As I have repeated often in these blogs, the true nature of number (as indeed all mathematical activity) is of an inherently dynamic interactive nature. Unfortunately Conventional Mathematics provides but a reduced and thereby distorted interpretation of number.
Firstly number inherently has both external (objective) and internal (subjective) aspects.
We cannot externally envisage a physical number “object” in the absence of the corresponding psychological mental perception of the number. So properly understood these two aspects necessarily continually interact in a relative manner with respect to experience.
Put another way, experience necessarily entails the interaction of two aspects of number that are physical and psychological with respect to each other.
Once again Conventional Mathematics gives but a reduced interpretation of this interaction.
Now a professional mathematician if sufficiently pressed might eventually concede that we cannot form knowledge of the number “object” in the absence of its corresponding mental perception. However the necessary interaction thereby involved is then completely ignored with interpretation taking place in a misleading absolute fashion. Thus the erroneous notion of numbers as abstract objective entities still dominates conventional thinking.
Because of its linear 1-dimensional nature Conventional Mathematics can only handle such dynamic interactions in a reduced manner whereby the subjective mental aspect is identified with the objective (which is predominant) or alternatively the objective aspect wirh its mental perception. In either case we then get an absolute rather than - more correctly - a truly relative interpretation of the nature of number.
Therefore to repeat once more the true dynamic nature of number necessarily entails twin interacting elements that are external (physical) and internal (psychological) with respect to each other.
This means in effect that once we identify for example – as recently with the (Type 1) non-trivial zeros – their physical similarity to certain quantum chaotic processes, this automatically entails that they must necessarily also have an equally important significance in complementary psychological terms.
However because Conventional Mathematics is completely lacking in dynamic interpretation it thereby places no emphasis on such complementary type relationships.
Therefore the extremely important psycho spiritual significance of the non-trivial zeros still remains completely unrecognised by the conventional mathematical community!
The other key distinction is with respect to the quantitative and qualitative aspects of number!
If we take the number “3” to illustrate we cannot experientially identify the cardinal nature of this number without implicitly recognising that it necessarily contains a 1st, a 2nd and 3rd member in ordinal terms. Thus the quantitative recognition of “3” implies corresponding ordinal recognition of its 1st, 2nd and 3rd members in a corresponding qualitative manner. And in reverse manner we cannot form knowledge of the ordinal members of a group without implicitly recognising its cardinal (quantitative) identity.
So all mathematical experience is fundamentally conditioned by the dynamic interaction as between opposite sets of polarities.
Chief among these are the external/internal that operate in a horizontal manner and the quantitative/qualitative that operates in a corresponding vertical manner.
In fact it may help to initially recognise the relationship as between them as like a compass with the four directions East and West along the horizontal axis and North and South along the vertical axis respectively.
However we can give a firmer mathematical rationale to these locations (in terms of the Type 2 aspect of the number system) by recognising these four equidistant points as corresponding to the four roots of 1. So the external and internal polarities are – relatively – complementary in a real manner (with directions that are + 1 and – 1 with respect to each other).
The qualitative therefore has likewise two directions that are + i and – i with respect to each other.
Quantitative and qualitative polarities are thereby real and imaginary with respect to each other.
What this means in effect is that the individual members of a number set have a unique qualitative meaning i.e. in their ordinal identity. However the overall set – which we initially identified in quantitative terms as cardinal likewise has an ordinal identity when related to other numbers.
So for example 2 and 3 are prime numbers (in a cardinal manner). However they equally enjoy a qualitative ordinal identity as the 1st and 2nd prime numbers respectively.
So the individual members of a cardinal prime number such as 3 enjoy a unique qualitative identity (in terms of its 1st, 2nd and 3rd members).
Thus the cardinal prime is ordinally defined in terms of its natural number members (in a corresponding qualitative manner).
However the same prime number 3 enjoys a unique collective qualitative identity as the 2nd prime in the natural number system. And all natural numbers represent a unique combination of these prime number factors.
Thus when properly understood, the all important relationship as between the primes and the natural numbers represents the fundamental manner by which their quantitative and qualitative aspects are related.
And as always – in dynamic interactive terms – there are two complementary ways in which this relationship can be understood:
1) whereby the natural number system as a whole is collectively defined through unique combinations of its prime number members.
2) whereby each prime number is uniquely defined through a collection of its natural number members.
Putting it bluntly therefore the conventional mathematical attempt to define the relationship as between the primes and the natural numbers misses the crucial point that this relationship entails a dynamic two-way complementarity as between its quantitative and qualitative aspects.
From one perspective, it is quite extraordinary how we have remained blind to this key relationship for so long!
We have tried to convince ourselves that Mathematics is solely concerned with the quantitative aspect of number. However strictly speaking we cannot even begin to identify the quantitative aspect without implicit recognition of its corresponding qualitative aspect.
Thus Mathematics is properly – in dynamic terms – as for example here with number, concerned with the relationship as between its quantitative and qualitative aspects.
Likewise – in relation to the other polarity set - we cannot form an objective knowledge of number (as external) without a corresponding mental interpretation (that is - relatively - internal).
Thus again in dynamic terms, Mathematics is properly about the relationship of objective type results to the corresponding interpretations (through which they are viewed). And from this perspective there is not just one absolute type interpretation that is valid but potentially an unlimited number (each enjoying a partial relative validity).
Once again the relationship as between the quantitative (analytic) and qualitative (holistic) aspects of number fundamentally points to the corresponding relationship as between the primes and natural numbers (and natural numbers and the primes).
And mediating this relationship are two important sets of zeta zeros (corresponding to the Type 1 and Type 2 number systems respectively).
As I have stated on a number of occasions, these zeros essentially can be viewed as the shadow of our one-sided quantitative view of number (i.e. where the qualitative aspect is directly confused with its quantitative expression).
So we can fruitfully view both sets of zeros as a means of giving two distinctive expressions (in a related complementary fashion) to the long unrecognised qualitative aspect of the number system.
Thursday, April 4, 2013
Saturday, March 30, 2013
More on Complementarity
I have continually referred on these blogs to the enormous central weakness underlying all conventional mathematical interpretation i.e. its complete absence of any genuine notion of interdependence.
It seems quite remarkabe that it is left to someone like myself - who would be considered a complete outsider by the mathematics profession - to address this all important issue that is steadfastly ignored by its own practitioners.
By its very nature Mathematics is built on linear rational notions of fixed relationships in a static independent manner. It therefore can only deal with interdependent notions – which essentially relate to the qualitative holistic aspect of mathematical activity – in a reduced and distorted quantitative fashion.
Now this problem cannot be addressed through the development of ever more complex specialised procedures within the present accepted framework of Mathematics. In fact these will only serve to blind us further to the fundamental problem which persistently is avoided i.e. that for all its great successes, such Mathematics is built on highly limited assumptions.
So what in truth represents but an extreme – though admittedly very important – special case has been misleadingly elevated to represent all valid Mathematics.
However – when properly appreciated – the true nature of Mathematics is dynamic in nature and incomparably greater that what has yet been imagined. In fact instead of the existence of just one valid system (in absolute terms), potentially an unlimited number of alternative dynamic mathematical systems exist (each possessing an important relative validity).
As we have seen Conventional Mathematics represents the special case where interpretation is - literally - 1-dimensional (linear) in nature. And the very nature of this system is that qualitative meaning such as holistic interdependence - in every context - is thereby reduced to quantitative interpretation.
The very way of properly dealing with interdependent relationships entails a circular – rather than linear – approach that is inherently based on the dynamic notion of complementarity.
I have made the important observation before that it is only when the dimension = 1 that an absolute – rather than relative – interpretation of mathematical symbols occurs. And here the qualitative aspect is thereby reduced to the quantitative in a static fixed manner.
For all other dimensions ≠ 1 a truly relative interpretation of mathematical symbols ensues where quantitative (analytic) and qualitative (holistic) aspects interact with each other in a dynamic complementary manner.
As is well known the only value for which the Riemann Zeta Function is undefined is where the dimensional value s = 1. For all other dimensional values of s, he function is indeed defined. What this implies again is that the Riemann Zeta Function – when correctly interpreted - entails the relationship as between the analytic and holistic aspects of number. Therefore the one point where it is undefined is where a pure analytic interpretation takes place (as in Conventional Mathematics).
This is why I repeatedly stress that the true nature of the Riemann Zeta Function (and its accompanying Riemann Hypothesis) cannot be properly understood within the framework of Conventional Mathematics (which uniquely does not formally allow for a holistic interpretation).
The simplest dynamic interpretation entails 2 dimensions i.e. a 1st and 2nd respectively).
The structure of these dimensions is inversely related to the quantitative nature of the two roots of i.e. + 1 and – 1 respectively.
The corresponding qualitative interpretation entails the complementary (interdependent) relationship of these two dimensions.
Now the 1st dimension (i.e. + 1) relates to the standard linear rational approach (which literally posits phenomena in experience in a rational conscious manner (which is of an analytic quantitative nature). The 2nd dimension – by contrast negates conscious interpretation in recognition of an alternative holistic qualitative type appreciation that is inherently of an unconscious (intuitive) nature.
What the latter truly entails is that conscious understanding is always – by definition – conditioned by polar opposites interpretations (that are equally valid in nature).
Therefore to affirm interpretation (in respect of one arbitrary pole) is always limited as interpretation according to the opposite pole is equally valid. Thus in order to switch reference frames as between poles – which is the very means through which experience becomes dynamically interactive – we must first negate identification with arbitrary interpretation (based on one pole as reference point).
I have illustrated this countless times before in relation to directions at a crossroads. If we approach the crossroads from one arbitrary polar direction, (e.g. travelling “up” a road) left and right have an unambiguous linear interpretation. Likewise if approached from the opposite direction (travelling “down” the road again left and right have an unambiguous linear interpretation.
However in terms of each other both of these interpretations are clearly paradoxical. So whereas linear understanding always implies interpretation in terms of one arbitrarily fixed reference frame, corresponding appreciation of interdependence entails the simultaneous recognition of both reference frames.
Now again this explains exactly why Conventional Mathematics is lacking any true notion of interdependence! By definition such recognition – at a minimum – entails two separate poles (as possible reference frames).
The opposite pole to quantitative is qualitative! Therefore in this regard before we can truly recognise notions of interdependence then we must recognise two separate poles (as reference frames) that are quantitative and qualitative with respect to each other!
Then within each frame (considered in relative separation) unambiguous type linear results will apply in rational terms. However when one then simultaneously combines both reference frames in an intuitive complementary manner, deep paradox results.
The important point therefore is that the unambiguous understanding of (single) independent reference frames conforms to linear interpretation in rational terms. By contrast the paradoxical appreciation of (multiple) interdependent reference frames conforms to a uniquely distinct type of appreciation which in direct is always of an (unconscious) intuitive nature. However indirectly it can be conveyed in a circular rational manner (based on the complementarity of opposite poles such as quantitative and qualitative).
Now my own approach is deliberately designed to conform to this model.
Thus I start by defining two aspects of the number system Type 1 and Type 2.
Initially quantitative results can be obtained under both types (in relative separation). However when we combine both in a truly complementary manner, their relative interdependence (which truly reveals their qualitative holistic nature) then becomes apparent.
So we can see in this two-dimensional approach that we continually alternate between the 1st dimension of interpretation that is linear and the 2nd dimension that is circular.
This enables us to properly preserve therefore (linear) analytic notions of quantitative independence with corresponding (circular) holistic notions of qualitative interdependence.
Now in an earlier blog I demonstrated what in fact is a striking example of this dynamic interactive approach with respect to both the Zeta 1 and Zeta 2 Functions.
Once again both of these Functions can be studied in relative separation from each other yielding important results (of a quantitative linear kind).
However when we attempt to combine both (in an interdependent manner) the truly complementary behaviour of both becomes readily apparent. One could validly say therefore that qualitative holistic understanding resides in the very ability to recognise this rich network of interconnecting complementary relationships with respect to the Functions.
I will just elaborate again on an especially striking example from previous blogs (Complementary Zeta Functions 5 & 6) .
I established that the terms in the harmonic series (i.e. for the Zeta 1 Function, where s = 1) is composed of the values corresponding to the Zeta 2 Functions (less 1) that range over the reciprocals of all natural number values of s ≠ 1.
So notice the complementarity here! Each term (as part of the infinite Zeta 1 expression) represents the whole of a corresponding infinite Zeta 2 expression (less 1).
Likewise whereas the Zeta 1 is defined for s = 1, the Zeta 2 is defined in terms of all other natural numbers ≠ 1. And finally we have complementarity where a whole number power (for Zeta 1) is paired with its reciprocal values for Zeta 2.
The requirement to subtract 1 from each Zeta 2 expression is explained by the difference between addition and multiplication with the additive identity (leaving an expression unchanged) = 0 and the multiplication identity (likewise leaving an expression unchanged = 1).
The deeper significance of this can be explained as follows.
In the cardinal (Type 1) approach each natural number is defined by a unique combination of prime number factors (in quantitative terms).
However when we understand more fully we realise that each of these prime numbers represents a hidden sub-atomic world as it were (in Type 2 terms) where each prime number is already defined in a unique ordinal manner by all its natural number members.
In this sense the corresponding quantitative uniqueness of every natural number except 1 (in terms of its prime factors) is only possible because each of these prime numbers is itself uniquely defined in a qualitative ordinal fashion by its prime number members!
However an equally fascinating reverse form of complementarity exists.
This time we start with the Zeta 2 expression where s = 1, which contains the infinite sequence of terms 1 + 1 + 1 + 1 ..
We now look at the corresponding Zeta 1 Functions which now in reverse complementary fashion range over all the natural number values for s (except 1).
Then when we sum up this infinite sequence of Zeta 1 expressions (again subtracting 1 in each case) the resulting sum = 1.
Therefore in this case the whole infinite sum of all the Zeta 1 Functions (with 1 subtracted in each case) = each individual term in the Zeta 2 expression.
Again the deeper significance of this is that just as the subatomic qualitative natural number ordinal structure is internally contained in each prime (in cardinal terms) thereby ensuring the subsequent unique quantitative structure of the natural numbers, this atomic quantitative prime number structure is already contained in each ordinal natural number member (thus ensuring the subsequent uniqueness of its internal qualitative structure). So each unique (qualitative) part is contained in the collective (quantitative) whole; also the collective (quantitative) whole is contained in each unique (qualitative) part!
In other words through this set of complementary relationships we grow in appreciation of the ultimate truly paradoxical nature of the relationship between the primes and natural numbers.
Thus the great question regarding the number system is ultimately resolved whereby the primes and natural numbers are seen as perfect reflections of each other in a pure ineffable manner.
So we are led through ever more refined rational reflection in attempting to grasp the two-way relationship of the primes and natural numbers with each other, to finally let go of all remaining remnants of thought through finding the answer revealed in total mystery.
It seems quite remarkabe that it is left to someone like myself - who would be considered a complete outsider by the mathematics profession - to address this all important issue that is steadfastly ignored by its own practitioners.
By its very nature Mathematics is built on linear rational notions of fixed relationships in a static independent manner. It therefore can only deal with interdependent notions – which essentially relate to the qualitative holistic aspect of mathematical activity – in a reduced and distorted quantitative fashion.
Now this problem cannot be addressed through the development of ever more complex specialised procedures within the present accepted framework of Mathematics. In fact these will only serve to blind us further to the fundamental problem which persistently is avoided i.e. that for all its great successes, such Mathematics is built on highly limited assumptions.
So what in truth represents but an extreme – though admittedly very important – special case has been misleadingly elevated to represent all valid Mathematics.
However – when properly appreciated – the true nature of Mathematics is dynamic in nature and incomparably greater that what has yet been imagined. In fact instead of the existence of just one valid system (in absolute terms), potentially an unlimited number of alternative dynamic mathematical systems exist (each possessing an important relative validity).
As we have seen Conventional Mathematics represents the special case where interpretation is - literally - 1-dimensional (linear) in nature. And the very nature of this system is that qualitative meaning such as holistic interdependence - in every context - is thereby reduced to quantitative interpretation.
The very way of properly dealing with interdependent relationships entails a circular – rather than linear – approach that is inherently based on the dynamic notion of complementarity.
I have made the important observation before that it is only when the dimension = 1 that an absolute – rather than relative – interpretation of mathematical symbols occurs. And here the qualitative aspect is thereby reduced to the quantitative in a static fixed manner.
For all other dimensions ≠ 1 a truly relative interpretation of mathematical symbols ensues where quantitative (analytic) and qualitative (holistic) aspects interact with each other in a dynamic complementary manner.
As is well known the only value for which the Riemann Zeta Function is undefined is where the dimensional value s = 1. For all other dimensional values of s, he function is indeed defined. What this implies again is that the Riemann Zeta Function – when correctly interpreted - entails the relationship as between the analytic and holistic aspects of number. Therefore the one point where it is undefined is where a pure analytic interpretation takes place (as in Conventional Mathematics).
This is why I repeatedly stress that the true nature of the Riemann Zeta Function (and its accompanying Riemann Hypothesis) cannot be properly understood within the framework of Conventional Mathematics (which uniquely does not formally allow for a holistic interpretation).
The simplest dynamic interpretation entails 2 dimensions i.e. a 1st and 2nd respectively).
The structure of these dimensions is inversely related to the quantitative nature of the two roots of i.e. + 1 and – 1 respectively.
The corresponding qualitative interpretation entails the complementary (interdependent) relationship of these two dimensions.
Now the 1st dimension (i.e. + 1) relates to the standard linear rational approach (which literally posits phenomena in experience in a rational conscious manner (which is of an analytic quantitative nature). The 2nd dimension – by contrast negates conscious interpretation in recognition of an alternative holistic qualitative type appreciation that is inherently of an unconscious (intuitive) nature.
What the latter truly entails is that conscious understanding is always – by definition – conditioned by polar opposites interpretations (that are equally valid in nature).
Therefore to affirm interpretation (in respect of one arbitrary pole) is always limited as interpretation according to the opposite pole is equally valid. Thus in order to switch reference frames as between poles – which is the very means through which experience becomes dynamically interactive – we must first negate identification with arbitrary interpretation (based on one pole as reference point).
I have illustrated this countless times before in relation to directions at a crossroads. If we approach the crossroads from one arbitrary polar direction, (e.g. travelling “up” a road) left and right have an unambiguous linear interpretation. Likewise if approached from the opposite direction (travelling “down” the road again left and right have an unambiguous linear interpretation.
However in terms of each other both of these interpretations are clearly paradoxical. So whereas linear understanding always implies interpretation in terms of one arbitrarily fixed reference frame, corresponding appreciation of interdependence entails the simultaneous recognition of both reference frames.
Now again this explains exactly why Conventional Mathematics is lacking any true notion of interdependence! By definition such recognition – at a minimum – entails two separate poles (as possible reference frames).
The opposite pole to quantitative is qualitative! Therefore in this regard before we can truly recognise notions of interdependence then we must recognise two separate poles (as reference frames) that are quantitative and qualitative with respect to each other!
Then within each frame (considered in relative separation) unambiguous type linear results will apply in rational terms. However when one then simultaneously combines both reference frames in an intuitive complementary manner, deep paradox results.
The important point therefore is that the unambiguous understanding of (single) independent reference frames conforms to linear interpretation in rational terms. By contrast the paradoxical appreciation of (multiple) interdependent reference frames conforms to a uniquely distinct type of appreciation which in direct is always of an (unconscious) intuitive nature. However indirectly it can be conveyed in a circular rational manner (based on the complementarity of opposite poles such as quantitative and qualitative).
Now my own approach is deliberately designed to conform to this model.
Thus I start by defining two aspects of the number system Type 1 and Type 2.
Initially quantitative results can be obtained under both types (in relative separation). However when we combine both in a truly complementary manner, their relative interdependence (which truly reveals their qualitative holistic nature) then becomes apparent.
So we can see in this two-dimensional approach that we continually alternate between the 1st dimension of interpretation that is linear and the 2nd dimension that is circular.
This enables us to properly preserve therefore (linear) analytic notions of quantitative independence with corresponding (circular) holistic notions of qualitative interdependence.
Now in an earlier blog I demonstrated what in fact is a striking example of this dynamic interactive approach with respect to both the Zeta 1 and Zeta 2 Functions.
Once again both of these Functions can be studied in relative separation from each other yielding important results (of a quantitative linear kind).
However when we attempt to combine both (in an interdependent manner) the truly complementary behaviour of both becomes readily apparent. One could validly say therefore that qualitative holistic understanding resides in the very ability to recognise this rich network of interconnecting complementary relationships with respect to the Functions.
I will just elaborate again on an especially striking example from previous blogs (Complementary Zeta Functions 5 & 6) .
I established that the terms in the harmonic series (i.e. for the Zeta 1 Function, where s = 1) is composed of the values corresponding to the Zeta 2 Functions (less 1) that range over the reciprocals of all natural number values of s ≠ 1.
So notice the complementarity here! Each term (as part of the infinite Zeta 1 expression) represents the whole of a corresponding infinite Zeta 2 expression (less 1).
Likewise whereas the Zeta 1 is defined for s = 1, the Zeta 2 is defined in terms of all other natural numbers ≠ 1. And finally we have complementarity where a whole number power (for Zeta 1) is paired with its reciprocal values for Zeta 2.
The requirement to subtract 1 from each Zeta 2 expression is explained by the difference between addition and multiplication with the additive identity (leaving an expression unchanged) = 0 and the multiplication identity (likewise leaving an expression unchanged = 1).
The deeper significance of this can be explained as follows.
In the cardinal (Type 1) approach each natural number is defined by a unique combination of prime number factors (in quantitative terms).
However when we understand more fully we realise that each of these prime numbers represents a hidden sub-atomic world as it were (in Type 2 terms) where each prime number is already defined in a unique ordinal manner by all its natural number members.
In this sense the corresponding quantitative uniqueness of every natural number except 1 (in terms of its prime factors) is only possible because each of these prime numbers is itself uniquely defined in a qualitative ordinal fashion by its prime number members!
However an equally fascinating reverse form of complementarity exists.
This time we start with the Zeta 2 expression where s = 1, which contains the infinite sequence of terms 1 + 1 + 1 + 1 ..
We now look at the corresponding Zeta 1 Functions which now in reverse complementary fashion range over all the natural number values for s (except 1).
Then when we sum up this infinite sequence of Zeta 1 expressions (again subtracting 1 in each case) the resulting sum = 1.
Therefore in this case the whole infinite sum of all the Zeta 1 Functions (with 1 subtracted in each case) = each individual term in the Zeta 2 expression.
Again the deeper significance of this is that just as the subatomic qualitative natural number ordinal structure is internally contained in each prime (in cardinal terms) thereby ensuring the subsequent unique quantitative structure of the natural numbers, this atomic quantitative prime number structure is already contained in each ordinal natural number member (thus ensuring the subsequent uniqueness of its internal qualitative structure). So each unique (qualitative) part is contained in the collective (quantitative) whole; also the collective (quantitative) whole is contained in each unique (qualitative) part!
In other words through this set of complementary relationships we grow in appreciation of the ultimate truly paradoxical nature of the relationship between the primes and natural numbers.
Thus the great question regarding the number system is ultimately resolved whereby the primes and natural numbers are seen as perfect reflections of each other in a pure ineffable manner.
So we are led through ever more refined rational reflection in attempting to grasp the two-way relationship of the primes and natural numbers with each other, to finally let go of all remaining remnants of thought through finding the answer revealed in total mystery.
Friday, March 29, 2013
Brief Interlude
It may perhaps help to place some of my recent blogs on the intimate complementarity of the Zeta 1 and Zeta 2 Functions in perspective.
My overriding purpose in these blogs is simple.
Over the past 50 years or so I have reached the firm conclusion that – strictly speaking - Mathematics, as we know it, is simply not fit for purpose!
So in this context I am using the Riemann Hypothesis to illustrate the nature of this dilemma.
The true significance of the Riemann Hypothesis could hardly be more significant as it relates directly to the fundamental nature of our number system.
Now the inherent nature of this system - as indeed all mathematical activity - is truly dynamic with twin analytic (quantitative) and holistic (qualitative) aspects that continually interact in a complementary fashion with respect to each other.
Now as Conventionally Mathematics is formally interpreted in a merely reduced manner (that gives sole recognition to its quantitative aspect) not alone is the Riemann Hypothesis incapable of proof from within this perspective, more importantly it cannot even be properly understood in this manner!
Right at the heart of the conventional understanding exists a basic form of reductionism where the ordinal (i.e. qualitative) interpretation of number is assumed to be implied directly by its corresponding cardinal (i.e. quantitative) interpretation!
So we tend to look at the cardinal numbers as independent units in quantitative terms. However momentary reflection on the matter would suggest that corresponding ordinal interpretation entails a relationship between numbers (which is necessarily of a qualitative nature).
In actual experience the notion of number (such as 2 in this example) continually alternates as between its cardinal (quantitative) and ordinal (qualitative) meanings.
However far from realising the significance of this relative dynamic interaction, a highly reduced - and thereby utterly distorted - interpretation of the nature of number has come to dominate Western culture (and indeed other cultures) whereby number is misleadingly interpreted in an absolute static manner (with respect to its quantitative attributes).
Thus we mistakenly believe that the ordinal notion of the natural numbers for example such as 1st, 2nd, 3rd, 4th and so on are directly implied by the corresponding cardinal notions of 1, 2, 3, 4!
Indeed very often cardinal numbers are used directly to refer to ordinal rankings. So Tiger Woods is now once again no. 1 (i.e. 1st) in the PGA professional golf rankings, illustrating how the cardinal notion of 1 as a quantitative number is readily interchanged with the corresponding ordinal notion of 1st, as its relational qualitative counterpart!
However this fallacy i.e. of the ordinal being directly implied by its cardinal aspect, is immediately exposed when we attempt to explain the relationship of the primes to the natural numbers (from the conventional mathematical perspective).
Here the primes are viewed as the cardinal building blocks of the natural numbers so that each natural number represents a unique combination of prime factors.
However if we reflect on it for a moment a prime number strictly has no meaning in the absence of its ordinal natural number counterpart.
So for example the very recognition of 2 and 3 and 5 as prime numbers implies a natural number ordinal ranking among the primes of 1st 2nd and 3rd respectively. And if we assume that the ordinal is implied by its cardinal aspect, then this means that we must already assume the pre-existent identity of the cardinal numbers before we can even begin to explain their derivation from the primes!
This observation in fact implies a related circular notion of number (the significance of which is effectively unrecognised in conventional terms) that represents the appropriate (unreduced) appreciation of ordinal meaning.
Again even a little reflection on the matter will reveal how ambiguous is the ordinal notion of number! Clearly the meaning of 1st, 2nd, 3rd etc is of a relative nature depending on the size of the number group to which it refers.
So for example the 2nd of a group of 2 has a relatively distinct meaning from the second of a group of 3, 4, 5 etc. with potentially an unlimited range applying.
Therefore all natural numbers (in ordinal terms) have an unlimited range of potential meanings (depending on the size of the finite group to which they belong).
A little further reflection would clearly indicate that since cardinal and ordinal meaning are thereby in dynamic terms interconnected, that indirectly the cardinal number system is also of a merely relative nature!
In other words, in dynamic terms, all numbers have two complementary aspects that are - relatively - independent and also - relatively - interdependent with each other.
The static notion of numbers as abstract entities is therefore just an illusion (based on a reduced - merely quantitative – interpretation).
In my own dynamic treatment of number, I am therefore at pains to demonstrate that the number system is composed of two interacting aspects - which I refer to as Type 1 and Type 2 respectively.
The Type 1 is directly associated with the quantitative aspect (though indirectly through interdependence equally possessing a qualitative aspect).
The Type 2 is then directly associated with the qualitative aspect (though again through interdependence) indirectly possessing a quantitative aspect.
Then I refer to the full combined interaction of both aspects (Type 1 and Type 2) as Type 3.
Basically the Type 1 aspect treats numbers as homogeneous collective units in quantitative terms (thus allowing for no qualitative distinction as between individual units). So 3 as a cardinal number from this perspective = 1 + 1 + 1 (in quantitative terms).
From the Type 1 cardinal aspect the prime numbers represent the basic building blocks of the natural number system. All natural numbers (except 1) from this perspective represent a unique combination of prime factors.
The Type 2 aspect by contrast is based the notion of a number group as composed of unique individual members (in qualitative terms). Thus from this perspective 3 relates to its distinctive 1st, 2nd and 3rd individual members as ordinally defined.
From the Type 2 aspect the natural numbers represent the basic building blocks of each prime number (in ordinal terms). So every prime number (p) from an ordinal perspective is composed of a unique set of natural numbers 1st, 2nd, 3rd,...pth!
Then using these distinctions, I go on to define both Zeta 1 and Zeta 2 Functions with respect to uncovering the true nature of the Riemann Hypothesis. This relates to the ultimate identity of both the quantitative and qualitative aspects of number.
From this enlarged perspective we now have two complementary sets of (non-trivial) zeta zeros.
Indeed perhaps the best explanation of these incredibly significant sets of numbers can be given as follows.
The Type 1 zeta zeros indirectly represent the holistic aspect of the cardinal number system, whereas the Type 2 zeta zeros directly represent its corresponding holistic ordinal aspect.
Ultimately - which can be approximated in Type 3 terms where both systems simultaneously interact - both Type 1 and Type 2 zeros can be seen as fully complementary and indeed identical in an ineffable manner with the natural number system (in both cardinal and ordinal terms).
The very manner by which the quantitative and qualitative aspects of the number system interact is though the two-way interaction of the primes with the natural numbers (and the natural numbers with the primes), both of which are mediated through the two sets of zeta zeros respectively.
This leads to the remarkable conclusion that the hidden holistic activity of both sets of zeta zeros necessarily underlies all human experience in an innate fashion in the seemingly obvious identification of cardinal with ordinal meaning (i.e. 1st with 1, 2nd with 2, 3rd with 3 etc.)
Indeed even more remarkably this innate activity necessarily underlies all natural processes as the very means by which they acquire a phenomenal identity!
So what seems most accessible at a conscious level (appearing totally obvious) therefore remains least accessible in corresponding unconscious terms.
Mathematics is now in urgent need of addressing its great shadow i.e. the unrecognised holistic aspect of interpretation. As such understanding intimately affects all of the sciences - now greatly in need of a more integrated understanding - our very civilisation may now well depend on such holistic realisation occurring rapidly in the very near future.
My overriding purpose in these blogs is simple.
Over the past 50 years or so I have reached the firm conclusion that – strictly speaking - Mathematics, as we know it, is simply not fit for purpose!
So in this context I am using the Riemann Hypothesis to illustrate the nature of this dilemma.
The true significance of the Riemann Hypothesis could hardly be more significant as it relates directly to the fundamental nature of our number system.
Now the inherent nature of this system - as indeed all mathematical activity - is truly dynamic with twin analytic (quantitative) and holistic (qualitative) aspects that continually interact in a complementary fashion with respect to each other.
Now as Conventionally Mathematics is formally interpreted in a merely reduced manner (that gives sole recognition to its quantitative aspect) not alone is the Riemann Hypothesis incapable of proof from within this perspective, more importantly it cannot even be properly understood in this manner!
Right at the heart of the conventional understanding exists a basic form of reductionism where the ordinal (i.e. qualitative) interpretation of number is assumed to be implied directly by its corresponding cardinal (i.e. quantitative) interpretation!
So we tend to look at the cardinal numbers as independent units in quantitative terms. However momentary reflection on the matter would suggest that corresponding ordinal interpretation entails a relationship between numbers (which is necessarily of a qualitative nature).
In actual experience the notion of number (such as 2 in this example) continually alternates as between its cardinal (quantitative) and ordinal (qualitative) meanings.
However far from realising the significance of this relative dynamic interaction, a highly reduced - and thereby utterly distorted - interpretation of the nature of number has come to dominate Western culture (and indeed other cultures) whereby number is misleadingly interpreted in an absolute static manner (with respect to its quantitative attributes).
Thus we mistakenly believe that the ordinal notion of the natural numbers for example such as 1st, 2nd, 3rd, 4th and so on are directly implied by the corresponding cardinal notions of 1, 2, 3, 4!
Indeed very often cardinal numbers are used directly to refer to ordinal rankings. So Tiger Woods is now once again no. 1 (i.e. 1st) in the PGA professional golf rankings, illustrating how the cardinal notion of 1 as a quantitative number is readily interchanged with the corresponding ordinal notion of 1st, as its relational qualitative counterpart!
However this fallacy i.e. of the ordinal being directly implied by its cardinal aspect, is immediately exposed when we attempt to explain the relationship of the primes to the natural numbers (from the conventional mathematical perspective).
Here the primes are viewed as the cardinal building blocks of the natural numbers so that each natural number represents a unique combination of prime factors.
However if we reflect on it for a moment a prime number strictly has no meaning in the absence of its ordinal natural number counterpart.
So for example the very recognition of 2 and 3 and 5 as prime numbers implies a natural number ordinal ranking among the primes of 1st 2nd and 3rd respectively. And if we assume that the ordinal is implied by its cardinal aspect, then this means that we must already assume the pre-existent identity of the cardinal numbers before we can even begin to explain their derivation from the primes!
This observation in fact implies a related circular notion of number (the significance of which is effectively unrecognised in conventional terms) that represents the appropriate (unreduced) appreciation of ordinal meaning.
Again even a little reflection on the matter will reveal how ambiguous is the ordinal notion of number! Clearly the meaning of 1st, 2nd, 3rd etc is of a relative nature depending on the size of the number group to which it refers.
So for example the 2nd of a group of 2 has a relatively distinct meaning from the second of a group of 3, 4, 5 etc. with potentially an unlimited range applying.
Therefore all natural numbers (in ordinal terms) have an unlimited range of potential meanings (depending on the size of the finite group to which they belong).
A little further reflection would clearly indicate that since cardinal and ordinal meaning are thereby in dynamic terms interconnected, that indirectly the cardinal number system is also of a merely relative nature!
In other words, in dynamic terms, all numbers have two complementary aspects that are - relatively - independent and also - relatively - interdependent with each other.
The static notion of numbers as abstract entities is therefore just an illusion (based on a reduced - merely quantitative – interpretation).
In my own dynamic treatment of number, I am therefore at pains to demonstrate that the number system is composed of two interacting aspects - which I refer to as Type 1 and Type 2 respectively.
The Type 1 is directly associated with the quantitative aspect (though indirectly through interdependence equally possessing a qualitative aspect).
The Type 2 is then directly associated with the qualitative aspect (though again through interdependence) indirectly possessing a quantitative aspect.
Then I refer to the full combined interaction of both aspects (Type 1 and Type 2) as Type 3.
Basically the Type 1 aspect treats numbers as homogeneous collective units in quantitative terms (thus allowing for no qualitative distinction as between individual units). So 3 as a cardinal number from this perspective = 1 + 1 + 1 (in quantitative terms).
From the Type 1 cardinal aspect the prime numbers represent the basic building blocks of the natural number system. All natural numbers (except 1) from this perspective represent a unique combination of prime factors.
The Type 2 aspect by contrast is based the notion of a number group as composed of unique individual members (in qualitative terms). Thus from this perspective 3 relates to its distinctive 1st, 2nd and 3rd individual members as ordinally defined.
From the Type 2 aspect the natural numbers represent the basic building blocks of each prime number (in ordinal terms). So every prime number (p) from an ordinal perspective is composed of a unique set of natural numbers 1st, 2nd, 3rd,...pth!
Then using these distinctions, I go on to define both Zeta 1 and Zeta 2 Functions with respect to uncovering the true nature of the Riemann Hypothesis. This relates to the ultimate identity of both the quantitative and qualitative aspects of number.
From this enlarged perspective we now have two complementary sets of (non-trivial) zeta zeros.
Indeed perhaps the best explanation of these incredibly significant sets of numbers can be given as follows.
The Type 1 zeta zeros indirectly represent the holistic aspect of the cardinal number system, whereas the Type 2 zeta zeros directly represent its corresponding holistic ordinal aspect.
Ultimately - which can be approximated in Type 3 terms where both systems simultaneously interact - both Type 1 and Type 2 zeros can be seen as fully complementary and indeed identical in an ineffable manner with the natural number system (in both cardinal and ordinal terms).
The very manner by which the quantitative and qualitative aspects of the number system interact is though the two-way interaction of the primes with the natural numbers (and the natural numbers with the primes), both of which are mediated through the two sets of zeta zeros respectively.
This leads to the remarkable conclusion that the hidden holistic activity of both sets of zeta zeros necessarily underlies all human experience in an innate fashion in the seemingly obvious identification of cardinal with ordinal meaning (i.e. 1st with 1, 2nd with 2, 3rd with 3 etc.)
Indeed even more remarkably this innate activity necessarily underlies all natural processes as the very means by which they acquire a phenomenal identity!
So what seems most accessible at a conscious level (appearing totally obvious) therefore remains least accessible in corresponding unconscious terms.
Mathematics is now in urgent need of addressing its great shadow i.e. the unrecognised holistic aspect of interpretation. As such understanding intimately affects all of the sciences - now greatly in need of a more integrated understanding - our very civilisation may now well depend on such holistic realisation occurring rapidly in the very near future.
Thursday, March 28, 2013
Two Complementary Zeta Functions (7)
We can also connect Euler's famous product formula to the Zeta 2 Function.
Indeed this can provide a new perspective on this formula (which I have not seen developed elsewhere).
For example when s = 2 in the Zeta 1 expression,
ζ1(2) = (1/1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .......
= 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...
So the product terms on the right hand side involve the expression p^2/(p^2 - 1) where p ranges over all the prime numbers!
However this same expression can be shown to be intimately related to the Zeta 2 Function.
Now in illustrating this it will perhaps be easier initially to start with the case corresponding in the Zeta 1 where s = 1.
Here according to Euler's formula
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 = 2/1 * 3/2 * 5/4 * 7/6 * ...
So the summation series on the LHS represents the well known harmonic series while the terms in the product series on the RHS conform to p/(p - 1) where again the value of p ranges over all the prime numbers! However because Euler's formula holds only for convergent series (and these diverge) we have an exception in this case. In fact in truth where s = 1, the R.H.S. conforms to p/(p + 1),
so that
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 ~ 3/2 * 4/3 * 6/5 * 8/7 i.e. (1 + 1/2) * (1 + 1/3) * (1 + 1/5) * (1 + 1/7) *....
However to illustrate our procedures we will illustrate initially by proceeding as if the Euler formula is true for the case s = 1!
As we have already seen the value of the infinite Zeta 2 Function (where s = 1/2)i.e. ζ2(1/2)
= 1 + 1/2 + 1/4 + 1/8 +..... = 2
Therefore the 1st term in the harmonic series = ζ2(1/2) - 1
Likewise the 2nd term in the harmonic series = ζ2(1/3) - 1, the 3rd term ζ2(1/4) - 1, the 4th term ζ2(1/5) - 1 etc.
Now the corresponding terms in the related product formula can be derived in a consistently simple manner with respect to the Zeta 2 expression.
So in each case where s represents the reciprocal of a prime value, we simply divide the value for the Function (representing the sum of an infinite series of terms) by the corresponding first term of the series.
So when s = 1/2 we divide 2 (the sum of the series) by 1 (the 1st term) to obtain 2/1 i.e. the 1st term in the product series
Then when s = 1/3 we divide 3/2 (the sum for the series by a) to obtain 3/2.
Then when s = 1/5 we divide 5/4 (the sum for the series by 1) to obtain 5/4.
Finally to illustrate when s = 1/7 we divide 7/6 (the sum of the series by 1) to obtain 7/6 and so on.
Now because the 1st term is always 1, the process seems somewhat trivial (and as we have seen does not actually apply in this case).
However the procedure assumes much more relevance when we deal with the product values corresponding to higher integer values of s (with respect to the Zeta 1).
So once again
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...
Now with respect to the Zeta 2 the additive terms on the LHS correspond now
to {ζ2(1/2) - 1}^2, {ζ2(1/3) - 1}^2, {ζ2(1/4) - 1}^2,{ζ2(1/5) - 1}^2 and so on!
Now this time because each terms is raised to the power of 2, to obtain the corresponding terms in the product formula we divide the sum of the Function (for each value in question) by the corresponding sum of the first two terms in the Function.
So once again where s = 1/2 the value of the Function = 2.
The sum of the corresponding 1st two terms of the Function = 1 + 1/2 = 3/2.
Then when we divide 2 by 3/2 (= 2 * 2/3) we get 4/3 i.e. the 1st term in the product formula!
The other terms in the product formula can be obtained in like manner
So for example when s = 1/3 the sum of the first two terms = 1 + 1/3 = 4/3.
When we divide the sum of the Function 3/2 by 4/3 we obtain 3/2 * 3/4 = 9/8 i.e. the 2nd value in the product formula.
The next relevant value (representing the reciprocal of a prime number) = 1/5.
Thus sum of the first two terms = 1 + 1/5 = 6/5 and the sum of the Function 5/4.
Then dividing 5/4 by 6/5 we get 5/4 * 5/6 = 25/24 i.e. the 3rd term in the product formula.
Finally to illustrate, the next relevant value for the Zeta 2 Function is 1/7
The value of the Function = 7/6 and the sum of first two terms = 1 + 1/7 = 8/7.
So 7/6 divided by 8/7 = 7/6 * 7/8 = 49/48 i.e. the next value in the product formula.
Now this process can be extended indefinitely for higher values of s.
For example when s = 3 (with respect to the Zeta 1 Function), the corresponding terms in additive series of terms in terms of the Zeta 2 will be
ζ2(1/2) - 1}^3, {ζ2(1/3) - 1}^3, {ζ2(1/4) - 1}^3,{ζ2(1/5) - 1}^3 and so on!
Then to get the corresponding terms in the product formula we divide the value of the Zeta 2 Function (ranging again over the reciprocals of all prime numbered values)by the sum of the first 3 terms of the Function in each case.
In this way both the additive and product parts of the Euler Formula can be intimately demonstrated in terms of the Zeta 2 Function (thus indicating its complementary nature to the Zeta 1).
In fact in many ways a more natural fit exists (especially for the multiplication part) in terms of the Zeta 2 expression.
Better notational representation and further illustrations of approach can be found under Section 2 at Two Zeta Functions.
Indeed this can provide a new perspective on this formula (which I have not seen developed elsewhere).
For example when s = 2 in the Zeta 1 expression,
ζ1(2) = (1/1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .......
= 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...
So the product terms on the right hand side involve the expression p^2/(p^2 - 1) where p ranges over all the prime numbers!
However this same expression can be shown to be intimately related to the Zeta 2 Function.
Now in illustrating this it will perhaps be easier initially to start with the case corresponding in the Zeta 1 where s = 1.
Here according to Euler's formula
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 = 2/1 * 3/2 * 5/4 * 7/6 * ...
So the summation series on the LHS represents the well known harmonic series while the terms in the product series on the RHS conform to p/(p - 1) where again the value of p ranges over all the prime numbers! However because Euler's formula holds only for convergent series (and these diverge) we have an exception in this case. In fact in truth where s = 1, the R.H.S. conforms to p/(p + 1),
so that
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 ~ 3/2 * 4/3 * 6/5 * 8/7 i.e. (1 + 1/2) * (1 + 1/3) * (1 + 1/5) * (1 + 1/7) *....
However to illustrate our procedures we will illustrate initially by proceeding as if the Euler formula is true for the case s = 1!
As we have already seen the value of the infinite Zeta 2 Function (where s = 1/2)i.e. ζ2(1/2)
= 1 + 1/2 + 1/4 + 1/8 +..... = 2
Therefore the 1st term in the harmonic series = ζ2(1/2) - 1
Likewise the 2nd term in the harmonic series = ζ2(1/3) - 1, the 3rd term ζ2(1/4) - 1, the 4th term ζ2(1/5) - 1 etc.
Now the corresponding terms in the related product formula can be derived in a consistently simple manner with respect to the Zeta 2 expression.
So in each case where s represents the reciprocal of a prime value, we simply divide the value for the Function (representing the sum of an infinite series of terms) by the corresponding first term of the series.
So when s = 1/2 we divide 2 (the sum of the series) by 1 (the 1st term) to obtain 2/1 i.e. the 1st term in the product series
Then when s = 1/3 we divide 3/2 (the sum for the series by a) to obtain 3/2.
Then when s = 1/5 we divide 5/4 (the sum for the series by 1) to obtain 5/4.
Finally to illustrate when s = 1/7 we divide 7/6 (the sum of the series by 1) to obtain 7/6 and so on.
Now because the 1st term is always 1, the process seems somewhat trivial (and as we have seen does not actually apply in this case).
However the procedure assumes much more relevance when we deal with the product values corresponding to higher integer values of s (with respect to the Zeta 1).
So once again
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...
Now with respect to the Zeta 2 the additive terms on the LHS correspond now
to {ζ2(1/2) - 1}^2, {ζ2(1/3) - 1}^2, {ζ2(1/4) - 1}^2,{ζ2(1/5) - 1}^2 and so on!
Now this time because each terms is raised to the power of 2, to obtain the corresponding terms in the product formula we divide the sum of the Function (for each value in question) by the corresponding sum of the first two terms in the Function.
So once again where s = 1/2 the value of the Function = 2.
The sum of the corresponding 1st two terms of the Function = 1 + 1/2 = 3/2.
Then when we divide 2 by 3/2 (= 2 * 2/3) we get 4/3 i.e. the 1st term in the product formula!
The other terms in the product formula can be obtained in like manner
So for example when s = 1/3 the sum of the first two terms = 1 + 1/3 = 4/3.
When we divide the sum of the Function 3/2 by 4/3 we obtain 3/2 * 3/4 = 9/8 i.e. the 2nd value in the product formula.
The next relevant value (representing the reciprocal of a prime number) = 1/5.
Thus sum of the first two terms = 1 + 1/5 = 6/5 and the sum of the Function 5/4.
Then dividing 5/4 by 6/5 we get 5/4 * 5/6 = 25/24 i.e. the 3rd term in the product formula.
Finally to illustrate, the next relevant value for the Zeta 2 Function is 1/7
The value of the Function = 7/6 and the sum of first two terms = 1 + 1/7 = 8/7.
So 7/6 divided by 8/7 = 7/6 * 7/8 = 49/48 i.e. the next value in the product formula.
Now this process can be extended indefinitely for higher values of s.
For example when s = 3 (with respect to the Zeta 1 Function), the corresponding terms in additive series of terms in terms of the Zeta 2 will be
ζ2(1/2) - 1}^3, {ζ2(1/3) - 1}^3, {ζ2(1/4) - 1}^3,{ζ2(1/5) - 1}^3 and so on!
Then to get the corresponding terms in the product formula we divide the value of the Zeta 2 Function (ranging again over the reciprocals of all prime numbered values)by the sum of the first 3 terms of the Function in each case.
In this way both the additive and product parts of the Euler Formula can be intimately demonstrated in terms of the Zeta 2 Function (thus indicating its complementary nature to the Zeta 1).
In fact in many ways a more natural fit exists (especially for the multiplication part) in terms of the Zeta 2 expression.
Better notational representation and further illustrations of approach can be found under Section 2 at Two Zeta Functions.
Wednesday, March 27, 2013
Two Complementary Zeta Functions (6)
Yesterday I illustrated just one important example of the intimate relationship exists as between the Zeta 1 and Zeta 2 Functions.
Once again - though this web-page does not ideally lend itself to sophisticated mathematical notation - I denote the Zeta 1 as ζ1(s) and the Zeta 2 as ζ2(s)respectively.
We saw then that that ζ1(1)i.e. the harmonic series = 1 + 1/2 + 1/3 + 1/4 + .... can be expressed in terms of a sum of Zeta 2 series where the value of s (for the Zeta 2) ranges over the reciprocals of all the natural numbers (except 1).
So the complementarity here relates to the fact that the value of s in the Zeta 1 (= 1) is related to all other natural numbers (except 1) with respect to the Zeta 2.
Thus ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, ....
Now a fascinating reverse form of complementarity also exists as between the Zeta 2 and the Zeta 1 Functions.
When s = 1
ζ2(1) = 1 + 1 + 1 + 1 +.....
Therefore for a finite range of values (i.e. for s = 1, up to n)
ζ2(1) = n
Now with respect to the corresponding Zeta 1 Function
ζ1(2) = (π^2)/6 = 1.6449340668...
ζ1(3) = 1.2020569032...
ζ1(4) = (π^4)/90 = 1.0823232337...
ζ1(5) = 1.0369277551...
ζ1(6) = (π^6)/945 = 1.0173430619...
ζ1(7) = 1.0083492774...
ζ1(8) = (π^8)/9450 = 1.0040773561...
ζ1(9) = 1.0020083928...
ζ1(10)= (π^10)/93555 = 1.0009945751...
and so on
Now if we subtract 1 from each value of the Zeta 1 and sum up the remaining values the total sum (for all natural number values of s > 1) = 1.
Indeed, when we sum up the first 10 values (with again 1 subtracted from each value) the total already converges very closely on 1 i.e. .9990146221...
Therefore when we obtain the total for all these Zeta 1 Functions over the finite range (this time from s = 2 up to n)
the answer converges on (n - 1) + 1 = n
So just as we have shown that,
ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, .... ,
Now in like reverse manner where the value of s with respect to Zeta 2 can range from 2 to n (with no finite upper limit on n)
ζ2(1) ~ ∑{ζ1(s)
Now in the former expression we subtracted 1 from the sum of each of the Zeta 2 values.
We do not do the same in the case of each Zeta 1 value as there is a lagged nature between both Functions differing by 1.
For example in the case where s = 0 with respect the the Zeta 1,
ζ1(0) = 1 + 1 + 1 + 1 +...
However this corresponds exactly with the Zeta 2 where now s = 1!
i.e. ζ2(1) = 1 + 1 + 1 + 1 +...
In fact other fascinating aspects with respect to Zeta 1 values can be briefly illustrated here!
Again when we subtract 1 from the sum of the Zeta 1 Function (ranging over the natural numbers from 2 upwards),
the sum of even numbered values converges to .75!
Again using the values listed above for the first 5 even values (up to 10), the relevant sum is
.6449340668 + ..08232332337 +.01734306198 + .00407735619 + .0009945751 = .74967229717
So the answer here has already converged very closely on .75!
This implies that the sum of the odd numbered values (excluding s = 1) converges on .25!
It equally implies that the alternating series (where each even is balanced by its succeeding odd term)
i.e. ζ1(2) - ζ1(3) + ζ1(4) - ζ1(5) +..... = .5
It is well known that with respect to the Zeta 1, for the sum of the harmonic series,
ζ1(1)~ log n + γ (where γ is the Euler-Mascheroni Constant = .5772156649..)
Fascinatingly γ is directly connected with the Zeta 1 Function (for all natural number values starting with 2) in the following manner
i.e. γ = ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....
Therefore
ζ1(1)~ log n + ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....
So,
log n ~ ζ1(1) - ζ1(2)/2 + ζ1(3)/3 - ζ1(4)/4 + ζ1(5)/5 - .... where the Zeta Functions are summed over a finite range of terms.
Alternatively,
γ ~ ζ1(1) – {ζ1(2)/2 + ζ(13)/3 + ζ1(4)/4 + ζ1(5)/5 +…..} again when summed to a finite n with approximation improving as n increases.
Thus,
log n ~ ζ1(2)/2 + ζ1(3)/3 + ζ1(4)/4 + ζ1(5)/5 +….. when summed to a finite n with approximation improving as n improves.
Once again - though this web-page does not ideally lend itself to sophisticated mathematical notation - I denote the Zeta 1 as ζ1(s) and the Zeta 2 as ζ2(s)respectively.
We saw then that that ζ1(1)i.e. the harmonic series = 1 + 1/2 + 1/3 + 1/4 + .... can be expressed in terms of a sum of Zeta 2 series where the value of s (for the Zeta 2) ranges over the reciprocals of all the natural numbers (except 1).
So the complementarity here relates to the fact that the value of s in the Zeta 1 (= 1) is related to all other natural numbers (except 1) with respect to the Zeta 2.
Thus ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, ....
Now a fascinating reverse form of complementarity also exists as between the Zeta 2 and the Zeta 1 Functions.
When s = 1
ζ2(1) = 1 + 1 + 1 + 1 +.....
Therefore for a finite range of values (i.e. for s = 1, up to n)
ζ2(1) = n
Now with respect to the corresponding Zeta 1 Function
ζ1(2) = (π^2)/6 = 1.6449340668...
ζ1(3) = 1.2020569032...
ζ1(4) = (π^4)/90 = 1.0823232337...
ζ1(5) = 1.0369277551...
ζ1(6) = (π^6)/945 = 1.0173430619...
ζ1(7) = 1.0083492774...
ζ1(8) = (π^8)/9450 = 1.0040773561...
ζ1(9) = 1.0020083928...
ζ1(10)= (π^10)/93555 = 1.0009945751...
and so on
Now if we subtract 1 from each value of the Zeta 1 and sum up the remaining values the total sum (for all natural number values of s > 1) = 1.
Indeed, when we sum up the first 10 values (with again 1 subtracted from each value) the total already converges very closely on 1 i.e. .9990146221...
Therefore when we obtain the total for all these Zeta 1 Functions over the finite range (this time from s = 2 up to n)
the answer converges on (n - 1) + 1 = n
So just as we have shown that,
ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, .... ,
Now in like reverse manner where the value of s with respect to Zeta 2 can range from 2 to n (with no finite upper limit on n)
ζ2(1) ~ ∑{ζ1(s)
Now in the former expression we subtracted 1 from the sum of each of the Zeta 2 values.
We do not do the same in the case of each Zeta 1 value as there is a lagged nature between both Functions differing by 1.
For example in the case where s = 0 with respect the the Zeta 1,
ζ1(0) = 1 + 1 + 1 + 1 +...
However this corresponds exactly with the Zeta 2 where now s = 1!
i.e. ζ2(1) = 1 + 1 + 1 + 1 +...
In fact other fascinating aspects with respect to Zeta 1 values can be briefly illustrated here!
Again when we subtract 1 from the sum of the Zeta 1 Function (ranging over the natural numbers from 2 upwards),
the sum of even numbered values converges to .75!
Again using the values listed above for the first 5 even values (up to 10), the relevant sum is
.6449340668 + ..08232332337 +.01734306198 + .00407735619 + .0009945751 = .74967229717
So the answer here has already converged very closely on .75!
This implies that the sum of the odd numbered values (excluding s = 1) converges on .25!
It equally implies that the alternating series (where each even is balanced by its succeeding odd term)
i.e. ζ1(2) - ζ1(3) + ζ1(4) - ζ1(5) +..... = .5
It is well known that with respect to the Zeta 1, for the sum of the harmonic series,
ζ1(1)~ log n + γ (where γ is the Euler-Mascheroni Constant = .5772156649..)
Fascinatingly γ is directly connected with the Zeta 1 Function (for all natural number values starting with 2) in the following manner
i.e. γ = ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....
Therefore
ζ1(1)~ log n + ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....
So,
log n ~ ζ1(1) - ζ1(2)/2 + ζ1(3)/3 - ζ1(4)/4 + ζ1(5)/5 - .... where the Zeta Functions are summed over a finite range of terms.
Alternatively,
γ ~ ζ1(1) – {ζ1(2)/2 + ζ(13)/3 + ζ1(4)/4 + ζ1(5)/5 +…..} again when summed to a finite n with approximation improving as n increases.
Thus,
log n ~ ζ1(2)/2 + ζ1(3)/3 + ζ1(4)/4 + ζ1(5)/5 +….. when summed to a finite n with approximation improving as n improves.
Tuesday, March 26, 2013
Two Complementary Zeta Functions (5)
Once again it is important to bear in mind that I define two complementary Zeta Functions.
The first relates to the recognised Riemann Zeta Function (Zeta 1) which is defined as an infinite series:
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….
Now if with respect to a number raised to a power we refer to the initial number as the base quantity and the exponent or power as the dimensional number,
for the Zeta 1 Function the natural numbers serve as the base quantities and s as the dimensional power - negative in this case - to which the base quantities are raised.
The Zeta 2 Function - the role of which is largely unrecognised - is defined as a finite series:
1 + s^1 + s^2 + ….+ s^(n – 1).
Here the role of base quantities and dimensional powers is reversed with s now serving as the base quantity which is defined with respect to the natural numbers (from 1 to n) where n has no finite limit.
In a qualified sense, the Zeta 2 can be defined in infinite terms as:
1 + s^1 + s^2 + s^3 + s^4 + ...
It is fascinating to probe the close connections between both Functions.
As is well known with respect to the recognised Riemann Zeta Function (i.e. Zeta 1). (I am inserting 1 and 2 after the ζ sign to distinguish the Zeta 1 and Zeta 2 expressions respectively).
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 + .......
Now each of these terms can be directly related to the Zeta 2 Function.
In terms of this latter Function (using the infinite expression),
ζ2(s) = 1 + s^1 + s^2 + s^3 + s^4 + ...
Now when s = 1/2 with respect to this series,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2
Therefore, s^1 + s^2 + s^3 + s^4 + ... = ζ2(s)- 1 = 1
Then when s = 1/3
ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 +... = 3/2
Therefore, s^1 + s^2 + s^3 + s^4 + ... = 3/2 - 1 = 1/2
In general s^1 + s^2 + s^3 + s^4 + ... = 1(1 - s)
So ζ2(1/4) = 1/3, ζ2(1/5)= 1/4 and so on!
Again,
ζ(1)1 = 1 + 1/2 + 1/3 + 1/4 + .......
= ∑(s^1 + s^2 + s^3 + s^4 + ... ) where the value of s ranges over the reciprocals of all natural numbers except 1).
and s^1 + s^2 + s^3 + s^4 + ... = ζ2(s) - 1.
Now with respect to the Zeta 1,
ζ1(k) = (1)^k + (1/2)^k + (1/3)^k + (1/4)^k + ...
So for example when k = 2.
ζ1(2) = (1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ...
= 1 + 1/4 + 1/9 + 1/16 + ....
Therefore in more general terms.
ζ1(k) = ∑(s^1k + s^2k + s^3k + s^4k + ... ),
where s^1k + s^2k + s^3k + s^4k + ... = {ζ2(s)^k} - 1
This relationship holds where the Zeta 1 ranges over real positive dimensional values (≥ 1), and the Zeta 2 over real fractional values for s < 1/2). So each term in the Zeta 1, is directly related to an entire Zeta 2 expression. I have mentioned before how each cardinal number is related (ordinally) to its natural number members. So we have an equivalent relationship here where the Zeta 1 expression is composed of a number of Zeta 2 members. Because of notational difficulties when writing on a web-page, I have produced some accompanying material in text! See Two Zeta Functions (Section 1).
The first relates to the recognised Riemann Zeta Function (Zeta 1) which is defined as an infinite series:
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….
Now if with respect to a number raised to a power we refer to the initial number as the base quantity and the exponent or power as the dimensional number,
for the Zeta 1 Function the natural numbers serve as the base quantities and s as the dimensional power - negative in this case - to which the base quantities are raised.
The Zeta 2 Function - the role of which is largely unrecognised - is defined as a finite series:
1 + s^1 + s^2 + ….+ s^(n – 1).
Here the role of base quantities and dimensional powers is reversed with s now serving as the base quantity which is defined with respect to the natural numbers (from 1 to n) where n has no finite limit.
In a qualified sense, the Zeta 2 can be defined in infinite terms as:
1 + s^1 + s^2 + s^3 + s^4 + ...
It is fascinating to probe the close connections between both Functions.
As is well known with respect to the recognised Riemann Zeta Function (i.e. Zeta 1). (I am inserting 1 and 2 after the ζ sign to distinguish the Zeta 1 and Zeta 2 expressions respectively).
ζ1(1) = 1 + 1/2 + 1/3 + 1/4 + .......
Now each of these terms can be directly related to the Zeta 2 Function.
In terms of this latter Function (using the infinite expression),
ζ2(s) = 1 + s^1 + s^2 + s^3 + s^4 + ...
Now when s = 1/2 with respect to this series,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2
Therefore, s^1 + s^2 + s^3 + s^4 + ... = ζ2(s)- 1 = 1
Then when s = 1/3
ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 +... = 3/2
Therefore, s^1 + s^2 + s^3 + s^4 + ... = 3/2 - 1 = 1/2
In general s^1 + s^2 + s^3 + s^4 + ... = 1(1 - s)
So ζ2(1/4) = 1/3, ζ2(1/5)= 1/4 and so on!
Again,
ζ(1)1 = 1 + 1/2 + 1/3 + 1/4 + .......
= ∑(s^1 + s^2 + s^3 + s^4 + ... ) where the value of s ranges over the reciprocals of all natural numbers except 1).
and s^1 + s^2 + s^3 + s^4 + ... = ζ2(s) - 1.
Now with respect to the Zeta 1,
ζ1(k) = (1)^k + (1/2)^k + (1/3)^k + (1/4)^k + ...
So for example when k = 2.
ζ1(2) = (1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ...
= 1 + 1/4 + 1/9 + 1/16 + ....
Therefore in more general terms.
ζ1(k) = ∑(s^1k + s^2k + s^3k + s^4k + ... ),
where s^1k + s^2k + s^3k + s^4k + ... = {ζ2(s)^k} - 1
This relationship holds where the Zeta 1 ranges over real positive dimensional values (≥ 1), and the Zeta 2 over real fractional values for s < 1/2). So each term in the Zeta 1, is directly related to an entire Zeta 2 expression. I have mentioned before how each cardinal number is related (ordinally) to its natural number members. So we have an equivalent relationship here where the Zeta 1 expression is composed of a number of Zeta 2 members. Because of notational difficulties when writing on a web-page, I have produced some accompanying material in text! See Two Zeta Functions (Section 1).
Saturday, March 23, 2013
The Shadow of Mathematics
I have already suggested that one revealing way of appreciating the nature of the non-trivial zeros (both for the Zeta 1 and Zeta 2 Functions) is as the shadow to the conventional natural system of number (in both cardinal and ordinal terms).
In Jungian psychological terms this would thereby imply that these two sets of non-trivial zeros represent the unconscious complement of what we conventionally understand as number (in conscious terms).
Likewise in physical terms it equally implies that the two sets of non-trivial zeros represent the holistic complement of what is conventionally interpreted in analytic terms with respect to the natural world.
Again from a Jungian perspective, true psychological integration requires the combined integration of both the (revealed) conscious and (hidden) unconscious aspects of personality.
Complementarity exists in the relationship between the natural number system (cardinal and ordinal) and the two sets of non-trivial zeros.
So from the conscious extreme the natural counting numbers seem fully accessible to understanding (even from a very early age).
However from the corresponding unconscious extreme, the two sets of non-trivial zeros – certainly in terms of what they inherently represent – seem almost entirely inaccessible to conventional understanding (at any age of development).
So what appears as completely obvious in terms of (revealed) conscious appears completely inaccessible (in terms of (hidden) unconscious behaviour. So identification with the conscious thereby blots out entire recognition with respect to the opposite unconscious extreme (relating to the hidden nature of the number system).
This thereby entails that ultimately true psychological integration requires appropriating this enormous shadow with respect to the nature of both sets of non-trivial zeros, so that eventually they can become fully accessible in intuitive terms to understanding.
The deeper significance is that Conventional Mathematics thereby is greatly lacking a true integral dimension with respect to overall understanding.
Once again this is due to a total failure in formal terms to recognise its hidden unconscious dimension.
Alternatively it represents the complete domination (in formal terms) of quantitative over qualitative type interpretation of mathematical symbols.
Expressed in yet another way it represents sole recognition of the analytic dimension of mathematical interpretation through the exclusion of its equally important holistic aspect.
Yet again it entails the remarkable fact that Mathematics as it currently stands is entirely lacking in any genuine notion of the nature of interdependent type relationships.
And the saddest part – though his is easily appreciated in Jungian terms – is that because of its extreme one-sided emphasis, it thereby is greatly lacking any insight into the nature of its own considerable shadow.
Thus what are understood as rigorous procedures (within its limited terms of reference) can equally be seen from an outside perspective as representing highly reduced understanding at every turn.
And by definition when one accepts such reductionism – as a recognised member of the mathematical profession - without question, one thereby becomes unable to tolerate or even appreciate any outside criticism of its inherent nature.
Though the Riemann Hypothesis is not necessary to identify the fundamental problem with present mathematical interpretation, it does however – when appropriately interpreted – provide a wonderful illustration of the precise nature of this problem.
So let me once more state it simply! Properly understood Mathematics is inherently dynamic in nature representing the two-way relative interaction of aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
However Conventional Mathematics is based on the special case where such interaction is effectively completely ignored. So the qualitative – though of equal importance - is thereby reduced to the quantitative.
Mathematics is then misleadingly viewed as relating merely to quantitative relationships in absolute terms.
The Riemann Hypothesis relates intrinsically to the fundamental nature of the number system and cannot thereby be properly understood – not alone solved – in conventional mathematical terms.
It thus reduces and thereby gravely distorts its inherent dynamic nature as the interaction of twin elements that are quantitative and qualitative (and equally physical and psychological) with respect to each other.
So number inherently represents the fundamental encoding of all physical and psychological phenomena in quantitative and qualitative terms.
Conventional Mathematics is thereby gravely in need of addressing its quite enormous shadow.
In physical terms this requires the recognition that all mathematical processes entail a qualitative holistic dimension that cannot be reduced in mere quantitative terms.
In complementary psychological terms it implies that that the understanding of such processes entails an unconscious (intuitive) aspect that cannot be reduced in a rational (linear) manner!
As I have repeatedly stated, because of the very nature of dynamic interaction, all interpretation that is directly analytic (i.e. quantitative) has a corresponding shadow – initially unrecognised - that is indirectly holistic and qualitative in nature.
Likewise all interpretation that is directly holistic (i.e. qualitative) has a corresponding unrecognised shadow that is indirectly of an analytic quantitative nature.
When we initially look at the number system it appears directly in a linear rational manner revealing its quantitative aspect. So here we literally view all numbers in quantitative terms as lying on a straight line!
However this number system has a corresponding shadow that is indirectly holistic and qualitative in nature.
Bringing this aspect to light depends much more on intuitive (unconscious) development rather than on linear rational understanding.
In fact indirectly it can be then given a paradoxical rational interpretation in a circular manner.
I have frequently over the past year or so in my blogs drawn direct attention to the nature of this holistic aspect identifying firstly the Type 2 aspect (of the number system) and then the corresponding Zeta 2 Function.
Not surprisingly, as Conventional Mathematics totally lacks in formal terms a holistic dimension, no recognition exists as yet of the enormous importance of what – I refer to as – as the Zeta 2 non-trivial zeros. These in fact are fully complementary with the Type 1 zeros.
The Type 2 non-trivial zeros essentially relate to the holistic nature of numbers (as dimensions).
As understanding becomes increasingly dynamic the nature of higher dimensions (the structure of which is related to the corresponding roots of the number in question) unfold in understanding.
Ultimately however as holistic appreciation deepens, remaining rigid elements of a linear rational level gradually are completely dissolved.
In psycho-spiritual terms this is identified with the continual deepening of a contemplative (intuitive) state that ultimately transcends all analytic interpretation (of a rigid linear kind). Increasingly however, dynamic rational structures of a circular (paradoxical) kind are required to mediate such understanding.
This represents the unconscious – as opposed to the conscious – extreme of interpretation.
So we start in development with the understanding of number with the direct linear rational understanding of number (i.e. its quantitative aspect).
Then when development proceeds to higher stages we gradually can uncover the shadow of this aspect of number (i.e. its initially unrecognised holistic qualitative dimension).
The full unfolding of this alternative holistic aspect leads to an increasingly dynamic appreciating of number that is directly intuitive but indirectly is conveyed in a refined circular rational fashion.
This understanding ultimately culminates in a spiritually contemplative appreciation of the transcendent nature of reality (i.e. the purely holistic appreciation of the infinite nature of the number system).
And the Type 2 zeros relate directly to this circular holistic aspect!
However the holistic also has a shadow aspect (in its unrecognised analytic aspect).
What this entails from a psycho spiritual perspective is that the pure holistic appreciation of the infinite nature of reality (as transcending all finite notions) must now be made immanent in all finite phenomena.
This means in effect that one now gradually learns to integrate the new qualitative aspect of appreciation with all scientific type phenomena (formerly understood in mere quantitative terms).
Let me briefly illustrate. From a conventional scientific perspective (i.e. quantitative) the recognition of object phenomena e.g. roses is of a reduced collective nature where we can impersonally classify all individual members as belonging to the same general class of rose!
Now the corresponding qualitative recognition requires that each rose be understood uniquely with respect to its distinctive attributes.
Remarkably it is the same with the number system.
The quantitative recognition of number is impersonal in nature where the qualitative distinction as between numbers is not taken into account.
Indeed the failure to appreciate the true nature of multiplication in Conventional Mathematics relates to this lack of qualitative distinction.
So 2^2 (represents a number expressed with respect to the second dimension). However the quantitative result = 4 (i.e. 4^1) represents a number expressed with respect to the (default) 1st dimension.
Thus the qualitative change in the nature of the number (which always results through multiplication) is thereby ignored. In other words the qualitative aspect of interpretation is reduced to the quantitative!
The very point about the Riemann Hypothesis is that it relates to this key problem of reconciling the quantitative with the qualitative nature of number.
The non-trivial zeros (i.e. Type 1) therefore relate to the holistic aspect of the number system now interpreted in an immanent manner, whereby the qualitative aspect can be properly incorporated with each finite member of the number system.
Put another way just as the conventional linear number system represents the interpretation of number with respect to its quantitative aspect, the non-trivial zeros (Type 1) represent the corresponding interpretation with respect to the qualitative aspect.
Now the conventional system (esp. primes and natural nos.) are understood as points on a real line. In qualitative terms “real” is synonymous with rational interpretation of a quantitative kind!
From a holistic perspective, “imaginary” relates to the indirect rational way of conveying meaning that is of an unconscious intuitive nature. So the recognition of “qualities” as associated with consciously recognised objects, always comes directly from the unconscious mind (in its interaction with the conscious).
In this sense such qualitative appreciation is of an imaginary nature. So whereas the quantitative number system is of a real, the corresponding qualitative appreciation of its nature is imaginary! So therefore the (Type 1) non-trivial zeros lie on a straight line that is imaginary.
If the unconscious remains undeveloped we thereby cannot properly identify true quality in any context!
Unfortunately this is a damning criticism that can be made of Conventional Mathematics (which fails to recognise its qualitative dimension)!
This therefore poses considerable difficulties in appreciating the nature of the non-trivial zeros (Type 1) which relate to the qualitative aspect of number (with respect to their finite nature). By contrast, the Type 2 non-trivial zeros relate to the qualitative aspect of number (with respect to their infinite nature).
Now I have already stated that all phenomena fundamentally represent the original dynamic interaction of the number system with respect to its quantitative and qualitative characteristics.
What is truly remarkable is that though remaining inaccessible to conventional understanding, the non-trivial zeros (Type 1 and Type 2) must necessarily implicitly exist in the unconscious mind of all human beings as the very means through which we are enabled to make qualitative distinctions with respect to objects.
What is even more remarkable is that these same zeros must innately exist with respect to all natural phenomena providing their very capacity to become manifest in nature (exhibiting both quantitative and qualitative characteristics).
In an important sense one key goal of evolution requires fully uncovering in understanding these hidden – though currently largely inaccessible - original aspects of our number system!
Indeed the very nature of both types of zeros is pointing clearly to the present inadequate nature of Conventional Mathematics.
Mathematics is not strictly about the quantitative but rather the dynamic interaction as between its quantitative and qualitative aspects. Therefore to properly understand this interaction we must face its persistent shadow in a continual failure to recognise the qualitative dimension.
In Jungian psychological terms this would thereby imply that these two sets of non-trivial zeros represent the unconscious complement of what we conventionally understand as number (in conscious terms).
Likewise in physical terms it equally implies that the two sets of non-trivial zeros represent the holistic complement of what is conventionally interpreted in analytic terms with respect to the natural world.
Again from a Jungian perspective, true psychological integration requires the combined integration of both the (revealed) conscious and (hidden) unconscious aspects of personality.
Complementarity exists in the relationship between the natural number system (cardinal and ordinal) and the two sets of non-trivial zeros.
So from the conscious extreme the natural counting numbers seem fully accessible to understanding (even from a very early age).
However from the corresponding unconscious extreme, the two sets of non-trivial zeros – certainly in terms of what they inherently represent – seem almost entirely inaccessible to conventional understanding (at any age of development).
So what appears as completely obvious in terms of (revealed) conscious appears completely inaccessible (in terms of (hidden) unconscious behaviour. So identification with the conscious thereby blots out entire recognition with respect to the opposite unconscious extreme (relating to the hidden nature of the number system).
This thereby entails that ultimately true psychological integration requires appropriating this enormous shadow with respect to the nature of both sets of non-trivial zeros, so that eventually they can become fully accessible in intuitive terms to understanding.
The deeper significance is that Conventional Mathematics thereby is greatly lacking a true integral dimension with respect to overall understanding.
Once again this is due to a total failure in formal terms to recognise its hidden unconscious dimension.
Alternatively it represents the complete domination (in formal terms) of quantitative over qualitative type interpretation of mathematical symbols.
Expressed in yet another way it represents sole recognition of the analytic dimension of mathematical interpretation through the exclusion of its equally important holistic aspect.
Yet again it entails the remarkable fact that Mathematics as it currently stands is entirely lacking in any genuine notion of the nature of interdependent type relationships.
And the saddest part – though his is easily appreciated in Jungian terms – is that because of its extreme one-sided emphasis, it thereby is greatly lacking any insight into the nature of its own considerable shadow.
Thus what are understood as rigorous procedures (within its limited terms of reference) can equally be seen from an outside perspective as representing highly reduced understanding at every turn.
And by definition when one accepts such reductionism – as a recognised member of the mathematical profession - without question, one thereby becomes unable to tolerate or even appreciate any outside criticism of its inherent nature.
Though the Riemann Hypothesis is not necessary to identify the fundamental problem with present mathematical interpretation, it does however – when appropriately interpreted – provide a wonderful illustration of the precise nature of this problem.
So let me once more state it simply! Properly understood Mathematics is inherently dynamic in nature representing the two-way relative interaction of aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.
However Conventional Mathematics is based on the special case where such interaction is effectively completely ignored. So the qualitative – though of equal importance - is thereby reduced to the quantitative.
Mathematics is then misleadingly viewed as relating merely to quantitative relationships in absolute terms.
The Riemann Hypothesis relates intrinsically to the fundamental nature of the number system and cannot thereby be properly understood – not alone solved – in conventional mathematical terms.
It thus reduces and thereby gravely distorts its inherent dynamic nature as the interaction of twin elements that are quantitative and qualitative (and equally physical and psychological) with respect to each other.
So number inherently represents the fundamental encoding of all physical and psychological phenomena in quantitative and qualitative terms.
Conventional Mathematics is thereby gravely in need of addressing its quite enormous shadow.
In physical terms this requires the recognition that all mathematical processes entail a qualitative holistic dimension that cannot be reduced in mere quantitative terms.
In complementary psychological terms it implies that that the understanding of such processes entails an unconscious (intuitive) aspect that cannot be reduced in a rational (linear) manner!
As I have repeatedly stated, because of the very nature of dynamic interaction, all interpretation that is directly analytic (i.e. quantitative) has a corresponding shadow – initially unrecognised - that is indirectly holistic and qualitative in nature.
Likewise all interpretation that is directly holistic (i.e. qualitative) has a corresponding unrecognised shadow that is indirectly of an analytic quantitative nature.
When we initially look at the number system it appears directly in a linear rational manner revealing its quantitative aspect. So here we literally view all numbers in quantitative terms as lying on a straight line!
However this number system has a corresponding shadow that is indirectly holistic and qualitative in nature.
Bringing this aspect to light depends much more on intuitive (unconscious) development rather than on linear rational understanding.
In fact indirectly it can be then given a paradoxical rational interpretation in a circular manner.
I have frequently over the past year or so in my blogs drawn direct attention to the nature of this holistic aspect identifying firstly the Type 2 aspect (of the number system) and then the corresponding Zeta 2 Function.
Not surprisingly, as Conventional Mathematics totally lacks in formal terms a holistic dimension, no recognition exists as yet of the enormous importance of what – I refer to as – as the Zeta 2 non-trivial zeros. These in fact are fully complementary with the Type 1 zeros.
The Type 2 non-trivial zeros essentially relate to the holistic nature of numbers (as dimensions).
As understanding becomes increasingly dynamic the nature of higher dimensions (the structure of which is related to the corresponding roots of the number in question) unfold in understanding.
Ultimately however as holistic appreciation deepens, remaining rigid elements of a linear rational level gradually are completely dissolved.
In psycho-spiritual terms this is identified with the continual deepening of a contemplative (intuitive) state that ultimately transcends all analytic interpretation (of a rigid linear kind). Increasingly however, dynamic rational structures of a circular (paradoxical) kind are required to mediate such understanding.
This represents the unconscious – as opposed to the conscious – extreme of interpretation.
So we start in development with the understanding of number with the direct linear rational understanding of number (i.e. its quantitative aspect).
Then when development proceeds to higher stages we gradually can uncover the shadow of this aspect of number (i.e. its initially unrecognised holistic qualitative dimension).
The full unfolding of this alternative holistic aspect leads to an increasingly dynamic appreciating of number that is directly intuitive but indirectly is conveyed in a refined circular rational fashion.
This understanding ultimately culminates in a spiritually contemplative appreciation of the transcendent nature of reality (i.e. the purely holistic appreciation of the infinite nature of the number system).
And the Type 2 zeros relate directly to this circular holistic aspect!
However the holistic also has a shadow aspect (in its unrecognised analytic aspect).
What this entails from a psycho spiritual perspective is that the pure holistic appreciation of the infinite nature of reality (as transcending all finite notions) must now be made immanent in all finite phenomena.
This means in effect that one now gradually learns to integrate the new qualitative aspect of appreciation with all scientific type phenomena (formerly understood in mere quantitative terms).
Let me briefly illustrate. From a conventional scientific perspective (i.e. quantitative) the recognition of object phenomena e.g. roses is of a reduced collective nature where we can impersonally classify all individual members as belonging to the same general class of rose!
Now the corresponding qualitative recognition requires that each rose be understood uniquely with respect to its distinctive attributes.
Remarkably it is the same with the number system.
The quantitative recognition of number is impersonal in nature where the qualitative distinction as between numbers is not taken into account.
Indeed the failure to appreciate the true nature of multiplication in Conventional Mathematics relates to this lack of qualitative distinction.
So 2^2 (represents a number expressed with respect to the second dimension). However the quantitative result = 4 (i.e. 4^1) represents a number expressed with respect to the (default) 1st dimension.
Thus the qualitative change in the nature of the number (which always results through multiplication) is thereby ignored. In other words the qualitative aspect of interpretation is reduced to the quantitative!
The very point about the Riemann Hypothesis is that it relates to this key problem of reconciling the quantitative with the qualitative nature of number.
The non-trivial zeros (i.e. Type 1) therefore relate to the holistic aspect of the number system now interpreted in an immanent manner, whereby the qualitative aspect can be properly incorporated with each finite member of the number system.
Put another way just as the conventional linear number system represents the interpretation of number with respect to its quantitative aspect, the non-trivial zeros (Type 1) represent the corresponding interpretation with respect to the qualitative aspect.
Now the conventional system (esp. primes and natural nos.) are understood as points on a real line. In qualitative terms “real” is synonymous with rational interpretation of a quantitative kind!
From a holistic perspective, “imaginary” relates to the indirect rational way of conveying meaning that is of an unconscious intuitive nature. So the recognition of “qualities” as associated with consciously recognised objects, always comes directly from the unconscious mind (in its interaction with the conscious).
In this sense such qualitative appreciation is of an imaginary nature. So whereas the quantitative number system is of a real, the corresponding qualitative appreciation of its nature is imaginary! So therefore the (Type 1) non-trivial zeros lie on a straight line that is imaginary.
If the unconscious remains undeveloped we thereby cannot properly identify true quality in any context!
Unfortunately this is a damning criticism that can be made of Conventional Mathematics (which fails to recognise its qualitative dimension)!
This therefore poses considerable difficulties in appreciating the nature of the non-trivial zeros (Type 1) which relate to the qualitative aspect of number (with respect to their finite nature). By contrast, the Type 2 non-trivial zeros relate to the qualitative aspect of number (with respect to their infinite nature).
Now I have already stated that all phenomena fundamentally represent the original dynamic interaction of the number system with respect to its quantitative and qualitative characteristics.
What is truly remarkable is that though remaining inaccessible to conventional understanding, the non-trivial zeros (Type 1 and Type 2) must necessarily implicitly exist in the unconscious mind of all human beings as the very means through which we are enabled to make qualitative distinctions with respect to objects.
What is even more remarkable is that these same zeros must innately exist with respect to all natural phenomena providing their very capacity to become manifest in nature (exhibiting both quantitative and qualitative characteristics).
In an important sense one key goal of evolution requires fully uncovering in understanding these hidden – though currently largely inaccessible - original aspects of our number system!
Indeed the very nature of both types of zeros is pointing clearly to the present inadequate nature of Conventional Mathematics.
Mathematics is not strictly about the quantitative but rather the dynamic interaction as between its quantitative and qualitative aspects. Therefore to properly understand this interaction we must face its persistent shadow in a continual failure to recognise the qualitative dimension.
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