = 1 + 1

i.e. expressed more fully

2¹ = 1¹ + 1¹

 

So the essence of this quantitative relationship is that default dimension to which the (base) number quantity (2) is expressed remains fixed as 1.

And as I have repeatedly stated, Conventional Mathematics, geared exclusively to the quantitative interpretation of relationships is precisely defined, from a qualitative perspective, in terms of its linear (i.e. 1-dimensional) mode of interpretation.

So in Conventional Mathematics with respect to multiplication a merely reduced (i.e. quantitative) interpretation is possible.

So in this context if we multiply 2 by 2 i.e. 2² , the answer will be given as 4 i.e. 4¹.

Thus the answer in reduced quantitative terms is expressed with respect to the default dimensional value of 1 (which again illustrates well the 1-dimensional nature of mathematical interpretation).

However if we think about it for a moment, clearly a dimensional transformation (of a qualitative nature) is likewise involved.

Thus in simple geometrical terms, 2 * 2 would be represented by a square (with side 2 units).

Therefore the area of this square is properly expressed in square (i.e. 2-dimensional units).

Thus a dimensional change of a qualitative nature is clearly involved through this simple multiplication process.

However in conventional mathematical terms the qualitative nature of this transformation is simply edited completely out of the process with the result expressed thereby in a reduced i.e. merely quantitative manner.

And such reductionism universally characterises the nature of multiplication from a conventional mathematical perspective.

It is hardly surprising therefore that mathematicians eventually realise that there is something fundamentally missing from their appreciation of the relationship as between addition and multiplication (as multiplication is inherently interpreted in a reduced merely quantitative manner).

This issue struck me so strongly at the age of 9 or 10, that I already realised then that there was - literally - a fundamental dimension with respect to conventional mathematical understanding that effectively was overlooked. In other words in the mere quantitative interpretation of number, its qualitative dimensional aspect is simply ignored.

So the conventional quantitative approach to number - where number is defined in a cardinal manner - I refer to as the Type 1 aspect of the number system.

So again in Type 1 terms,

2¹ = 1¹ + 1¹

However in the corresponding Type 2 aspect, the number 2 is expressed in terms of a (pure) multiplication process.

So in Type 2 terms,

2 = 1 * 1

In other words,

1² = 1¹ * 1¹

Now 2 in our definition of number has been inverted, with 2 representing the dimensional qualitative nature of number, which is defined with respect to the default (base) quantity of 1.

So again if we think of this in geometrical terms, when we square 1, we obtain a 2-dimensional figure (with area 1 sq. units)

Therefore, though in quantitative terms nothing has changed (with the 2-dimensional area the same as its 1-dimesnional side), clearly a qualitative change in the nature of units has been involved.

 

Thus, the very purpose of the Type 2 approach is to isolate the qualitative aspect of number transformation, which is related directly to the pure nature of multiplication.

As I have repeatedly stated in my blog entries, the mystery of the relationship of addition and multiplication is the same mystery as the relationship of the quantitative and qualitative aspects of number.

Once again there are insuperable difficulties in attempting to understand this relationship in a mere quantitative manner!

However having isolated the qualitative aspect of number transformation (through its Type 2 aspect), the next problem is in attempting to give expression in an indirect quantitative manner to this aspect.

The deeper reason for this is that when one accepts that there are necessarily two distinctive aspects to number (which are quantitative and qualitative) with respect to each other, then the ultimate consistency of both aspects becomes the indisputable key mathematical requirement.

So demonstrating such consistency - which clearly cannot be proved within the reduced conventional mathematical perspective - requires the ability to (i) indirectly convert from the qualitative to the quantitative aspect and (ii) the ability in reverse manner to convert from the quantitative to the qualitative aspect, while establishing complete harmony between both aspects.

Put another way it requires establishing the interdependence of both quantitative and qualitative aspects (from two complementary perspectives).

Now once again, by its very nature, this is not strictly possible within the conventional mathematical perspective (defined as it is in a mere quantitative manner).

Such interdependence relates directly to a holistic - rather than analytic - type appreciation of mathematical relationships.

It is central to appreciation of the true nature of the number system and yet in formal mathematical terms currently does not even exist.

 

To sum up this blog entry, properly understood in dynamic interactive terms, there are two distinctive ways of interpreting every number.

The Type 1 relates to its quantitative aspect, directly associated with the (pure) notion of addition.

The Type 2 relates to its qualitative aspect directly associated with the (pure) nature of multiplication.

Both of these continually interact in a - necessarily - relative manner in experience and are directly associated with the cardinal and ordinal aspects of number interpretation respectively.

Conventional mathematical interpretation therefore represents but a highly reduced - and greatly confused - interpretation of the number system.

We will have more to say about the - much misunderstood - ordinal nature of number in the next entry.

The Holistic Nature of the Number System (2)

Once again the root cause of qualitative confusion with respect to Conventional Mathematics relates directly to the manner in which the infinite notion is reduced in finite terms.

Appreciating this in turn helps one to appreciate the two uses of number i.e. as a number quantity (that can be raised to a number dimension) and in reverse a number dimension (to which a given number quantity can be raised).

When one uses 1 in actual terms to represent a quantity, clearly it relates to an actual specific finite notion; however when one uses 1 to represent a dimensional quality strictly it relates to an infinite notion (of a potential nature).

So for example the line is 1-dimensional and this implies that it is potentially unlimited with respect to extension.

Now the direct confusion that is involved is that when we attempt to apply this notion in an actual quantitative sense it is always necessarily limited in a finite manner.

So if I draw a straight line, regardless of how far it is extended it will always necessarily be of a merely finite length!

However because of the quantitative bias of Conventional Mathematics, the attempt is then made to reduce the true potential nature of the infinite notion in an actual finite manner.

Thus the totally misleading impression is then given that the infinite somehow represents the ultimate limit resulting from finite extension.

In common sense terms this could be expressed by saying that if one extends the line far enough, its length will approach infinity.

Now to put it bluntly, this is utter nonsense; and yet it is such a reduced notion of the infinite that pervades mathematical thinking.

Therefore though the infinite properly relates to a qualitative notion, conventionally it is treated in a merely reduced quantitative manner.

In understanding, intuition directly relates to the (qualitative) infinite and reason to the (quantitative) finite aspect respectively.

So intuition strictly relates to the potential infinite aspect that is inherent in all actual finite circumstances.

However once again, because Conventional Mathematics is formally defined in a merely linear rational manner, the potential infinite aspect is necessarily reduced in an actual finite manner.

Therefore the holistic aspect of mathematical understanding - which essentially relates to the (infinite) qualitative aspect of relationships - formally, is completely unrecognised by the profession.

Thus when I repeatedly state that Conventional Mathematics is totally unbalanced, I mean precisely what I say.

 

Properly understood in dynamic interactive terms, we have two aspects quantitative and qualitative (of equal relevance) that define the nature of all mathematical relationships. Yet, quite incredibly, only one of these is formally recognised.

To use an analogy is it like maintaining that water (that comprises both hydrogen and oxygen molecules) is comprised solely of oxygen. However in truth it is much more serious than that!

Coming back to my original point, the use of number in dynamic interactive terms, as representing base quantities and dimensional exponents respectively, relates to two quite distinctive aspects.

As we saw in yesterday’s blog entry, whereas the cardinal aspect is properly associated with the former, the ordinal aspect directly relates to the latter.

Therefore in actual mathematical experience, we have the continual dynamic interaction of both cardinal and ordinal aspects (which are quantitative and qualitative with respect to each other).

 

The deepest issue with respect to the number system is therefore the key requirement of achieving consistency as between these two distinctive aspects.

And once again when properly understood, this is what the Riemann Zeta Function (and its associated Riemann Hypothesis) is all about.

However when appreciated in such terms, it is somewhat ludicrous to approach the Riemann Hypothesis from a merely quantitative perspective.

In the truest sense this represents the "reductio ad absurdum" which crucially exposes the limits of the conventional mathematical approach.

Remember again that the Riemann Zeta Function remains uniquely undefined for just one value where s (the dimensional value)  = 1.

Now Conventional Mathematics limits itself entirely to the quantitative interpretation of this statement.

However the corresponding qualitative interpretation is that the Riemann Zeta Function remains uniquely undefined in conventional mathematical terms (defined as it is, in a 1-dimensional manner).

So for all other values of s, a dynamic interaction as between both cardinal (quantitative) and ordinal (qualitative) type meanings necessarily exist.

Therefore through the Functional Equation, we can always match cardinal type interpretation on the RHS of the Function for ζ(s) to a corresponding ordinal type interpretation on the LHS for ζ(1 – s).

And so the condition for the mutual coincidence of cardinal and ordinal values is that s = .5.

So the requirement that all the non-trivial zeros lie on an imaginary line drawn through .5 (which is the Riemann Hypothesis) is in fact the condition for ensuring that both the cardinal (quantitative) and ordinal (qualitative) aspects of number are mutually identical.

So again the Riemann Hypothesis - when appropriately understood in dynamic interactive terms - serves as the key requirement for ensuring the subsequent consistency of quantitative and qualitative meaning with respect to all mathematical relationships.
Now clearly such a proposition cannot be proved (or disproved) with reference to a system defined with respect to mere quantitative interpretation!

Indeed the truth of Riemann Hypothesis is already necessarily assumed in the very use of conventional mathematical axioms.  

So the zeta zeros inherently relate to an interdependence with respect to both quantitative and qualitative aspects of mathematical meaning.

Now yesterday, I was at pains to show that Conventional Mathematics necessarily is defined in terms of independent reference frames, in what represents analytic interpretation.

However the very nature of the zeta zeros relates to the interdependence of both quantitative and qualitative aspects.

And the appropriate manner for interpreting such interdependence (of complementary poles) relates to holistic -  as opposed to analytic - understanding.

Once again this creates insuperable problem for Conventional Mathematics (which in formal terms is completely lacking a holistic aspect).

Indeed put simply the (non-trivial) zeta zeros (Zeta 1 and Zeta 2) represent the holistic aspect of the number system, which necessarily underpins (in a dynamic interactive manner) our everyday analytic appreciation of number.  

 

The seta zeros therefore play an indispensable role in our number system.

However perhaps the most important implication of the nature of these zeros is that Mathematics itself now needs to be completely reformulated in an appropriate dynamic manner. 

The Holistic Nature of the Number System (1)

The obsession with the number line - as the means of representing real numbers – simply reflects the 1-dimensional (i.e. linear) paradigm on which Conventional Mathematics is based.

Now as I have expressed repeatedly in these blog entries the deeper basis of this (1-dimensional) paradigm is rooted in an approach whereby mathematical interpretation is conducted absolutely in terms of just one polar reference frame. This in turn leads (i) to the notion of mathematical “objects” that are unaffected through (subjective) mental interaction and (ii) an exclusive focus on the quantitative aspect of mathematical relationships (with no reference to a qualitative context of meaning).

Of course momentary reflection on the matter will quickly show that our actual experience of Mathematics is inherently of a dynamic relative nature entailing the continual interaction of both internal and external aspects and likewise the continual interaction of part (quantitative) and whole (qualitative) notions.

In psychological terms this equally entails both conscious and unconscious aspects of understanding through the corresponding interaction of rational (conscious) and intuitive (unconscious) processes.

Momentary reflection on the matter should also show that the accepted method adopted by Conventional Mathematics reduces the truly dynamic nature of actual mathematical experience in a greatly reduced absolute static manner.

So the necessary dynamic interaction of opposite poles of experience is formally expressed statically in terms of just one pole!

Likewise the dynamic interaction psychologically of (conscious) reason and (unconscious) intuition is rigidly expressed solely in terms of (conscious) reason.

 

It is therefore hardly surprising in this context that our present understanding of the number system should be fatally flawed.

Certainly from my perspective it has long been obvious that - when appropriately understood i.e. in accordance with actual experience - that the number system is necessarily of a dynamic interactive nature.

And of course this equally implies that it should be interpreted in a relative - rather than absolute - manner.

So we cannot therefore ultimately divorce the (apparent) objective nature of number from the subjective mental means of its interpretation.

Therefore in this relative sense when we change the nature of interpretation the objective nature of number objects likewise changes!

 

Likewise - and perhaps even more tellingly - we cannot ultimately divorce the (apparent) quantitative aspect of number from its corresponding relational context (which is of a qualitative dimensional nature).

As I expressed in yesterday’s entry, once we accept that numbers can indeed be related to each other then this implies that their independent status (in specific cardinal terms) is necessarily of a relative - rather than absolute - nature.

So in this dynamic context the quantitative nature of number relates to its cardinal status as relatively independent.

In other words, without an implicit acceptance that to have meaning, numbers must be placed in an ordered manner with respect to other numbers, it would be impossible to locate cardinals on the number scale! So the quantitative definition of number thereby implicitly implies a distinctive qualitative aspect.

 

Also, in this dynamic context, the qualitative nature of number relates to its ordinal status as relatively interdependent (with respect to other numbers).

So once again the very notion of ordinal requires placing a number in a group context (with respect to other numbers) with its identity thereby defined with respect to the group members involved.

Thus for example to define 2^nd I must place the number 2 in relation to other group members. The simplest case would involve a group of 2 members. Therefore in this context, I can unambiguously define a 2^nd member. However the definition of 2^nd clearly changes when the number of group members increases. So 2^nd in the context of 200 members is clearly distinct from 2^nd in the context of 2.

So the ordinal identity of a number (reflecting its qualitative identity) springs from a relationship of interdependence with a wider group of numbers.

However, once again we must define interdependence in a merely relative sense here, as we start with the cardinal notion before we can establish its ordinal identity.

 

So the notion of 2^nd (as ordinal) already dynamically implies 2 (as cardinal). Likewise the notion of 2 (as cardinal) already dynamically implies 2^nd (as ordinal).

However dynamic interaction necessarily implies a phenomenal context of space and time. Though we may appreciate from what has been said, that ultimately cardinal and ordinal identity must be identical, clearly this cannot be the case in a phenomenal context (which necessarily implies a degree of relative separation). Thus total unification of both cardinal and ordinal aspects of the number system points to an ineffable state.

Now this is all deeply relevant for appreciation of the true nature of the Riemann Hypothesis (which postulates the very condition for this ultimate identity)!

The vital fact to grasp is that cardinal and ordinal aspects require two distinctive interpretations respectively.

Thus great confusion pervades Conventional Mathematics. Because it is 1-dimensional in nature, it must necessarily attempt to interpret both cardinal and ordinal aspects from the same quantitative perspective.

Therefore though the ordinal aspect properly relates to the qualitative aspect of number, it is mistakenly dealt with in a merely quantitative manner.

This is a key problem of the very first magnitude with respect to the number system, which is at present just totally ignored!

As I was stating yesterday the quantitative (cardinal) aspect directly relates to the analytic interpretation of number; however the qualitative (ordinal) aspect properly relates to the (as yet unrecognised) holistic interpretation.    

Whereas the cardinal notion relates to the number line, the ordinal notion - by contrast - relates to the circle.   

So cardinal and ordinal notions are linear and circular with respect to each other.

Solving the ordinal problem (i.e. relating to the qualitative nature of ordinal numbers) comes through the Type 2 number system.

Therefore the qualitative i.e. dimensional notion of 2 is expressed as 1².   

As we have seen the corresponding cardinal notion is expressed in terms of the Type 1 number system as 2¹.

However to give indirect quantitative expression to the qualitative notion of number we convert to a circular format through obtaining the corresponding 2 roots of 1.

So + 1 and – 1 are these 2 roots which again - in this context - provide the indirect quantitative means of expressing the two ordinal members of a group of 2.

However with holistic understanding both quantitative and qualitative appreciation must be balanced with each other.

So the true qualitative appreciation requires intuitively being able to see + 1 and – 1 as interdependent with each other.

Put another way, this is the appreciation of ordinal rankings as purely relative (depending on context).
 

I have explained before how implicitly this is what one does for example in recognising that turns at crossroads can be either left or right (depending on context).

So if approaching to crossroads from one direction we label the 1^st turn + 1 (a left turn) and the 2^nd, – 1 (i.e. not a left turn) then clearly if the crossroads is approached from the opposite direction the 2^nd will now be labelled ,+ 1 (a left turn) and the 1^st, – 1 (i.e. not a left turn).

So each turn can be both left and right (depending on context).

This understanding therefore illustrates the very nature of interdependence which requires the ability to see simultaneously from - at a minimum - two opposite reference frames.

Now clearly we cannot make such paradoxical connections in the context of just one polar reference frame.

However Conventional Mathematics is formally defined in terms of just one such frame (where relationships appear unambiguous and linear).

Therefore by its very nature, Conventional Mathematics is not equipped to deal with the key notion of interdependence (except in a misleading reduced sense).

And as ordinal meaning directly relates to number interdependence, Conventional Mathematics cannot properly deal with the ordinal notion.

It is quite remarkable. Though the use of the circular number system (defined in the complex plane with circle of unit radius) provides the appropriate means for defining the ordinal nature of number, I have yet to see it mentioned anywhere!

We have something simple, right under our noses as it were and we fail to see it! This is directly due to the pronounced linear bias of Conventional Mathematics (based on mere analytic type understanding).

Now of course the unit circle is well recognised in Conventional Mathematics. However appreciation of its true role remains greatly limited due to the inevitable attempt to understand its nature in a merely linear manner. However to properly understand the circle, we require circular type interpretation!

Thus to appreciate the nature of ordinal interpretation we must adopt holistic understanding that requires at a minimum 2-dimensional interpretation (entailing two interacting polar reference frames).

And again such holistic understanding is totally missing from what is formally accepted as Mathematics!

Illustrating the Holistic Approach

As I have frequently stated, when understood in appropriate dynamic interactive terms, Mathematics entails two aspects of equal importance i.e. analytic and holistic respectively.

Though it is not quite as clear-cut as this, the analytic - for convenience - can be identified with the quantitative and the holistic with the qualitative aspect of interpretation respectively.

Indeed right away we have a problem with the very use of the word “analytic” which is a specialised more limited interpretation within Conventional Mathematics.

Here analytic relates to the study of infinite series (real and complex), limits, the use calculus notions with respect to such series etc.

However in the wider more universal scientific use of the term, analytic - in any context - applies to the breaking down of a whole into its component parts.

And because of the very nature of present scientific method this implies a reduced (i.e. quantitative) notion of a whole which is seen merely as the sum of its constituent parts.

So this is the sense in which I use the word analytic in a mathematical context (which would include all analysis in the narrow more specialised sense in which the terms is used).

 

Indeed the analytic aspect of understanding concurs perfectly with what I characterise as the 1-dimensional approach.

So once again what this precisely means is that where the interpretation of any relationship in concerned that only one polar frame of reference is used.

Therefore all of current Mathematics - at least what is formally accepted as valid Mathematics - is one dimensional (in this qualitative interpretative sense).

Thus in relation to the first key polarity set i.e. external and internal, mathematical symbols and relationship are given a mere external (i.e. objective) identity in absolute terms that is not influenced through internal (subjective) interpretation.

Therefore though in dynamic interactive terms external and internal polarities of experience are positive (+) and negative (–) with respect to each other. However the very essence of absolute interpretation is to ignore this distinction in effect treating meaning in a merely positive (+) fashion.

So this is a perfect example of what is meant by the 1-dimensional approach where rational interpretation is linear (and unambiguous) in just one positive direction!

This 1-dimensional approach equally characterises treatment in conventional terms with respect to the second key polarity set i.e. quantitative and qualitative.

Here symbols and relationships are given a mere quantitative identity, again in absolute terms that is not influenced through qualitative interaction.

There is huge confusion in present Mathematics with respect to this fundamental point.

For example the natural numbers 1, 2, 3, 4,  etc are defined as independent with respect to their mere cardinal identity in quantitative terms.

However the very ordering of numbers (whereby we meaningfully can place numbers in relation to each other) requires a distinctive ordinal identity (of a qualitative nature). In other words in ordinal terms a number only has meaning in relation to other members of a number group.

So the meaning of 3^rd for example only has meaning in the context of a number group (which can arbitrarily vary in size). Therefore, 3^rd in the context of 3 numbers clearly carries a very distinct meaning from 3^rd in the context of 300!

Thus ordinal identity therefore relates to the notion of an interdependent - rather than independent - number identity.

So the huge reductionist assumption that is made in Conventional numbers is that we can order the cardinal numbers in a merely quantitative manner.

However once we depict the cardinal numbers e.g. as successive points on a number line, we are thereby assuming a dimensional context that properly relates to a qualitative ordinal identity!

Thus the very essence of 1-dimensional interpretation is that it directly confuses the quantitative notion of individual numbers (as independent units) with the qualitative dimensional notion of the overall general order or collective interdependence as between various numbers.

And let’s be utterly frank here. Our cherished notions of the number system (that have developed now over several millennia) are thereby based on a fundamental confusion (i.e. where qualitative is reduced to quantitative interpretation).

So the qualitative nature of number is not just something vague, as I have often read, such as number personalities or even number archetypes, but as referring directly to the fundamental issue of the means by which we are enabled to achieve an overall order with respect to the number system.

Therefore without properly recognising a qualitative dimension we cannot strictly derive meaningful quantitative notions of number.

Put another way the reduced (i.e. merely quantitative) notions of number we have inherited are defined in a merely 1-dimensional analytic manner.   

Now if pressed sufficiently professional mathematicians may eventually concede that the mental constructs we use to interpret mathematical reality are strictly of a subjective rather than objective nature; with much greatly difficulty they may even concede that ordinal notions strictly refer to qualitative rather than quantitative meaning.

However they will then go on to happily assume a direct absolute correspondence as between (i) objective mathematical reality and subjective mental interpretation and (ii) quantitative objects of an independent nature and a overall qualitative dimensional context (that implies interdependence between these objects).

So in both cases a basic reduction in meaning is involved. This does not entail that no useful benefit can be achieved through such reductionism. Clearly the development of conventional Mathematics proves otherwise. However it does entail that mathematical edifice has been built on a limited and - ultimately - faulty foundation.

 

Therefore the essence of the analytic approach is to clearly attempt to separate on the one hand

(i) objective truth from subjective mental interpretation and

(ii) quantitative (independent) objects from a qualitative (relational) context.

Then having attempted this clear separation a direct correspondence is assumed as between both in absolute terms.

So once again such analysis entails the implicit belief that mental constructs directly correspond with the mathematical reality (thereby interpreted) and also that general relationships between objects correspond with the (assumed) independent identities of these objects.

Now if one thinks clearly about it such assumptions are untenable and even farcical.

Surely it offends common sense to maintain for example that numbers are independent (in an absolute sense) when clearly numbers can be placed in relationship with other numbers!

The essence of the holistic approach by contrast is that it is inherently dynamic and interactive in nature.

Now the remarkable fact that immediately arises is that all numbers (and indeed all mathematical symbols) can be given a holistic - as well as analytic - identity.

Thus associated with each number for example can be defined a unique mathematical reality in holistic terms. Therefore for example, when the lens of interpretation keeps changing the corresponding mathematical reality to which it corresponds likewise keeps changing (in a relative manner).

So all numbers in holistic terms (except 1) are associated with unique holistic interpretations of mathematical reality (defined in a merely relative manner).

The significance of the number 1 in this context is that it represents the special limiting case where mathematical reality is defined in an absolute manner.

And this is the number that defines all conventional mathematical interpretation. So once again Conventional Mathematics can be precisely characterised as 1-dimensional (in qualitative terms).

So with all other dimensional numbers (other than 1) we have a unique dynamic interaction as between opposite polarities ,constituting in turn unique holistic interpretations.

 

Now the significance of this for interpretation of the Riemann Zeta Function is immense.

As we know , the one value for which the Function remains undefined in analytic (quantitative) terms is where s (the dimensional value) = 1.

However we now have a corresponding qualitative interpretation where the Function likewise remains undefined in holistic (qualitative) terms for s = 1.

What this means quite remarkably is that the Riemann Zeta Function remains uniquely undefined when we seek to interpret it in a mere analytic (i.e. conventional mathematical) manner.

This therefore implies that the Riemann Zeta Function - when appropriately interpreted in a dynamic interactive fashion  - provides an ingenious means of reconciling analytic and holistic interpretation of the number system i.e. its cardinal and ordinal aspects.

And clearly again this cannot be done in an absolute analytic manner!

 

Now to keep it simple, I will illustrate briefly the holistic approach with reference to  the (qualitative) dimensional notion of 2 (which in a very special manner is the most important).

Now the essence of 2 in this sense is that it is defined by a 1^st and 2^nd dimension.

I have explained before how indirectly we give quantitative expression to such ordinal notions through obtaining the corresponding two roots of 1 i.e. (of 1¹ and 1²).

So writing these roots (in reverse order) we obtain + 1 and – 1.

Now the 1^st dimension relates to analytic interpretation. So we interpret both numbers in an independent quantitative manner.

However the 2^nd (new) dimension relates directly to qualitative interpretation (of an interdependent nature).

So + 1 and – 1 now represent the dynamic manner in which we switch as between the opposite poles of experience (with 1 representing each pole).

 

For example with relation to the first polarity set ,we posit the external pole (i.e. by making it conscious) to get objective knowledge of mathematical relationships.

However in order to switch to the internal pole of mental interpretation (of these objects) we must negate the external pole (thereby rendering it unconscious).

If in reverse, we start by positing the internal pole, we then switch to the external (by negating the internal pole).

So there is a dynamic interdependence as between both poles with - what is labelled as - positive or negative depending purely on context.

And as all mathematical experience necessarily entails both external objects and internal interpretation, then the dynamic interpretation of this experience requires 2-dimensional - rather than 1-dimensional – interpretation.

So in this context the very number 2 takes on a remarkable new holistic meaning.

Thus once again in analytic (cardinal) terms 2 = 1 + 1 (with no qualitative distinction as between units).

However in holistic (ordinal) terms 2 = 1^st + 2^nd which indirectly is represented as (+) 1 – 1 = 0. Thus the qualitative interdependent nature of the relationship is exemplified by the absence of a quantitative result!

 

In actual experience the analytic (cardinal) understanding of numbers relates directly to rational interpretation (of a linear kind).

The holistic (ordinal) understanding relates however directly to intuitive appreciation (which indirectly is expressed in a circular i.e. paradoxical rational manner).

Therefore in actual experience (including of course mathematical) there is always a sense in which opposite poles remain separate. However there equally is an important sense in which they overlap as interdependent.   

Whereas rational understanding relates directly to their separation, intuitive appreciation arises from realisation of their mutual interdependence.

So the analytic aspect of understanding is directly of a linear rational nature (though indirectly requiring, especially in creative work, a degree of intuition).  

However the holistic aspect of understanding is directly of an intuitive nature (though indirectly requiring rational expression in a circular paradoxical manner).

Once again we can see the highly reduced nature of Conventional Mathematics, which is formally defined merely in terms of linear rational notions. So intuition - which is directly associated with qualitative type understanding - is formally reduced to (linear) reason. And this again is why qualitative meaning is reduced to quantitative!  

From this reduced perspective, mathematical symbols and relationships are given but one unambiguous interpretation.  

However when we move - at a minimum - to 2-dimensional interpretation, all mathematical symbols and relationships are given two unique interpretations (in both analytic and holistic terms).

I have illustrated, in this entry, the nature of both interpretations - analytic and holistic - for the number 2.

However in principle every mathematical symbol and relationship can equally be defined for both, with comprehensive mathematical understanding arising from the interaction of twin aspects.