It is now recognised that the non-trivial zeros (i.e. the Type 1 zeros) in all probability have a spectral interpretation relating to the frequencies of some dynamic system of a physical nature.
Now this comes as no surprise to me at all, as I would have long suspected that this was necessarily the case. Indeed I would go considerably further in emphasising that this spectral interpretation relates to a dynamic system with twin physical and psychological aspects that are complementary.
However my own interest in the Riemann Hypothesis initially arose from a prolonged interest in the nature of the (unrecognised) non-trivial zeros (Type 2).
And in this context I had already made use of a spectral interpretation as just one illuminating pathway towards appreciating their true nature.
Now the electromagnetic spectrum is perhaps the best-known physical spectrum.
So natural light comprises just one narrow band on this overall spectrum. Below this band lies electromagnetic energy of higher frequency and shorter wavelength than light (e.g. x-rays and gamma rays). Above this band lie corresponding energies of lower frequency and longer wavelength (e.g. microwave and radio).
Because in dynamic terms of complementarity, a psychological aspect necessarily exists to such a spectrum with fascinating implications for the spiritual contemplative journey.
The term “light” of course is equally used in a spiritual as well as physical sense where it refers to the holistic aspect of intuition or illumination which necessarily interacts with (analytic) reason.
Just as natural light inhabits the middle band of the physical spectrum likewise the intuition that characterises conventional (linear) reason likewise inhabits the middle of the psychological spectrum.
However it has been long known that through authentic contemplative development that higher bands on this spectrum are attainable where the intuitive light becomes ever more refined and thereby capable of interacting with paradoxical (i.e. circular) rational structures.
Indeed ultimately these structures become so unrestricted and transparent that they no longer appear to possess any remaining rigid structure becoming fully integrated with the very light through which they are mediated.
Now associated with the deepest level of contemplation is a quality of peace and utter stillness. This illumination can therefore be readily characterised as (intuitive) light of exceptionally long wavelength (and corresponding low frequency).
By contrast in the spiritual literature the other extreme (of short wavelength and high frequency) is often related to periods of purgation where an incessant inner bombardment takes place exposing one’s hidden faults and failings!
One sobering point however to consider is this!
Just as energies outside the natural light band are not visible to the naked eye, likewise all these energies (and the understanding associated with them) cannot be appropriated through linear reason (informed by common intuition).
This therefore creates insuperable difficulties for Conventional Mathematics in trying to appropriate such understanding within its accepted axioms and definitions!
Now again in the spiritual esoteric traditions, these higher bands of understanding are often referred to as higher “dimensions”.
Now the key here is to recognise that number is used to represent dimensions in Mathematics.
So therefore associated with each number – now understood in a qualitative holistic sense – is a unique dynamic form of understanding where a more refined quality of intuition interacts with the ever more circular (paradoxical) use of reason!
I have often mentioned in this context the fact that Conventional Mathematics is based on linear reason i.e. literally in a 1-dimensional manner.
Therefore though the quantitative expression of other dimensional numbers (as powers or exponents) is indeed accommodated, this all takes place within a default qualitative interpretation (= 1).
And the very nature of such interpretation is that distinctive holistic meaning (in any relevant context) is necessarily reduced in a mere quantitative manner.
Using the language of string theory, we could say therefore that Conventional Mathematics represents the lowest possible energy state (with respect to its understanding) where entities such as numbers thereby assume a (misleading) absolute identity.
More accurately we could say that Conventional Mathematics is exclusively defined in 1-dimensional terms (where 1 is defined in a qualitative holistic manner).
However the remarkable truth that needs to be embraced is that mathematical relationships can be validly defined in an unlimited number of possible ways (where each number as dimension represents a unique overall manner of interpretation).
Now in all cases (except 1) these dimensional interpretations imply unique configurations with respect to the dynamic interaction of both the analytic (particle) and holistic (wave) aspects of mathematical understanding.
So in making this very point, I am illustrating how the well-known fact that the Zeta Function remains uniquely undefined on the complex plane for s = 1, can be given both a quantitative (analytic) and qualitative (holistic) interpretation.
If you can appreciate this then you can understand immediately why the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted from within the standpoint of Conventional Mathematics.
As – when correctly interpreted - it fundamental relates to the dynamic interaction of the quantitative (analytic) and qualitative (holistic) aspects of the number system, this remains uniquely undefined in linear (1-dimensional) terms where the qualitative aspect formally is not recognised.
And of course from this enlarged perspective, as the Riemann Hypothesis relates to the fundamental condition for the ultimate identity of both quantitative (analytic) and qualitative (holistic) aspects, it is futile trying to prove it through the axioms of Conventional Mathematics (which give no formal recognition to the qualitative aspect).
From what we have said it is but a short journey to appreciating that the number system itself represents the ultimate spectrum (in both physical and psychological terms).
So the vibrations of all dynamic systems (physical and psychological) can thereby be ultimately understood as the original vibrations of this number system with respect to both its quantitative and qualitative aspects.
Therefore all natural phenomena thereby represent but a secondary expression of an original system where number - when dynamically interpreted with respect to both analytic and holistic aspects – inherently exists as their deepest possible means of encoding.
Once again the recognised non-trivial zeros (of the Zeta 1 Function) are - when properly understood - fully complementary with the Zeta 2 zeros (which I have been explaining).
Whereas the Zeta 2 can be identified initially with each natural number except 1 (that constitutes each prime) in ordinal terms, The Zeta 1 inversely can be identified initially overall with the entire group of prime numbers (that comprise the natural number system except 1) in cardinal terms.
The solutions from both perspectives establish the identity of the quantitative and qualitative aspects of the number system. Ultimately through simultaneous appreciation of Zeta 1 and Zeta 2, we can perfectly see (in Zeta 3 terms) the primes and natural numbers (and indeed both sets of non-trivial zeros) as perfect mirrors of each other with an ineffable identity.
Wednesday, March 20, 2013
Sunday, March 17, 2013
Two Complementary Zeta Functions (3)
In the latter part of my last blog entry I emphasised the inherent holistic nature of the Type 2 aspect of the number system. Ultimately this relates to the notion of number with respect to its qualitative - rather than quantitative - identity.
Alternatively, it relates to the understanding of number as interdependent (which conflicts strongly with the cultural dominance of numbers as representing solely independent entities).
The clue again with respect to this latter holistic emphasis is to view a number i.e. natural, with respect to the relationship as between its individual ordinal members.
So for example the analytic linear notion of 2 as cardinal relates to this number in static absolute terms as representing a collective whole (whose individuals units are necessarily of a homogenous - and thereby - indistinguishable nature).
However the corresponding holistic circular notion of 2 as ordinal relates to this same number, now understood in a dynamic relative fashion, as representing the interaction of its individual members (i.e. 1st and 2nd) which can be uniquely distinguished in qualitative terms.
Once again, this interaction entails two polar directions (i.e. 1st and 2nd dimensions which are inversely associated with the 1st and 2nd roots of 1) that are positive and negative with respect to each other represented as + 1 and – 1 respectively.
So the combined fusion of both as interdependent is represented as (+) 1 – 1 = 0.
Therefore from this holistic perspective every number can be given a qualitative meaning as a pure energy state (= 0 from a quantitative perspective). Of course in actual experience, this holistic energy state interacts with the customary rational understanding of number (as discrete form).
Then in formal interpretation of a conventional kind, the holistic aspect (which is of a uniquely distinct nature) is reduced in a mere rational fashion! We are thereby conditioned to view numbers as representing distinct quantitative entities with no recognition therefore of their corresponding holistic qualitative aspect as energy states (in both physical and psychological terms).
Thus, to emphasise once more, the holistic notion of any number relates to the dynamic interaction of its individual ordinal members (which are uniquely defined for the number).
Through reference to the corresponding roots of 1 for each number, this does indeed enable one to demonstrate the holistic aspect of all individual numbers. However, in itself it does not directly lend itself to quantitative type analysis, as by definition the sum of all these roots (demonstrating the qualitative relationship aspect of ordinal members) = 0.
However just as the Type 1 (analytic) aspect indirectly can be given a holistic interpretation (with respect to the nature of Zeta 1 trivial zeros) equally the Type 2 (holistic) aspect indirectly can be given an analytic expression (with respect to the Zeta 2 trivial zeros)!
Once again a fascinating form of complementarity is at work. The Zeta 1 non-trivial zeros are expressed in quantitative terms in a linear (1-dimensional) manner. The key to the holistic understanding is then to provide a “higher” dimensional interpretation for these zeros in a qualitative manner.
In reverse fashion, the Zeta 2 non-trivial zeros are expressed - representing their qualitative meaning - directly in a circular (higher dimensional) manner.
So again for example, I have been at pains to express the precise qualitative meaning of 2 as a dimensional number (i.e. with respect to its 1st and 2nd ordinal members).
Therefore the key here is to represent higher dimensional roots in a reduced linear (1-dimensional) fashion from a quantitative perspective.
In effect this implies treating negative signs as positive and imaginary numbers as real!
So when we convert the various roots of 1 in this quantitative manner, a fascinating new set of analytic type relationships emerge, leading both to an alternative Type 2 Prime Number Theorem and Riemann Hypothesis respectively.
I have expressed the nature of this alternative formulation on many occasions previously in my blog entries.
Once again we initially treat both the cos and sin parts of the roots of 1 (now expressed in this reduced linear quantitative fashion) obtaining the average for each prime number grouping (representing all of its ordinal number members)
So to illustrate where p = 3 the three roots (1st, 2nd and 3rd) are 1, .5 + .866 and .5 + .866.
So the sum of the 3 cos parts = 2 and three sin parts (the first of which is zero) = 1.732.
The average of the cos parts therefore = .666.. and the sin parts = .5773.
Now when we obtain the average of cos and sin parts for higher prime numbered root,s the answer quickly converges for both parts (with a small remaining deviation) on the value 2/π = .6366….
The alternative Prime Number Theorem relates to the fact that 2/π = i/log i.
So i/log i with respect to the Type 2 aspect (as the mean reduced average for both parts of the ordinal number roots of p, as p increases without limit) plays a complementary role to that which n/log n plays with respect to the Type 1 (representing the average number of primes among the natural numbers as n increases without limit).
Also it quickly becomes apparent that the average for the cos part always exceeds 2/π (= i/log i) while the corresponding sin part always is less than 2/π (= i/log i).
Indeed the absolute value of the ratio of the deviation of the cos part to the deviation of the sin part from 2/π (= i/log i) quickly converges towards .5 (which operates as the alternative Riemann Hypothesis with respect to the Type 2 aspect).
Even here further complementarity is in evidence: From the Type 1 perspective, n/log n is an approximate varying quantity, whereas .5 according to the Riemann Hypothesis is the fixed point on the real axis through which the imaginary line on which all the non-trivial zeros lie; however from the Type 2 perspective i/log i is a fixed value with .5 as the absolute ratio of cos and sin deviations by contrast an approximate varying quantity.
Now here is the important point!
As we saw in the last blog entry, in the Zeta 1 function, both the prime numbers (with respect to the natural numbers) and trivial zeros each possess extreme analytic properties (representing independence) and extreme holistic properties (representing interdependence) respectively depending on the frame of reference from which they are viewed.
So from one perspective we can use the non-trivial zeros to eliminate all remaining (independent) deviations of the prime numbers with respect to their overall collective (interdependent) relationship with the real natural numbers.
Equally from the other perspective we can use the prime numbers (as a group) to eliminate all (independent) deviations of non-trivial zeros with respect to their overall collective (interdependent) relationship with the imaginary number system.
Again a complementary type connection exists with respect to the Zeta 2 function. Here the two aspects (independence and interdependence respectively) are reconciled within the same distribution
I have already illustrated how the prime numbers (and by extension all the natural numbers) can be used to explain to a very high degree of accuracy the deviations of the average of both cos and sin parts from 2/π (= i/log i).
However what is perhaps even more interesting is that these deviations can in principle be used in such a manner that the frequency of primes within the natural numbers can be approximated to any degree of accuracy.
It would work something like this. In the normal general calculation of prime number frequency (among the primes) each natural number is given a weighting of 1.
However using the (reduced) Zeta 2 deviations a slightly modified weighting would be given with earliest natural numbers differing most. So for example when predicting the frequency of primes relating to the first 100 natural numbers (i.e. n = 100) we would substitute a replacement number based on the total representing the sum of each slightly modified weighted natural number (up to 100) which could then correctly predict from the general formula the actual frequency associated with 100!
So whereas with the Zeta 1 function, the role of deviations associated with the non-trivial zeros must be understood in the context of the number system as a whole, here in the Zeta 2 approach we would have a one-to one relationship of a new weighted number (differing slightly from 1) with each natural number (based on the default weighting of 1).
Once again the greatest deviations would occur - with respect to these newly weighted members - amongst the earliest natural numbers. Then as we ascend up the natural number scale the weightings would approach ever closer to the default natural number weightings of 1.
Alternatively, it relates to the understanding of number as interdependent (which conflicts strongly with the cultural dominance of numbers as representing solely independent entities).
The clue again with respect to this latter holistic emphasis is to view a number i.e. natural, with respect to the relationship as between its individual ordinal members.
So for example the analytic linear notion of 2 as cardinal relates to this number in static absolute terms as representing a collective whole (whose individuals units are necessarily of a homogenous - and thereby - indistinguishable nature).
However the corresponding holistic circular notion of 2 as ordinal relates to this same number, now understood in a dynamic relative fashion, as representing the interaction of its individual members (i.e. 1st and 2nd) which can be uniquely distinguished in qualitative terms.
Once again, this interaction entails two polar directions (i.e. 1st and 2nd dimensions which are inversely associated with the 1st and 2nd roots of 1) that are positive and negative with respect to each other represented as + 1 and – 1 respectively.
So the combined fusion of both as interdependent is represented as (+) 1 – 1 = 0.
Therefore from this holistic perspective every number can be given a qualitative meaning as a pure energy state (= 0 from a quantitative perspective). Of course in actual experience, this holistic energy state interacts with the customary rational understanding of number (as discrete form).
Then in formal interpretation of a conventional kind, the holistic aspect (which is of a uniquely distinct nature) is reduced in a mere rational fashion! We are thereby conditioned to view numbers as representing distinct quantitative entities with no recognition therefore of their corresponding holistic qualitative aspect as energy states (in both physical and psychological terms).
Thus, to emphasise once more, the holistic notion of any number relates to the dynamic interaction of its individual ordinal members (which are uniquely defined for the number).
Through reference to the corresponding roots of 1 for each number, this does indeed enable one to demonstrate the holistic aspect of all individual numbers. However, in itself it does not directly lend itself to quantitative type analysis, as by definition the sum of all these roots (demonstrating the qualitative relationship aspect of ordinal members) = 0.
However just as the Type 1 (analytic) aspect indirectly can be given a holistic interpretation (with respect to the nature of Zeta 1 trivial zeros) equally the Type 2 (holistic) aspect indirectly can be given an analytic expression (with respect to the Zeta 2 trivial zeros)!
Once again a fascinating form of complementarity is at work. The Zeta 1 non-trivial zeros are expressed in quantitative terms in a linear (1-dimensional) manner. The key to the holistic understanding is then to provide a “higher” dimensional interpretation for these zeros in a qualitative manner.
In reverse fashion, the Zeta 2 non-trivial zeros are expressed - representing their qualitative meaning - directly in a circular (higher dimensional) manner.
So again for example, I have been at pains to express the precise qualitative meaning of 2 as a dimensional number (i.e. with respect to its 1st and 2nd ordinal members).
Therefore the key here is to represent higher dimensional roots in a reduced linear (1-dimensional) fashion from a quantitative perspective.
In effect this implies treating negative signs as positive and imaginary numbers as real!
So when we convert the various roots of 1 in this quantitative manner, a fascinating new set of analytic type relationships emerge, leading both to an alternative Type 2 Prime Number Theorem and Riemann Hypothesis respectively.
I have expressed the nature of this alternative formulation on many occasions previously in my blog entries.
Once again we initially treat both the cos and sin parts of the roots of 1 (now expressed in this reduced linear quantitative fashion) obtaining the average for each prime number grouping (representing all of its ordinal number members)
So to illustrate where p = 3 the three roots (1st, 2nd and 3rd) are 1, .5 + .866 and .5 + .866.
So the sum of the 3 cos parts = 2 and three sin parts (the first of which is zero) = 1.732.
The average of the cos parts therefore = .666.. and the sin parts = .5773.
Now when we obtain the average of cos and sin parts for higher prime numbered root,s the answer quickly converges for both parts (with a small remaining deviation) on the value 2/π = .6366….
The alternative Prime Number Theorem relates to the fact that 2/π = i/log i.
So i/log i with respect to the Type 2 aspect (as the mean reduced average for both parts of the ordinal number roots of p, as p increases without limit) plays a complementary role to that which n/log n plays with respect to the Type 1 (representing the average number of primes among the natural numbers as n increases without limit).
Also it quickly becomes apparent that the average for the cos part always exceeds 2/π (= i/log i) while the corresponding sin part always is less than 2/π (= i/log i).
Indeed the absolute value of the ratio of the deviation of the cos part to the deviation of the sin part from 2/π (= i/log i) quickly converges towards .5 (which operates as the alternative Riemann Hypothesis with respect to the Type 2 aspect).
Even here further complementarity is in evidence: From the Type 1 perspective, n/log n is an approximate varying quantity, whereas .5 according to the Riemann Hypothesis is the fixed point on the real axis through which the imaginary line on which all the non-trivial zeros lie; however from the Type 2 perspective i/log i is a fixed value with .5 as the absolute ratio of cos and sin deviations by contrast an approximate varying quantity.
Now here is the important point!
As we saw in the last blog entry, in the Zeta 1 function, both the prime numbers (with respect to the natural numbers) and trivial zeros each possess extreme analytic properties (representing independence) and extreme holistic properties (representing interdependence) respectively depending on the frame of reference from which they are viewed.
So from one perspective we can use the non-trivial zeros to eliminate all remaining (independent) deviations of the prime numbers with respect to their overall collective (interdependent) relationship with the real natural numbers.
Equally from the other perspective we can use the prime numbers (as a group) to eliminate all (independent) deviations of non-trivial zeros with respect to their overall collective (interdependent) relationship with the imaginary number system.
Again a complementary type connection exists with respect to the Zeta 2 function. Here the two aspects (independence and interdependence respectively) are reconciled within the same distribution
I have already illustrated how the prime numbers (and by extension all the natural numbers) can be used to explain to a very high degree of accuracy the deviations of the average of both cos and sin parts from 2/π (= i/log i).
However what is perhaps even more interesting is that these deviations can in principle be used in such a manner that the frequency of primes within the natural numbers can be approximated to any degree of accuracy.
It would work something like this. In the normal general calculation of prime number frequency (among the primes) each natural number is given a weighting of 1.
However using the (reduced) Zeta 2 deviations a slightly modified weighting would be given with earliest natural numbers differing most. So for example when predicting the frequency of primes relating to the first 100 natural numbers (i.e. n = 100) we would substitute a replacement number based on the total representing the sum of each slightly modified weighted natural number (up to 100) which could then correctly predict from the general formula the actual frequency associated with 100!
So whereas with the Zeta 1 function, the role of deviations associated with the non-trivial zeros must be understood in the context of the number system as a whole, here in the Zeta 2 approach we would have a one-to one relationship of a new weighted number (differing slightly from 1) with each natural number (based on the default weighting of 1).
Once again the greatest deviations would occur - with respect to these newly weighted members - amongst the earliest natural numbers. Then as we ascend up the natural number scale the weightings would approach ever closer to the default natural number weightings of 1.
Thursday, March 14, 2013
Two Complementary Zeta Functions (2)
As we have seen, all mathematical symbols can be given two complementary interpretations (analytic and holistic) respectively.
This is then reflected in the number system which itself has Type 1 (analytic) and Type 2 (holistic) aspects which are complementary. Then indirectly the Type 1 aspect can be given a Type 2 interpretation while then Type 2 indirectly can be given a Type 1 formulation!
Then with respect to the famed complex zeta function, it too has too complementary aspects.
So corresponding to Type 1 we have the recognised Riemann Zeta Function (which I refer to as Zeta 1).
Then corresponding to Type 2 we have a - largely unrecognised – complementary function (which I refer to as Zeta 2). Strictly, though the Zeta 2 Function is in fact well known; its true holistic significance is not at all appreciated.
Once again the Zeta 1 is defined as the infinite series:
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….
And the non-trivial (pair) solutions of the form s = a + it and a – it (where a according to the Riemann Hypothesis = ½) occur when
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + …. = 0
The Zeta 2 is then in inverse complementary terms (i.e. where the natural numbers now represent the powers and s the base quantities) as the finite series:
1 + s^1 + s^2 + ….+ s^(n – 1).
And the non-trivial solutions of the form s = a + it arise from the solutions of the equation,
1 + s^1 + s^2 + ….+ s^(n – 1) = 0
Now in a qualified sense we can equally define this second function in an infinite manner!
So if we take terms in in a strict cyclical order in accordance with the value of n, then the infinite equation will hold.
Thus for example in the simplest case where n = 2, the non-trivial 2nd root as solution to
1 + s^1 = 0 gives s = – 1.
So when we take terms two at a time the value of the infinite series
1 + s^1 + s^2 + s^3 + ….. = 0.
Now in this blog entry I will try to highlight some of the key complementary relations existing between both series and their implications.
Indeed when one begins to appreciate the very nature of the number system from this dynamic complementary perspective it leads to a completely new type of insight that itself directly fosters appreciation of its holistic nature.
Customarily we tend to view the number system in (real) linear terms with the prime numbers viewed as its independent building blocks! So using the terminology that I customarily employ this reflects an analytic (quantitative) interpretation.
Then parallel with this finding we find that the system of non-trivial zeros relating to the Zeta 1 in turn - assuming the truth of the Riemann Hypothesis – represents a corresponding linear number system (this time however in imaginary terms).
Now properly understood this indicates a direct complementary relationship.
So the numbers on the real number line can represented as separate (independent) points i.e. in analytic terms within this frame of reference.
Likewise the non-trivial zeros on the imaginary number line equally can be represented as separate (independent) points i.e. in analytic terms within this different frame of reference.
However the clear implication here is that when we seek to switch reference frames so as to interpret the imaginary points (from a real perspective) they now assume a holistic – rather than an analytic – identity.
In other words the significance of the non-trivial zeros, from a real number perspective, is that they can be used as an entire group to reconcile the unique individual identity of each prime (which is random with respect to the natural numbers) with an overall collective nature (where they are perfectly synchronised with the natural number system).
In other words, from the perspective of the imaginary number line, each non-trivial zero has an (extreme) analytic identity; however from the perspective of the real number line the non-trivial zeros as an entire group (which is always ultimately indeterminate in finite terms) have a complementary (extreme) holistic identity!
The same in fact applies to the prime numbers.
From the perspective of the real number line, each prime number has an (extreme) analytic identity i.e. as a unique building block of the natural number system!
However from the complementary perspective of the imaginary number line, the prime numbers as an entire group (the ultimate nature of which is necessarily indeterminate in finite terms) play a remarkable holistic role in serving to completely reconcile the random individual identity of each non-trivial zero with their overall collective identity (i.e. through being perfectly synchronised with the imaginary number system).
So we see that when we adopt both perspectives that:
1) Each individual prime number has an extreme analytic identity as separate (in real number terms).
2) That the prime numbers as an entire collective group have a corresponding extreme holistic identity (in imaginary number terms).
3) That each individual non-trivial zero likewise has an extreme analytic identity (in imaginary number terms).
4) That the non-trivial zeros as an entire collective group have a corresponding extreme holistic identity (from the real number perspective).
However to properly appreciate such relationships we must preserve both analytic (Type 1) and holistic (Type 2) perspectives with respect to the number system.
Clearly this cannot be achieved in conventional mathematical terms, as it is based formally on mere Type 1 interpretation.
So each number type contains have both particle and wave identities with an individual (analytic) and collective (holistic) identity respectively.
We can then approach the same number relationship issues in an inverted fashion through the Type 2 aspect of the number system.
Now whereas the Type 1 is directly geared to analytic type appreciation (and indirectly holistic), Type 2 is - by contrast - directly geared to holistic type appreciation (and indirectly analytic).
With Type 2, rather than a linear, we adopt a circular notion of number.
Therefore, from a real perspective instead of representing the natural numbers as equidistant points on the number line, we represent them in alternative fashion as equidistant points on the unit circle (where now they strictly represent an ordinal - rather than cardinal - identity)!
So in this approach the prime number 3 - for example – can be represented by the 3 equidistant points labelled 1, 2 and 3 respectively relating to the three corresponding ordinal members of 3 i.e. 1st, 2nd and 3rd respectively.
Thus once again whereas in the cardinal (Type 1) approach the natural numbers, (as an entire collective group) are all uniquely derived from individual prime constituents, in the complementary ordinal (Type 2) approach each prime number (as an individual group) is uniquely defined by its individual natural number members.
This in turn leads to a fascinating alternative explanation of prime number identity.
From the Type 1 perspective, a prime number has no factors (other than itself and 1).
From the Type 2 perspective a prime number - by definition - will always be a factor of the sum of its individual members.
For example 3 as prime number is necessarily a factor of 1 + 2 + 3 = 6 (and this relationship will always hold where the number is prime). However this is not unique for prime numbers and in fact is shared by all odd numbers.
Put another way, expressing this in modular (clock arithmetic) if the modulus is prime, then the sum of the individual natural number members of this prime = 0.
So with the Type 1 a prime number is defined by having no non-trivial factor in natural number terms (i.e. other than itself and 1).
In Type 2 terms, in inverse complementary fashion, a prime number is defined as always constituting a non-trivial factor of the sum of its natural number members!
This in fact therefore provides an alternative means in practice for testing for primeness (which I have already illustrated in other contexts). See "Interesting Prime Result"
However this circular (Type 2) system of (real) numbers has a counterpart system of (complex) numbers as the non-trivial zeros arising as solutions to the equation,
1 + s^1 + s^2 + ….+ s^(n – 1) = 0, where n is prime.
In contrast to Type 1, appreciation of the nature of these zeros is directly of a holistic nature.
Though 1 is always included as one of the prime roots of 1, in a certain sense it is trivial in that it is thereby not unique!
So the non-trivial strictly refer to the remaining n – 1 roots!
We could perhaps illustrate the true holistic nature of such solutions with respect to the simplest case of the 2 roots of 1.
So once again these are 1 and – 1 respectively.
The holistic interdependence of these roots can be illustrated by the fact that the sum of these roots i.e. 1 – 1 = 0.
Now this interdependence strictly relates to an energy state!
For example at the sub-atomic level, a particle and anti-particle are + 1 and – 1 with respect to each other. Then when combined their separate identities are annihilated resulting in a pure energy state.
So in a very literal sense such a pure energy state thereby characterises the holistic nature of 2.
Indeed if we were to equally consider virtual (imaginary) as well as real particles the combined annihilation with anti-particles at both levels would characterise the true holistic nature of 4!
Though other numbers are indeed more difficult to illustrate, the basic principle is clear in that the holistic nature of number in fact relates directly to a physical energy state!
And as physical and psychological aspects are complementary, this equally implies that associated with each number in holistic terms is a corresponding psychological energy state relating to the (unconscious) intuitive aspect of number experience.
For example, in the dynamics of experience external and internal polarities of understanding are positive and negative with respect to each other.
As the very process of understanding necessarily entails integrating both of these polarities + 1 and – 1 as directions (i.e. dimensions) the holistic understanding of 2 from a psychological perspective necessarily relates to a psycho-spiritual energy state! So intuitive recognition (as a psychological eneregy state) always reflects the holistic aspect of understanding. Unfortunately such intuitive recognition is then reduced to merely rational (analytic) interpretation in formal conventional mathematical terms.
So when one looks at the issue from a number perspective it should come as no surprise that likewise the holistic appreciation of the non-trivial zeros (with respect to Type 1) equally relate to energy states.
Though a degree of recognition of this fact has admittedly been obtained in recent years (with reference to the behaviour of certain quantum chaotic physical systems) Conventional Mathematics still lacks the means to intuitively explain why this is so (due to the complete absence in formal terms of a holistic aspect to its means of interpretation).
In my own case, as I had been specialising for some years in the holistic appreciation of number, I had long reached the conclusion that the Riemann zeros necessarily had quantum mechanical implications.
However even more, I had firmly reached the conclusion that these same zeros necessarily possess extremely important psycho-spiritual implications (which is completely missing as yet from conventional type understanding).
Furthermore the appropriate way to view both these aspects - physical and psychological - is in a dynamic complementary manner!
Moreover, correctly understood, we have in fact two sets of zeros (in accordance with Type 1 and Type 2 appreciation respectively).
Whereas Type 1 directly relates to the understanding of prime numbers as discrete phenomenal entities, by contrast Type 2 directly relates to the same prime numbers as (formless) energy states in both physical and psychological terms.
The deeper implications of all this is that - correctly understood - number itself must be viewed as inherent in all physical and psychological processes as the most fundamental way in which their quantitative and qualitative attributes are encoded.
In this sense, we can truly say that all living phenomenal forms at their most fundamental level represent but the dynamic interaction of number processes with respect to both their quantitative and qualitative aspects!
This is then reflected in the number system which itself has Type 1 (analytic) and Type 2 (holistic) aspects which are complementary. Then indirectly the Type 1 aspect can be given a Type 2 interpretation while then Type 2 indirectly can be given a Type 1 formulation!
Then with respect to the famed complex zeta function, it too has too complementary aspects.
So corresponding to Type 1 we have the recognised Riemann Zeta Function (which I refer to as Zeta 1).
Then corresponding to Type 2 we have a - largely unrecognised – complementary function (which I refer to as Zeta 2). Strictly, though the Zeta 2 Function is in fact well known; its true holistic significance is not at all appreciated.
Once again the Zeta 1 is defined as the infinite series:
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….
And the non-trivial (pair) solutions of the form s = a + it and a – it (where a according to the Riemann Hypothesis = ½) occur when
1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + …. = 0
The Zeta 2 is then in inverse complementary terms (i.e. where the natural numbers now represent the powers and s the base quantities) as the finite series:
1 + s^1 + s^2 + ….+ s^(n – 1).
And the non-trivial solutions of the form s = a + it arise from the solutions of the equation,
1 + s^1 + s^2 + ….+ s^(n – 1) = 0
Now in a qualified sense we can equally define this second function in an infinite manner!
So if we take terms in in a strict cyclical order in accordance with the value of n, then the infinite equation will hold.
Thus for example in the simplest case where n = 2, the non-trivial 2nd root as solution to
1 + s^1 = 0 gives s = – 1.
So when we take terms two at a time the value of the infinite series
1 + s^1 + s^2 + s^3 + ….. = 0.
Now in this blog entry I will try to highlight some of the key complementary relations existing between both series and their implications.
Indeed when one begins to appreciate the very nature of the number system from this dynamic complementary perspective it leads to a completely new type of insight that itself directly fosters appreciation of its holistic nature.
Customarily we tend to view the number system in (real) linear terms with the prime numbers viewed as its independent building blocks! So using the terminology that I customarily employ this reflects an analytic (quantitative) interpretation.
Then parallel with this finding we find that the system of non-trivial zeros relating to the Zeta 1 in turn - assuming the truth of the Riemann Hypothesis – represents a corresponding linear number system (this time however in imaginary terms).
Now properly understood this indicates a direct complementary relationship.
So the numbers on the real number line can represented as separate (independent) points i.e. in analytic terms within this frame of reference.
Likewise the non-trivial zeros on the imaginary number line equally can be represented as separate (independent) points i.e. in analytic terms within this different frame of reference.
However the clear implication here is that when we seek to switch reference frames so as to interpret the imaginary points (from a real perspective) they now assume a holistic – rather than an analytic – identity.
In other words the significance of the non-trivial zeros, from a real number perspective, is that they can be used as an entire group to reconcile the unique individual identity of each prime (which is random with respect to the natural numbers) with an overall collective nature (where they are perfectly synchronised with the natural number system).
In other words, from the perspective of the imaginary number line, each non-trivial zero has an (extreme) analytic identity; however from the perspective of the real number line the non-trivial zeros as an entire group (which is always ultimately indeterminate in finite terms) have a complementary (extreme) holistic identity!
The same in fact applies to the prime numbers.
From the perspective of the real number line, each prime number has an (extreme) analytic identity i.e. as a unique building block of the natural number system!
However from the complementary perspective of the imaginary number line, the prime numbers as an entire group (the ultimate nature of which is necessarily indeterminate in finite terms) play a remarkable holistic role in serving to completely reconcile the random individual identity of each non-trivial zero with their overall collective identity (i.e. through being perfectly synchronised with the imaginary number system).
So we see that when we adopt both perspectives that:
1) Each individual prime number has an extreme analytic identity as separate (in real number terms).
2) That the prime numbers as an entire collective group have a corresponding extreme holistic identity (in imaginary number terms).
3) That each individual non-trivial zero likewise has an extreme analytic identity (in imaginary number terms).
4) That the non-trivial zeros as an entire collective group have a corresponding extreme holistic identity (from the real number perspective).
However to properly appreciate such relationships we must preserve both analytic (Type 1) and holistic (Type 2) perspectives with respect to the number system.
Clearly this cannot be achieved in conventional mathematical terms, as it is based formally on mere Type 1 interpretation.
So each number type contains have both particle and wave identities with an individual (analytic) and collective (holistic) identity respectively.
We can then approach the same number relationship issues in an inverted fashion through the Type 2 aspect of the number system.
Now whereas the Type 1 is directly geared to analytic type appreciation (and indirectly holistic), Type 2 is - by contrast - directly geared to holistic type appreciation (and indirectly analytic).
With Type 2, rather than a linear, we adopt a circular notion of number.
Therefore, from a real perspective instead of representing the natural numbers as equidistant points on the number line, we represent them in alternative fashion as equidistant points on the unit circle (where now they strictly represent an ordinal - rather than cardinal - identity)!
So in this approach the prime number 3 - for example – can be represented by the 3 equidistant points labelled 1, 2 and 3 respectively relating to the three corresponding ordinal members of 3 i.e. 1st, 2nd and 3rd respectively.
Thus once again whereas in the cardinal (Type 1) approach the natural numbers, (as an entire collective group) are all uniquely derived from individual prime constituents, in the complementary ordinal (Type 2) approach each prime number (as an individual group) is uniquely defined by its individual natural number members.
This in turn leads to a fascinating alternative explanation of prime number identity.
From the Type 1 perspective, a prime number has no factors (other than itself and 1).
From the Type 2 perspective a prime number - by definition - will always be a factor of the sum of its individual members.
For example 3 as prime number is necessarily a factor of 1 + 2 + 3 = 6 (and this relationship will always hold where the number is prime). However this is not unique for prime numbers and in fact is shared by all odd numbers.
Put another way, expressing this in modular (clock arithmetic) if the modulus is prime, then the sum of the individual natural number members of this prime = 0.
So with the Type 1 a prime number is defined by having no non-trivial factor in natural number terms (i.e. other than itself and 1).
In Type 2 terms, in inverse complementary fashion, a prime number is defined as always constituting a non-trivial factor of the sum of its natural number members!
This in fact therefore provides an alternative means in practice for testing for primeness (which I have already illustrated in other contexts). See "Interesting Prime Result"
However this circular (Type 2) system of (real) numbers has a counterpart system of (complex) numbers as the non-trivial zeros arising as solutions to the equation,
1 + s^1 + s^2 + ….+ s^(n – 1) = 0, where n is prime.
In contrast to Type 1, appreciation of the nature of these zeros is directly of a holistic nature.
Though 1 is always included as one of the prime roots of 1, in a certain sense it is trivial in that it is thereby not unique!
So the non-trivial strictly refer to the remaining n – 1 roots!
We could perhaps illustrate the true holistic nature of such solutions with respect to the simplest case of the 2 roots of 1.
So once again these are 1 and – 1 respectively.
The holistic interdependence of these roots can be illustrated by the fact that the sum of these roots i.e. 1 – 1 = 0.
Now this interdependence strictly relates to an energy state!
For example at the sub-atomic level, a particle and anti-particle are + 1 and – 1 with respect to each other. Then when combined their separate identities are annihilated resulting in a pure energy state.
So in a very literal sense such a pure energy state thereby characterises the holistic nature of 2.
Indeed if we were to equally consider virtual (imaginary) as well as real particles the combined annihilation with anti-particles at both levels would characterise the true holistic nature of 4!
Though other numbers are indeed more difficult to illustrate, the basic principle is clear in that the holistic nature of number in fact relates directly to a physical energy state!
And as physical and psychological aspects are complementary, this equally implies that associated with each number in holistic terms is a corresponding psychological energy state relating to the (unconscious) intuitive aspect of number experience.
For example, in the dynamics of experience external and internal polarities of understanding are positive and negative with respect to each other.
As the very process of understanding necessarily entails integrating both of these polarities + 1 and – 1 as directions (i.e. dimensions) the holistic understanding of 2 from a psychological perspective necessarily relates to a psycho-spiritual energy state! So intuitive recognition (as a psychological eneregy state) always reflects the holistic aspect of understanding. Unfortunately such intuitive recognition is then reduced to merely rational (analytic) interpretation in formal conventional mathematical terms.
So when one looks at the issue from a number perspective it should come as no surprise that likewise the holistic appreciation of the non-trivial zeros (with respect to Type 1) equally relate to energy states.
Though a degree of recognition of this fact has admittedly been obtained in recent years (with reference to the behaviour of certain quantum chaotic physical systems) Conventional Mathematics still lacks the means to intuitively explain why this is so (due to the complete absence in formal terms of a holistic aspect to its means of interpretation).
In my own case, as I had been specialising for some years in the holistic appreciation of number, I had long reached the conclusion that the Riemann zeros necessarily had quantum mechanical implications.
However even more, I had firmly reached the conclusion that these same zeros necessarily possess extremely important psycho-spiritual implications (which is completely missing as yet from conventional type understanding).
Furthermore the appropriate way to view both these aspects - physical and psychological - is in a dynamic complementary manner!
Moreover, correctly understood, we have in fact two sets of zeros (in accordance with Type 1 and Type 2 appreciation respectively).
Whereas Type 1 directly relates to the understanding of prime numbers as discrete phenomenal entities, by contrast Type 2 directly relates to the same prime numbers as (formless) energy states in both physical and psychological terms.
The deeper implications of all this is that - correctly understood - number itself must be viewed as inherent in all physical and psychological processes as the most fundamental way in which their quantitative and qualitative attributes are encoded.
In this sense, we can truly say that all living phenomenal forms at their most fundamental level represent but the dynamic interaction of number processes with respect to both their quantitative and qualitative aspects!
Monday, March 11, 2013
Two Complementary Zeta Functions (1)
Once again my basic contention is that – properly understood – there are two distinct aspects to the number system.
1) An analytic aspect where the fundamental poles of understanding (i.e. internal and internal and quantitative and qualitative are clearly separated). This indeed is what makes such Mathematics linear (i.e. 1-dimensional) in nature as it literally interprets its symbols exclusively – in any context - in terms of the dominance of just one pole of understanding.
So relationships are considered absolutely in terms of the objective nature of symbols (where the external pole dominates); equally relationships are considered absolutely in terms of the quantitative nature of symbols (where the qualitative is thereby reduced to the quantitative).
(It is important to appreciate that when I use the word analytic, typically I use it is this general context of absolute uni-polar interpretation of symbols and not in the narrower more specialised conventional mathematical sense relating to infinite series).
I refer to this aspect of Mathematics as Type 1.
2) A holistic aspect where the same fundamental poles of understanding are now considered as dynamically related. In this new distinctive interpretation the objective nature of mathematical symbols strictly has no meaning in the absence of their corresponding means of (subjective) mental interpretation.
And as a wide variety of partially valid interpretations can be employed the objective nature of such symbols is thereby necessarily of a merely relative nature.
Likewise the quantitative interpretation of symbols strictly has no means in the absence of qualitative appreciation (of a distinctive holistic variety).
To be more accurate the holistic aspect properly refers to the dynamic interaction of poles that are (a) external and internal and (b) quantitative and qualitative with respect to each other.
Likewise the analytic aspect refers to the extreme situation where such interaction is ignored through reducing one entirely in terms of the other. And again in Conventional Mathematics the internal aspect is reduced in terms of the external and the qualitative in terms of the quantitative!
However for convenience I continually contrast analytic and holistic (as shorthand for there respectively different ways of treating such polar interaction)!
I refer to this aspect of Mathematics as Type 2.
3) The actual experience of Mathematics entails the mutual dynamic interaction of both Type 1 and Type 2 understanding. Though customary objective quantitative distinctions of a valid nature can indeed be made, as the polar reference frames to which they relate continually change they are now understood in a merely relative manner.
I refer to this most comprehensive aspect of Mathematics as Type 3.
So properly understood (i.e. in a comprehensive manner) all mathematical relationships are necessarily understood in merely relative terms. The customary fixation with absolute type relationships in Conventional Mathematics reflects its reduced – and thereby limited – (1-dimensional) manner of interpretation.
The distinction as between the Type 1 and Type 2 aspects Mathematics leads to a corresponding distinction with respect to the number system.
For example the natural numbers in Type 1 terms are understood as representing cardinal whole units. So 2 in cardinal terms represent a collective whole unit (i.e. an integer).
The problem with this quantitative approach however is that when one attempts to describe the individual members of 2, one must treat them in homogenous terms as without qualitative distinction. So 2 = 1 + 1.
However this leaves one with the considerable problem of having no means of making a meaningful distinction as between the 1st and 2nd members of 2 (in ordinal terms).
The fact that this fundamental issue is completely glossed over in Conventional Mathematics clearly reflects its reduced nature i.e. where the qualitative (ordinal) nature of 1st and 2nd is misleadingly assumed to be implied by the quantitative (cardinal) understanding of 1 and 2!
So the Type 2 understanding of number resembles – as it were - a sub-atomic approach where each of the individual members of a number group are now given a unique individual identity in relative terms.
So once again from a Type 1 approach, 2 is treated from a cardinal perspective as a collective whole unit (in quantitative terms).
From the complementary Type 2 approach, 2 is treated from an ordinal perspective as comprising unique individual units i.e. 1st and 2nd (in qualitative terms).
Now Type 1 is – literally - defined in 1-dimensional terms, where each natural number is implicitly defined with respect to 1 as exponent (i.e. dimensional number).
So the natural numbers 1, 2, 3, 4,…. are defined in Type 1 terms as
1^1, 2^1, 3^ 1. 4^1,….
The 1-dimensional nature of this approach is clearly illustrated by the fact that any number initially raised to another exponent (dimensional number) will be given a reduced linear interpretation i.e. where its ultimate value is expressed in reduced quantitative terms.
So 2^2 in Type 1 = 4^1!
The Type 1 approach is properly geared to explain the fundamental nature of addition where quantitative change occurs (without qualitative transformation)!
So 2 + 3 in this additive system = 2^ 1 + 3^ 1 = 5^1.
In the Type 2 aspect of the number system, 2 is by contrast is defined with respect to 1 (representing a default base number quantity).
So the natural number 1, 2, 3, 4,….. are defined from a Type 2 perspective as
1^1, 1^2, 1^3, 1^4,…..
Just as the Type 1 approach is geared to the pure nature of addition (where no qualitative transformation in the variables takes place), The Type 2 approach is then geared to the pure nature of multiplication (where no quantitative transformation in the variables takes place).
So 2 + 3 in this system represents 1^2 * 1^ 3 = 1 ^ (2 + 3) = 1^5.
So what appears as multiplication from the Type 1 perspective represents addition from the corresponding Type 2 perspective.
However to properly express the nature of the Type 2 aspect we need to move to a circular number system (based on successive roots of unity).
The key to understanding the nature of this system is that a two-way interaction is necessarily entailed as between variables given both an analytic and holistic interpretation respectively.
Thus for example the number 2 in type 2 terms = 1^2.
2 here is identified with the two individual unique dimensions of 2 i.e. its 1st and 2nd dimensions (with a holistic interpretation as interdependent).
Now in analytic terms these two dimensions can be given expression as the 1st and 2nd roots of 1 respectively or more correctly the roots of 1^2 and 1^1 respectively.
So the first root of 1 is therefore (1^2) raised to the power ½ = + 1 (i.e. 1^1).
The second root of 1 is (1^1) raised to the power of ½ = – 1.
However in dynamic terms these two separate values (as analytically understood) are in continual relationship with the corresponding interdependent value (as holistically understood).
So the quantitative independent interpretation of 1 and – 1 as separate strictly have no meaning in dynamic relative terms apart from the combined holistic interpretation of 1 and – 1 as interdependent.
Once again I have repeatedly used the example of a crossroads to illustrate this type of understanding. When understood (within independent reference frames) left and right turns at a crossroads can be given a – relatively – separate interpretation which are + 1 and – 1 with respect to each other. (1 refers here to the dimensional unit i.e. representing a direction).
However the appreciation that both left and right ultimately have a purely relative i.e. paradoxical meaning (when both frames are simultaneously combined) represents the true qualitative holistic nature of 2!
And this could be represented quantitatively as 1 – 1 = 0, i.e. as strictly without quantitative significance!
So in the actual understanding of the nature of a direction at a crossroads, both types of understanding (analytic and holistic) are necessarily combined.
Indeed this 2-dimensional type of dynamic understanding is well-recognised in spiritual literature (e.g. Taoism and Buddhism) in philosophy (e.g. Heraclitus and Hegel) in psychology (e.g. Jung) and indeed even indirectly in quantum physics (e.g. wave-particle duality).
What is vital to appreciate however is that it necessarily combines both analytic and holistic aspects of understanding in a complementary fashion.
Now clearly this cannot be appreciated in Type 1 terms (where holistic notions are reduced to analytic in a static absolute manner).
As Conventional Mathematics is fundamentally based on such absolute notions it is severely limited in terms of dealing with the very notion of interdependence.
And as the nature of prime numbers likewise involves such interdependence e.g. in the relationship of the primes - constituting the natural number system in cardinal terms - to the non-trivial zeta zeros, it is fatally flawed in terms of appreciating the true nature of the number system.
Another key point is that this interpretation of the number 2 from a Type 2 perspective represents but the simplest version of Type 2 understanding.
Indeed associated with every number as an exponent (i.e. dimensional number) is a unique means of configuring the dynamic interaction as between its analytic and holistic elements! Or using the shorthand means that I have been employing every number has a unique holistic significance serving as a distinct means of interpretation of the dynamic interaction of polarities (underlying all mathematical understanding).
In this holistic context 1 (as 1-dimensional) has an extreme limiting interpretation where the holistic aspect is directly reduced to analytic interpretation.
And this is what precisely defines the nature of Conventional Mathematics which is based on absolute objective interpretation of its symbols in a merely (reduced) quantitative manner!
However in truth in a more comprehensive vision of Mathematics, an unlimited number of possible interpretations exist (all with a partial relative validity). And in all these other systems (based on a dimensional number ≠ 1).
And if you understand this you will immediately understand why from a holistic perspective, the Riemann zeta function is uniquely undefined for s = 1 (as this is the one value where the inherent dynamic interaction as between the analytic and holistic aspects of the number system is broken).
So once again, putting it bluntly the Riemann zeta function cannot be properly interpreted in conventional mathematical terms. And of course neither can the Riemann Hypothesis (which is fundamentally based on true appropriate interpretation of the Riemann Function).
Finally it is futile therefore attempting to prove the Riemann Hypothesis from a conventional (1-dimensional) perspective!
However fortunately, the key to appreciating the nature of the Riemann Hypothesis can be related to the simplest case of Type 2 understanding (with respect to the number 2).
Once again the most comprehensive form of mathematical understanding involves the interaction of both Type 1 and Type 2 aspects where both analytic (as separate) and holistic appreciation can reach a high level of refinement.
In fact whenever a non-unitary base number is raised to a non-unitary power, both Type 1 and Type 2 aspects of interpretation are both required.
So in the simplest case of 2^2, strictly both a quantitative (analytic) and qualitative (holistic) transformation is entailed the mutual interaction of which entails Type 3 understanding.
Next we find that associated with the Type 1 and Type 2 aspects of the number system are corresponding Type 1 and Type 2 zeta functions.
Now the Riemann zeta function refers strictly to the Type 1 aspect; however proper appreciation of the Riemann zeta function and its associated Riemann Hypothesis requires incorporation of a complementary Type 2 zeta function.
Though the conventional Riemann zeta function can indeed be given an analytic Type 1 formulation (as the Zeta 1 function), strictly this has no proper meaning in the absence of corresponding holistic Type 2 appreciation.
In like manner through the Riemann zeta function can be given an – unrecognised – Type 2 formulation (as the Zeta 2 function), strictly again this has no proper meaning in the absence of corresponding analytic Type 1 formulation.
Finally the most comprehensive understanding – in what I refer to as the Type 3 zeta function - entails the simultaneous interaction of both Type 1 and Type 2 formulations in a manner ultimately approaching purely ineffable understanding!
1) An analytic aspect where the fundamental poles of understanding (i.e. internal and internal and quantitative and qualitative are clearly separated). This indeed is what makes such Mathematics linear (i.e. 1-dimensional) in nature as it literally interprets its symbols exclusively – in any context - in terms of the dominance of just one pole of understanding.
So relationships are considered absolutely in terms of the objective nature of symbols (where the external pole dominates); equally relationships are considered absolutely in terms of the quantitative nature of symbols (where the qualitative is thereby reduced to the quantitative).
(It is important to appreciate that when I use the word analytic, typically I use it is this general context of absolute uni-polar interpretation of symbols and not in the narrower more specialised conventional mathematical sense relating to infinite series).
I refer to this aspect of Mathematics as Type 1.
2) A holistic aspect where the same fundamental poles of understanding are now considered as dynamically related. In this new distinctive interpretation the objective nature of mathematical symbols strictly has no meaning in the absence of their corresponding means of (subjective) mental interpretation.
And as a wide variety of partially valid interpretations can be employed the objective nature of such symbols is thereby necessarily of a merely relative nature.
Likewise the quantitative interpretation of symbols strictly has no means in the absence of qualitative appreciation (of a distinctive holistic variety).
To be more accurate the holistic aspect properly refers to the dynamic interaction of poles that are (a) external and internal and (b) quantitative and qualitative with respect to each other.
Likewise the analytic aspect refers to the extreme situation where such interaction is ignored through reducing one entirely in terms of the other. And again in Conventional Mathematics the internal aspect is reduced in terms of the external and the qualitative in terms of the quantitative!
However for convenience I continually contrast analytic and holistic (as shorthand for there respectively different ways of treating such polar interaction)!
I refer to this aspect of Mathematics as Type 2.
3) The actual experience of Mathematics entails the mutual dynamic interaction of both Type 1 and Type 2 understanding. Though customary objective quantitative distinctions of a valid nature can indeed be made, as the polar reference frames to which they relate continually change they are now understood in a merely relative manner.
I refer to this most comprehensive aspect of Mathematics as Type 3.
So properly understood (i.e. in a comprehensive manner) all mathematical relationships are necessarily understood in merely relative terms. The customary fixation with absolute type relationships in Conventional Mathematics reflects its reduced – and thereby limited – (1-dimensional) manner of interpretation.
The distinction as between the Type 1 and Type 2 aspects Mathematics leads to a corresponding distinction with respect to the number system.
For example the natural numbers in Type 1 terms are understood as representing cardinal whole units. So 2 in cardinal terms represent a collective whole unit (i.e. an integer).
The problem with this quantitative approach however is that when one attempts to describe the individual members of 2, one must treat them in homogenous terms as without qualitative distinction. So 2 = 1 + 1.
However this leaves one with the considerable problem of having no means of making a meaningful distinction as between the 1st and 2nd members of 2 (in ordinal terms).
The fact that this fundamental issue is completely glossed over in Conventional Mathematics clearly reflects its reduced nature i.e. where the qualitative (ordinal) nature of 1st and 2nd is misleadingly assumed to be implied by the quantitative (cardinal) understanding of 1 and 2!
So the Type 2 understanding of number resembles – as it were - a sub-atomic approach where each of the individual members of a number group are now given a unique individual identity in relative terms.
So once again from a Type 1 approach, 2 is treated from a cardinal perspective as a collective whole unit (in quantitative terms).
From the complementary Type 2 approach, 2 is treated from an ordinal perspective as comprising unique individual units i.e. 1st and 2nd (in qualitative terms).
Now Type 1 is – literally - defined in 1-dimensional terms, where each natural number is implicitly defined with respect to 1 as exponent (i.e. dimensional number).
So the natural numbers 1, 2, 3, 4,…. are defined in Type 1 terms as
1^1, 2^1, 3^ 1. 4^1,….
The 1-dimensional nature of this approach is clearly illustrated by the fact that any number initially raised to another exponent (dimensional number) will be given a reduced linear interpretation i.e. where its ultimate value is expressed in reduced quantitative terms.
So 2^2 in Type 1 = 4^1!
The Type 1 approach is properly geared to explain the fundamental nature of addition where quantitative change occurs (without qualitative transformation)!
So 2 + 3 in this additive system = 2^ 1 + 3^ 1 = 5^1.
In the Type 2 aspect of the number system, 2 is by contrast is defined with respect to 1 (representing a default base number quantity).
So the natural number 1, 2, 3, 4,….. are defined from a Type 2 perspective as
1^1, 1^2, 1^3, 1^4,…..
Just as the Type 1 approach is geared to the pure nature of addition (where no qualitative transformation in the variables takes place), The Type 2 approach is then geared to the pure nature of multiplication (where no quantitative transformation in the variables takes place).
So 2 + 3 in this system represents 1^2 * 1^ 3 = 1 ^ (2 + 3) = 1^5.
So what appears as multiplication from the Type 1 perspective represents addition from the corresponding Type 2 perspective.
However to properly express the nature of the Type 2 aspect we need to move to a circular number system (based on successive roots of unity).
The key to understanding the nature of this system is that a two-way interaction is necessarily entailed as between variables given both an analytic and holistic interpretation respectively.
Thus for example the number 2 in type 2 terms = 1^2.
2 here is identified with the two individual unique dimensions of 2 i.e. its 1st and 2nd dimensions (with a holistic interpretation as interdependent).
Now in analytic terms these two dimensions can be given expression as the 1st and 2nd roots of 1 respectively or more correctly the roots of 1^2 and 1^1 respectively.
So the first root of 1 is therefore (1^2) raised to the power ½ = + 1 (i.e. 1^1).
The second root of 1 is (1^1) raised to the power of ½ = – 1.
However in dynamic terms these two separate values (as analytically understood) are in continual relationship with the corresponding interdependent value (as holistically understood).
So the quantitative independent interpretation of 1 and – 1 as separate strictly have no meaning in dynamic relative terms apart from the combined holistic interpretation of 1 and – 1 as interdependent.
Once again I have repeatedly used the example of a crossroads to illustrate this type of understanding. When understood (within independent reference frames) left and right turns at a crossroads can be given a – relatively – separate interpretation which are + 1 and – 1 with respect to each other. (1 refers here to the dimensional unit i.e. representing a direction).
However the appreciation that both left and right ultimately have a purely relative i.e. paradoxical meaning (when both frames are simultaneously combined) represents the true qualitative holistic nature of 2!
And this could be represented quantitatively as 1 – 1 = 0, i.e. as strictly without quantitative significance!
So in the actual understanding of the nature of a direction at a crossroads, both types of understanding (analytic and holistic) are necessarily combined.
Indeed this 2-dimensional type of dynamic understanding is well-recognised in spiritual literature (e.g. Taoism and Buddhism) in philosophy (e.g. Heraclitus and Hegel) in psychology (e.g. Jung) and indeed even indirectly in quantum physics (e.g. wave-particle duality).
What is vital to appreciate however is that it necessarily combines both analytic and holistic aspects of understanding in a complementary fashion.
Now clearly this cannot be appreciated in Type 1 terms (where holistic notions are reduced to analytic in a static absolute manner).
As Conventional Mathematics is fundamentally based on such absolute notions it is severely limited in terms of dealing with the very notion of interdependence.
And as the nature of prime numbers likewise involves such interdependence e.g. in the relationship of the primes - constituting the natural number system in cardinal terms - to the non-trivial zeta zeros, it is fatally flawed in terms of appreciating the true nature of the number system.
Another key point is that this interpretation of the number 2 from a Type 2 perspective represents but the simplest version of Type 2 understanding.
Indeed associated with every number as an exponent (i.e. dimensional number) is a unique means of configuring the dynamic interaction as between its analytic and holistic elements! Or using the shorthand means that I have been employing every number has a unique holistic significance serving as a distinct means of interpretation of the dynamic interaction of polarities (underlying all mathematical understanding).
In this holistic context 1 (as 1-dimensional) has an extreme limiting interpretation where the holistic aspect is directly reduced to analytic interpretation.
And this is what precisely defines the nature of Conventional Mathematics which is based on absolute objective interpretation of its symbols in a merely (reduced) quantitative manner!
However in truth in a more comprehensive vision of Mathematics, an unlimited number of possible interpretations exist (all with a partial relative validity). And in all these other systems (based on a dimensional number ≠ 1).
And if you understand this you will immediately understand why from a holistic perspective, the Riemann zeta function is uniquely undefined for s = 1 (as this is the one value where the inherent dynamic interaction as between the analytic and holistic aspects of the number system is broken).
So once again, putting it bluntly the Riemann zeta function cannot be properly interpreted in conventional mathematical terms. And of course neither can the Riemann Hypothesis (which is fundamentally based on true appropriate interpretation of the Riemann Function).
Finally it is futile therefore attempting to prove the Riemann Hypothesis from a conventional (1-dimensional) perspective!
However fortunately, the key to appreciating the nature of the Riemann Hypothesis can be related to the simplest case of Type 2 understanding (with respect to the number 2).
Once again the most comprehensive form of mathematical understanding involves the interaction of both Type 1 and Type 2 aspects where both analytic (as separate) and holistic appreciation can reach a high level of refinement.
In fact whenever a non-unitary base number is raised to a non-unitary power, both Type 1 and Type 2 aspects of interpretation are both required.
So in the simplest case of 2^2, strictly both a quantitative (analytic) and qualitative (holistic) transformation is entailed the mutual interaction of which entails Type 3 understanding.
Next we find that associated with the Type 1 and Type 2 aspects of the number system are corresponding Type 1 and Type 2 zeta functions.
Now the Riemann zeta function refers strictly to the Type 1 aspect; however proper appreciation of the Riemann zeta function and its associated Riemann Hypothesis requires incorporation of a complementary Type 2 zeta function.
Though the conventional Riemann zeta function can indeed be given an analytic Type 1 formulation (as the Zeta 1 function), strictly this has no proper meaning in the absence of corresponding holistic Type 2 appreciation.
In like manner through the Riemann zeta function can be given an – unrecognised – Type 2 formulation (as the Zeta 2 function), strictly again this has no proper meaning in the absence of corresponding analytic Type 1 formulation.
Finally the most comprehensive understanding – in what I refer to as the Type 3 zeta function - entails the simultaneous interaction of both Type 1 and Type 2 formulations in a manner ultimately approaching purely ineffable understanding!
Thursday, March 7, 2013
In a Nutshell!
As we have seen natural numbers can be given two distinct definitions (in relative isolation from each other):
1. In cardinal terms as a collective homogeneous group (i.e. where individual units have no unique identity). This relates to the analytic (quantitative) aspect of number which I refer to as the Type 1 interpretation.
2. In ordinal terms as the relationship between unique individual members of a number group (where the collective group - from this perspective - has no distinct identity). This relates to the holistic (qualitative) aspect of number which I refer to as the Type 2 interpretation.
In actual experience, both of these aspects of number continually interact. Strictly speaking therefore, we cannot form an analytic appreciation of the nature of number without its corresponding holistic aspect; likewise we cannot form a holistic appreciation of number without its corresponding analytic aspect.
Ultimately both analytic and holistic aspects are of a purely relative nature.
Though such pure relativity relates to an absolute ineffable state, in the phenomenal experience of number it can only be approximated.
And properly understood this is what the Riemann Hypothesis relates to as the closest approximation in phenomenal terms to the ultimate ineffable state where both the analytic (quantitative) and holistic (qualitative) aspects of number are fully identical.
I have already explained in an earlier blog how the non-trivial zeros - corresponding to the Type 1 interpretation - represent in fact the perfect shadow number system to the primes (that comprise the natural number system - except 1 - in cardinal terms).
Equally the non-trivial zeros – corresponding to the Type 2 interpretation – represent the perfect shadow number system to the natural numbers (that comprise each prime group - except 1 - in ordinal terms).
Properly understood therefore, both the cardinal and ordinal numbers have no meaning in the absence of their corresponding shadow (wave-like) number systems as represented by the non-trivial zeros in both cases.
So both these particle-like features of number (as independent) and their corresponding wave-like properties (as interdependent) are simultaneously co-determined in a manner that is ultimately ineffable.
Put another way both the Type 1 and Type 2 aspects of the number system have a particle and wave-like identity reflecting their analytic and holistic aspects respectively.
From a psychological perspective this implies that all numbers have both conscious aspects (as analytic) and unconscious aspects (as holistic) respectively.
Now just as in quantum physics through interaction, every particle possesses wave-like properties and every wave particle-like properties, likewise with respect to number, the analytic (Type 1) interpretation of number possesses holistic (Type 2) properties and the holistic (Type 2) interpretation analytic properties respectively.
This means - as I have repeatedly stated – that we cannot possibly hope to understand the nature of the Type 1 (quantitative) non-trivial zeros - to which the Riemann Hypothesis directly relates - in the absence of Type 2 (qualitative) interpretation.
Equally, we cannot possibly hope to understand the nature of the Type 2 (qualitative) non-trivial zeros - to which an unrecognised counterpart Riemann Type Hypothesis relates - in the absence of Type 1 (quantitative) interpretation.
And I have indicated in several of my blogs what such a quantitative interpretation entails.
The basic message is that the inherent nature of the number system (and indeed of all mathematical activity) is of a dynamic relative nature entailing the interaction of both its quantitative (analytic) and qualitative (holistic) aspects.
It therefore cannot possibly be understood within the current mathematical paradigm, which is geared formally to - mere - quantitative interpretation!
In a nutshell this is the essential message underlying all my blog entries and urgently needs to be grasped!
So every mathematical symbol with a standard (Type 1) quantitative interpretation in conventional mathematical terms possesses an equally important (Type 2) qualitative interpretation from a holistic perspective.
Properly understood therefore mathematical meaning necessarily represents the combined interaction of both perspectives.
Thus once again to illustrate the number 2 has a standard quantitative interpretation in linear mathematical terms (representing the separate collective whole identity of a homogeneous set of units in a cardinal manner).
However the number 2 equally has a qualitative interpretation in circular terms (representing the interdependence of the two unique individual members of a group in an ordinal manner).
Likewise the operations + and – have a recognised quantitative meaning in standard mathematical terms; however they equally have a - largely unrecognised - qualitative meaning! So + in this context relates to the direct positing of meaning in a (conscious) rational fashion. – relates to the corresponding negation of such rational meaning in an (unconscious) intuitive manner.
So deeply implicit in the holistic notion of 2 is the interaction of two polar directions of experience that are positive and negative with respect to each other.
Indeed deeply implicit in the holistic notion of 1 is the existence of just one direction of experience that is positive (in an absolute manner).
So linear (1-dimensional) rational understanding with just one direction, necessarily leads to the reduction of holistic to analytic (and unconscious to conscious) interpretation respectively.
Once again though indeed it represents an important special case, in holistic terms, 1 represents the only dimension that interprets mathematical meaning in an absolute manner.
For all other numbers (as dimensions) a dynamic relationship as between analytic and holistic (conscious and unconscious) is implied.
Once again this is the Type 2 explanation as to why the Riemann Zeta Function remains uniquely undefined for s = 1!
As the Function - when properly interpreted - relates to the relationship as between analytic and holistic type meaning, it thereby remains undefined in conventional (1-dimensional) terms, where analytic type meaning is solely recognised in an absolute manner!
Therefore it is ultimately futile - not only in trying to prove - but more importantly in even trying to understand the Riemann Hypothesis from this limited perspective.
1. In cardinal terms as a collective homogeneous group (i.e. where individual units have no unique identity). This relates to the analytic (quantitative) aspect of number which I refer to as the Type 1 interpretation.
2. In ordinal terms as the relationship between unique individual members of a number group (where the collective group - from this perspective - has no distinct identity). This relates to the holistic (qualitative) aspect of number which I refer to as the Type 2 interpretation.
In actual experience, both of these aspects of number continually interact. Strictly speaking therefore, we cannot form an analytic appreciation of the nature of number without its corresponding holistic aspect; likewise we cannot form a holistic appreciation of number without its corresponding analytic aspect.
Ultimately both analytic and holistic aspects are of a purely relative nature.
Though such pure relativity relates to an absolute ineffable state, in the phenomenal experience of number it can only be approximated.
And properly understood this is what the Riemann Hypothesis relates to as the closest approximation in phenomenal terms to the ultimate ineffable state where both the analytic (quantitative) and holistic (qualitative) aspects of number are fully identical.
I have already explained in an earlier blog how the non-trivial zeros - corresponding to the Type 1 interpretation - represent in fact the perfect shadow number system to the primes (that comprise the natural number system - except 1 - in cardinal terms).
Equally the non-trivial zeros – corresponding to the Type 2 interpretation – represent the perfect shadow number system to the natural numbers (that comprise each prime group - except 1 - in ordinal terms).
Properly understood therefore, both the cardinal and ordinal numbers have no meaning in the absence of their corresponding shadow (wave-like) number systems as represented by the non-trivial zeros in both cases.
So both these particle-like features of number (as independent) and their corresponding wave-like properties (as interdependent) are simultaneously co-determined in a manner that is ultimately ineffable.
Put another way both the Type 1 and Type 2 aspects of the number system have a particle and wave-like identity reflecting their analytic and holistic aspects respectively.
From a psychological perspective this implies that all numbers have both conscious aspects (as analytic) and unconscious aspects (as holistic) respectively.
Now just as in quantum physics through interaction, every particle possesses wave-like properties and every wave particle-like properties, likewise with respect to number, the analytic (Type 1) interpretation of number possesses holistic (Type 2) properties and the holistic (Type 2) interpretation analytic properties respectively.
This means - as I have repeatedly stated – that we cannot possibly hope to understand the nature of the Type 1 (quantitative) non-trivial zeros - to which the Riemann Hypothesis directly relates - in the absence of Type 2 (qualitative) interpretation.
Equally, we cannot possibly hope to understand the nature of the Type 2 (qualitative) non-trivial zeros - to which an unrecognised counterpart Riemann Type Hypothesis relates - in the absence of Type 1 (quantitative) interpretation.
And I have indicated in several of my blogs what such a quantitative interpretation entails.
The basic message is that the inherent nature of the number system (and indeed of all mathematical activity) is of a dynamic relative nature entailing the interaction of both its quantitative (analytic) and qualitative (holistic) aspects.
It therefore cannot possibly be understood within the current mathematical paradigm, which is geared formally to - mere - quantitative interpretation!
In a nutshell this is the essential message underlying all my blog entries and urgently needs to be grasped!
So every mathematical symbol with a standard (Type 1) quantitative interpretation in conventional mathematical terms possesses an equally important (Type 2) qualitative interpretation from a holistic perspective.
Properly understood therefore mathematical meaning necessarily represents the combined interaction of both perspectives.
Thus once again to illustrate the number 2 has a standard quantitative interpretation in linear mathematical terms (representing the separate collective whole identity of a homogeneous set of units in a cardinal manner).
However the number 2 equally has a qualitative interpretation in circular terms (representing the interdependence of the two unique individual members of a group in an ordinal manner).
Likewise the operations + and – have a recognised quantitative meaning in standard mathematical terms; however they equally have a - largely unrecognised - qualitative meaning! So + in this context relates to the direct positing of meaning in a (conscious) rational fashion. – relates to the corresponding negation of such rational meaning in an (unconscious) intuitive manner.
So deeply implicit in the holistic notion of 2 is the interaction of two polar directions of experience that are positive and negative with respect to each other.
Indeed deeply implicit in the holistic notion of 1 is the existence of just one direction of experience that is positive (in an absolute manner).
So linear (1-dimensional) rational understanding with just one direction, necessarily leads to the reduction of holistic to analytic (and unconscious to conscious) interpretation respectively.
Once again though indeed it represents an important special case, in holistic terms, 1 represents the only dimension that interprets mathematical meaning in an absolute manner.
For all other numbers (as dimensions) a dynamic relationship as between analytic and holistic (conscious and unconscious) is implied.
Once again this is the Type 2 explanation as to why the Riemann Zeta Function remains uniquely undefined for s = 1!
As the Function - when properly interpreted - relates to the relationship as between analytic and holistic type meaning, it thereby remains undefined in conventional (1-dimensional) terms, where analytic type meaning is solely recognised in an absolute manner!
Therefore it is ultimately futile - not only in trying to prove - but more importantly in even trying to understand the Riemann Hypothesis from this limited perspective.
Monday, March 4, 2013
The Number 2
We have repeatedly seen that there are two distinct aspects to number that are quantitative (analytic) and qualitative (holistic) with respect to each other.
Expressed another way, one aspect is properly geared towards interpretation of the cardinal aspect and the other towards the ordinal aspect of number respectively.
And as stated here this requires two distinct interpretations with respect to the number system which I refer to as Type 1 and Type 2 respectively.
Whereas the Type 1 relates directly to the (recognised) conscious aspect of linear rational interpretation, the Type 2 relates to the (unrecognised) unconscious aspect, which indirectly can be given a rational interpretation in a circular logical manner.
Properly understood therefore the actual experience of number entails the dynamic interaction of both conscious and unconscious in experience entailing in turn the dynamic interaction of the analytic and holistic aspects of number (entailing both linear and circular logical type interpretation).
As we have seen the inherent nature of addition and multiplication relate to both Type 1 and Type 2 aspects of number understanding respectively.
Therefore the key point once again is that we cannot possibly come to grips with the deeper issues underlying prime numbers and the Riemann Hypothesis from within the conventional mathematical perspective (which is based solely on a reduced absolute Type 1 interpretation).
Now I will seek here to illustrate these points with respect to the number 2, appreciation of which is sufficient to grasp the essential nature of the Riemann Hypothesis.
Now the standard Type 1 interpretation is based on the notion of 2 as representing a collective whole unit in a merely quantitative manner.
So if we try here to subdivide into its individual units we get 1 + 1, where the separate units are treated as fully homogenous (i.e. lacking any qualitative distinction).
So 2 in this context more comprehensively means 2 ^ 1 i.e. where 2 is defined in linear terms with respect to the default 1st dimension.
Therefore geometrically, numbers in Type 1 terms are represented as points plotted on the 1-dimensional line. So when we consider a number initially expressed with respect to a different dimensional value, its ultimate value is expressed in a merely reduced linear (1-dimensional) fashion.
For example from this perspective 2^2 = 4 i.e. (4 ^ 1).
In another words, though strictly a qualitative distinction is now involved with 2 ^ 2 representing 4 square (2-dimensional) units, this qualitative transformation is simply ignored with the result expressed in a reduced linear (i.e. merely quantitative) manner.
Now I have raised the fundamental problem with this approach in that in ignoring the qualitative distinction (expressed through the dimensional number) that no means in fact exist for making ordinal distinctions as between numbers.
In other words if we seek to interpret numbers in a merely quantitative manner, then we have no basis for making ordinal rankings (which assumes a qualitative relationship as between numbers).
Again though this as I say is absolutely fundamental and strictly undermines the very basis of all conventional mathematical interpretation, it remains completely ignored within Mathematics. This is why I have long been of the view that such Mathematics - despite its admitted great advances - is simply not fit for purpose (as it is based on highly reduced and ultimately untenable assumption regarding the very nature of its relationships).
So the alternative Type 2 interpretation of number starts from the opposite basis of viewing each number with respect to its individual members in distinct ordinal terms.
So from this perspective the number 2 comprises a 1st and 2nd member (each of which is qualitatively distinct).
However as this understanding is of a very distinct nature it cannot be reflected through Type 1 understanding.
So here we reverse the relationship as between number (as base quantity) and number (as dimension).
Thus in the Type 1 system where 2 = 2 ^ 1, 2 represents the base quantity and 1 the (default) dimension.
However in the Type 2 system, 2 = 1 ^ 2 so that now 1 represents the (default) base quantity and 2 the dimensional number (expressing qualitative distinction).
So again for example if we square the number 1, no quantitative change in the unit takes place; however a qualitative change to square (2-dimensional) units occurs.
And in the Type 2 we concentrate on the nature of such qualitative change!
To highlight the qualitative nature of this system we must switch to a circular type representation of number.
So the number 2 as dimensional number comprises two distinct ordinal members (i.e. 1st and 2nd).
Now an inverse relationship exists as between the qualitative nature of these units and the indirect (reduced) quantitative interpretation.
So the inverse of the 1st member as first root (where the number 1 now represents ordinal identity) remains unchanged as 1 ^ 1 = 1; however the inverse of the 2nd member as 2nd root (where 2 now represents ordinal identity) 1^ (1/2) = – 1.
So what we have achieved here - through the two roots of 1 - is an indirect quantitative means (on a circular number scale) of expressing the unique identity of the two ordinal members of 2.
And of course by extension we can express - again in an indirect quantitative manner - the unique identity of all the ordinal members of any number n by taking the corresponding n roots of 1!
So the various roots of 1 possess an as yet unrecognised significance in providing the indirect quantitative means of uniquely identifying the ordinal nature of number with respect to any group.
I have raised this key issue before as to how the meaning of 2nd for example is purely relative depending on the size of the group to which it belongs! So this problem of giving 2 a distinctive ordinal meaning is solved by considering the value of the 2nd of n roots (where n can be any natural number).
So each ordinal member of a group is given a - relatively - separate (indirect) quantitative identity; however the true holistic nature of the group requires the interdependence of these members in a qualitative manner.
And as we have seen the sum of all n roots of 1 (except where n = 1) = 0. So the qualitative interdependence of a group requires the (circular) combination of all its separate members.
Again I have illustrated before the nature of this interdependence with respect to road directions.
However as it fruitfully illustrates the true Type 2 nature of number, I will do so again.
Clearly left and right designations at a crossroads have a merely relative identity depending on the direction from which the crossroads is defined.
So if we approach a crossroads travelling towards it in an upward northerly direction, we can unambiguously identify for example, a left turn (which we can designate as + 1).
In this context the opposite right turn can be designated as – 1.
However this designation has a merely relative identity (depending on the arbitrary direction from which the crossroads is approached). So from the opposite direction (approaching the crossroads while moving in a downwards southerly direction) once again a left turn has an unambiguous meaning (as + 1) and a right (as – 1).
However this designation of left and right will be precisely the opposite as in the first case.
So the pure holistic appreciation of the interdependence of the directions (left and right) requires the ability to simultaneous combine both reference frames where what is left from one perspective is right from the other; and what is right from one is left from the other.
So in number terms, the simultaneous qualitative interdependence of both can be expressed as (+) 1 – 1 = 0 (or alternatively – 1 + 1 = 0).
So this apparently simple example of a crossroads implicitly entails the Type 2 appreciation of the nature of 2.
So each member of the group can be given a relatively separate ordinal identity in an indirect quantitative manner (i.e. within a given frame of reference) as + 1 and – 1 respectively.
However the overall holistic identity of the two members in qualitative terms requires the simultaneous recognition of both reference frames (creating paradox from a linear rational perspective).
In this way the quantitative recognition of the two members of 2 as separate, is cancelled out in pure holistic recognition of their qualitative interdependent nature.
Again I have often used the example of a crossroads to illustrate the nature of polar reference frames.
However such polarised reference frames are the very nature of experience (which is inevitably conditioned with respect to all phenomenal understanding by external and internal, and equally quantitative and qualitative polarities).
So when we look at the number system in an appropriate dynamic manner, we recognise that a ceaseless interaction as between these opposite polarities is necessarily involved.
Therefore any distinct knowledge of a quantitative nature is merely relative, and ultimately becomes completely cancelled out through a pure holistic qualitative recognition.
In psychological terms this entails the ceaseless interaction of both (conscious) rational and (unconscious) intuitive aspects of understanding which cannot be reduced in terms of each other.
However the unconscious qualitative aspect can be indirectly translated in a circular rational fashion that seems paradoxical in terms of conventional (linear) reason.
Even more remarkably, this circular understanding can be further indirectly translated in a linear rational fashion (that is imaginary).
So, just as we have real and imaginary aspects to number in quantitative terms; equally we have real and imaginary aspects to number (in qualitative terms).
As I have stated the key aspect with respect to number thereby points to the ultimate nature of the relationship of its quantitative (analytic) to its qualitative (holistic) aspect and this is what the Riemann Hypothesis is essentially all about!
The Riemann Zeta Function is defined, as we know in conventional mathematics terms, with respect to the complex plane (where real and imaginary possess a merely quantitative meaning).
However properly understood this complex plane should be equally defined so that real and imaginary are also given distinctive qualitative meanings.
Now from the quantitative perspective the one value where the Function remains underfined is for s (the dimensional value) = 1.
However this equally has an interpretation in qualitative terms which means that the Riemann Zeta Function remains uniquely undefined fo s = 1 (i.e. in terms of conventional linear rational understanding).
What this implies is that the Riemann Function entails the relationship - with respect to the Functional Equation - of - relatively - quantitative (analytic) on one side of the Zeta Function > .5 with qualitative (holistic) meaning on the other side < .5.
So the Riemann Hypothesis where the real part of s = .5 relates to the point where analytic (quantitative) and holistic (qualitative) meaning directly coincides.
So again the reason why the Zeta Function remains uniquely undefined in qualitative 1-dimensional terms is because this represents the one (and only) dimension where mathematical meaning is exclusively defined in a merely analytic (quantitative) manner.
Expressed another way, one aspect is properly geared towards interpretation of the cardinal aspect and the other towards the ordinal aspect of number respectively.
And as stated here this requires two distinct interpretations with respect to the number system which I refer to as Type 1 and Type 2 respectively.
Whereas the Type 1 relates directly to the (recognised) conscious aspect of linear rational interpretation, the Type 2 relates to the (unrecognised) unconscious aspect, which indirectly can be given a rational interpretation in a circular logical manner.
Properly understood therefore the actual experience of number entails the dynamic interaction of both conscious and unconscious in experience entailing in turn the dynamic interaction of the analytic and holistic aspects of number (entailing both linear and circular logical type interpretation).
As we have seen the inherent nature of addition and multiplication relate to both Type 1 and Type 2 aspects of number understanding respectively.
Therefore the key point once again is that we cannot possibly come to grips with the deeper issues underlying prime numbers and the Riemann Hypothesis from within the conventional mathematical perspective (which is based solely on a reduced absolute Type 1 interpretation).
Now I will seek here to illustrate these points with respect to the number 2, appreciation of which is sufficient to grasp the essential nature of the Riemann Hypothesis.
Now the standard Type 1 interpretation is based on the notion of 2 as representing a collective whole unit in a merely quantitative manner.
So if we try here to subdivide into its individual units we get 1 + 1, where the separate units are treated as fully homogenous (i.e. lacking any qualitative distinction).
So 2 in this context more comprehensively means 2 ^ 1 i.e. where 2 is defined in linear terms with respect to the default 1st dimension.
Therefore geometrically, numbers in Type 1 terms are represented as points plotted on the 1-dimensional line. So when we consider a number initially expressed with respect to a different dimensional value, its ultimate value is expressed in a merely reduced linear (1-dimensional) fashion.
For example from this perspective 2^2 = 4 i.e. (4 ^ 1).
In another words, though strictly a qualitative distinction is now involved with 2 ^ 2 representing 4 square (2-dimensional) units, this qualitative transformation is simply ignored with the result expressed in a reduced linear (i.e. merely quantitative) manner.
Now I have raised the fundamental problem with this approach in that in ignoring the qualitative distinction (expressed through the dimensional number) that no means in fact exist for making ordinal distinctions as between numbers.
In other words if we seek to interpret numbers in a merely quantitative manner, then we have no basis for making ordinal rankings (which assumes a qualitative relationship as between numbers).
Again though this as I say is absolutely fundamental and strictly undermines the very basis of all conventional mathematical interpretation, it remains completely ignored within Mathematics. This is why I have long been of the view that such Mathematics - despite its admitted great advances - is simply not fit for purpose (as it is based on highly reduced and ultimately untenable assumption regarding the very nature of its relationships).
So the alternative Type 2 interpretation of number starts from the opposite basis of viewing each number with respect to its individual members in distinct ordinal terms.
So from this perspective the number 2 comprises a 1st and 2nd member (each of which is qualitatively distinct).
However as this understanding is of a very distinct nature it cannot be reflected through Type 1 understanding.
So here we reverse the relationship as between number (as base quantity) and number (as dimension).
Thus in the Type 1 system where 2 = 2 ^ 1, 2 represents the base quantity and 1 the (default) dimension.
However in the Type 2 system, 2 = 1 ^ 2 so that now 1 represents the (default) base quantity and 2 the dimensional number (expressing qualitative distinction).
So again for example if we square the number 1, no quantitative change in the unit takes place; however a qualitative change to square (2-dimensional) units occurs.
And in the Type 2 we concentrate on the nature of such qualitative change!
To highlight the qualitative nature of this system we must switch to a circular type representation of number.
So the number 2 as dimensional number comprises two distinct ordinal members (i.e. 1st and 2nd).
Now an inverse relationship exists as between the qualitative nature of these units and the indirect (reduced) quantitative interpretation.
So the inverse of the 1st member as first root (where the number 1 now represents ordinal identity) remains unchanged as 1 ^ 1 = 1; however the inverse of the 2nd member as 2nd root (where 2 now represents ordinal identity) 1^ (1/2) = – 1.
So what we have achieved here - through the two roots of 1 - is an indirect quantitative means (on a circular number scale) of expressing the unique identity of the two ordinal members of 2.
And of course by extension we can express - again in an indirect quantitative manner - the unique identity of all the ordinal members of any number n by taking the corresponding n roots of 1!
So the various roots of 1 possess an as yet unrecognised significance in providing the indirect quantitative means of uniquely identifying the ordinal nature of number with respect to any group.
I have raised this key issue before as to how the meaning of 2nd for example is purely relative depending on the size of the group to which it belongs! So this problem of giving 2 a distinctive ordinal meaning is solved by considering the value of the 2nd of n roots (where n can be any natural number).
So each ordinal member of a group is given a - relatively - separate (indirect) quantitative identity; however the true holistic nature of the group requires the interdependence of these members in a qualitative manner.
And as we have seen the sum of all n roots of 1 (except where n = 1) = 0. So the qualitative interdependence of a group requires the (circular) combination of all its separate members.
Again I have illustrated before the nature of this interdependence with respect to road directions.
However as it fruitfully illustrates the true Type 2 nature of number, I will do so again.
Clearly left and right designations at a crossroads have a merely relative identity depending on the direction from which the crossroads is defined.
So if we approach a crossroads travelling towards it in an upward northerly direction, we can unambiguously identify for example, a left turn (which we can designate as + 1).
In this context the opposite right turn can be designated as – 1.
However this designation has a merely relative identity (depending on the arbitrary direction from which the crossroads is approached). So from the opposite direction (approaching the crossroads while moving in a downwards southerly direction) once again a left turn has an unambiguous meaning (as + 1) and a right (as – 1).
However this designation of left and right will be precisely the opposite as in the first case.
So the pure holistic appreciation of the interdependence of the directions (left and right) requires the ability to simultaneous combine both reference frames where what is left from one perspective is right from the other; and what is right from one is left from the other.
So in number terms, the simultaneous qualitative interdependence of both can be expressed as (+) 1 – 1 = 0 (or alternatively – 1 + 1 = 0).
So this apparently simple example of a crossroads implicitly entails the Type 2 appreciation of the nature of 2.
So each member of the group can be given a relatively separate ordinal identity in an indirect quantitative manner (i.e. within a given frame of reference) as + 1 and – 1 respectively.
However the overall holistic identity of the two members in qualitative terms requires the simultaneous recognition of both reference frames (creating paradox from a linear rational perspective).
In this way the quantitative recognition of the two members of 2 as separate, is cancelled out in pure holistic recognition of their qualitative interdependent nature.
Again I have often used the example of a crossroads to illustrate the nature of polar reference frames.
However such polarised reference frames are the very nature of experience (which is inevitably conditioned with respect to all phenomenal understanding by external and internal, and equally quantitative and qualitative polarities).
So when we look at the number system in an appropriate dynamic manner, we recognise that a ceaseless interaction as between these opposite polarities is necessarily involved.
Therefore any distinct knowledge of a quantitative nature is merely relative, and ultimately becomes completely cancelled out through a pure holistic qualitative recognition.
In psychological terms this entails the ceaseless interaction of both (conscious) rational and (unconscious) intuitive aspects of understanding which cannot be reduced in terms of each other.
However the unconscious qualitative aspect can be indirectly translated in a circular rational fashion that seems paradoxical in terms of conventional (linear) reason.
Even more remarkably, this circular understanding can be further indirectly translated in a linear rational fashion (that is imaginary).
So, just as we have real and imaginary aspects to number in quantitative terms; equally we have real and imaginary aspects to number (in qualitative terms).
As I have stated the key aspect with respect to number thereby points to the ultimate nature of the relationship of its quantitative (analytic) to its qualitative (holistic) aspect and this is what the Riemann Hypothesis is essentially all about!
The Riemann Zeta Function is defined, as we know in conventional mathematics terms, with respect to the complex plane (where real and imaginary possess a merely quantitative meaning).
However properly understood this complex plane should be equally defined so that real and imaginary are also given distinctive qualitative meanings.
Now from the quantitative perspective the one value where the Function remains underfined is for s (the dimensional value) = 1.
However this equally has an interpretation in qualitative terms which means that the Riemann Zeta Function remains uniquely undefined fo s = 1 (i.e. in terms of conventional linear rational understanding).
What this implies is that the Riemann Function entails the relationship - with respect to the Functional Equation - of - relatively - quantitative (analytic) on one side of the Zeta Function > .5 with qualitative (holistic) meaning on the other side < .5.
So the Riemann Hypothesis where the real part of s = .5 relates to the point where analytic (quantitative) and holistic (qualitative) meaning directly coincides.
So again the reason why the Zeta Function remains uniquely undefined in qualitative 1-dimensional terms is because this represents the one (and only) dimension where mathematical meaning is exclusively defined in a merely analytic (quantitative) manner.
Sunday, March 3, 2013
Mysterious Role of 6 and 12!
I mentioned in a previous entry how a very simple formula could be used to calculate to a very high degree of accuracy the deviation from 2/π (= i/ ln i) of both the (reduced) average cos and sin values of the n roots of 1.
I then illustrated this with respect to the prime number n = 53!
So the formula again for the cos part deviation
I then illustrated this with respect to the prime number n = 53!
So the formula again for the cos part deviation
= π/12(n ^ 2) and the sin part = π/6(n ^ 2).
Having written this piece I then searched for an underlying explanation of this phenomenon and now can provide a clue to what is involved.
As we know for non-trivial zeros with respect to the Zeta 1 Function that the Riemann Hypothesis maintains that the real part = 1/2.
Interestingly when we obtain the 6th root of 1, the real part of the non-trivial zero for the Zeta 2 (i.e. cos 60) likewise = 1/2.
Then when we obtain the 12th root of 1, the imaginary part of the non-trivial zero (i.e. sin 30) = 1/2!
So it does seem that this very fact (which is unique in the case of both the 6th and 12th roots of 1 respectively) indeed represents the underlying explanation with a mirror significance to that of the Zeta 1 Function.
Some time ago I explained how the Zeta 2 distribution can be used to explain values of the Zeta 1 distribution, where real non-intuitive values for the Function arise with the negative odd integers where s = – 1, – 3, – 5, – 7, – 9 etc.
For example when. ζ (– 1) = – 1/12 the denominator is divisible by 12 (and of course 6).
When ζ (– 3) = 1/120 the denominator is again divisible by 12 (and 6).
When ζ (– 5) = – 1/252 the denominator is again divisible by 12 (and 6).
When ζ (– 7) = 1/240 the denominator is divisible by 12 (and 6).
And when ζ (– 9) = – 1/132 with once again the denominator divisible by 12 (and 6).
Indeed this is a feature that seems to universally hold for all denominator values (where s is a negative odd integer).
This would not only strongly support my contention that the Zeta 2 Function is embedded as it were in the Zeta 1 (for values of s < 0). It would also strongly suggest that the underlying explanation lies in the unique nature for the 6th and 12th root of 1 (where the real and the imaginary parts = 1/2 respectively).
For some time now I have also been aware of the fact that apart from the first two (3 and 5) the sum of twin primes always appear to be divisible by 12 (and thereby of course 6).
For example 5 + 7 = 12 (divisible by 12 and 6)
11 + 13 = 24 (divisible by 12 and 6)
17 + 19 = 36 (divisible by 12 and 6)
29 + 31 = 60 (divisible by 12 and 6)
41 + 43 = 84 (divisible by 12 and 6).
As I say this pattern continues and appears to be universal.
This would again suggest that perhaps the same underlying reason applies (ultimately tied to the fact that the real and imaginary parts of the 6th and 12th roots of 1 = 1/2 respectively).
This would also strongly indicate (as I have long suspected) that the Twin Prime Hypothesis is intimately tied to the Riemann Hypothesis.
Therefore as the Riemann Hypothesis can have no solution (in conventional mathematical terms) this would imply that likewise the Twin Prime Hypothesis can have no solution!
Having written this piece I then searched for an underlying explanation of this phenomenon and now can provide a clue to what is involved.
As we know for non-trivial zeros with respect to the Zeta 1 Function that the Riemann Hypothesis maintains that the real part = 1/2.
Interestingly when we obtain the 6th root of 1, the real part of the non-trivial zero for the Zeta 2 (i.e. cos 60) likewise = 1/2.
Then when we obtain the 12th root of 1, the imaginary part of the non-trivial zero (i.e. sin 30) = 1/2!
So it does seem that this very fact (which is unique in the case of both the 6th and 12th roots of 1 respectively) indeed represents the underlying explanation with a mirror significance to that of the Zeta 1 Function.
Some time ago I explained how the Zeta 2 distribution can be used to explain values of the Zeta 1 distribution, where real non-intuitive values for the Function arise with the negative odd integers where s = – 1, – 3, – 5, – 7, – 9 etc.
For example when. ζ (– 1) = – 1/12 the denominator is divisible by 12 (and of course 6).
When ζ (– 3) = 1/120 the denominator is again divisible by 12 (and 6).
When ζ (– 5) = – 1/252 the denominator is again divisible by 12 (and 6).
When ζ (– 7) = 1/240 the denominator is divisible by 12 (and 6).
And when ζ (– 9) = – 1/132 with once again the denominator divisible by 12 (and 6).
Indeed this is a feature that seems to universally hold for all denominator values (where s is a negative odd integer).
This would not only strongly support my contention that the Zeta 2 Function is embedded as it were in the Zeta 1 (for values of s < 0). It would also strongly suggest that the underlying explanation lies in the unique nature for the 6th and 12th root of 1 (where the real and the imaginary parts = 1/2 respectively).
For some time now I have also been aware of the fact that apart from the first two (3 and 5) the sum of twin primes always appear to be divisible by 12 (and thereby of course 6).
For example 5 + 7 = 12 (divisible by 12 and 6)
11 + 13 = 24 (divisible by 12 and 6)
17 + 19 = 36 (divisible by 12 and 6)
29 + 31 = 60 (divisible by 12 and 6)
41 + 43 = 84 (divisible by 12 and 6).
As I say this pattern continues and appears to be universal.
This would again suggest that perhaps the same underlying reason applies (ultimately tied to the fact that the real and imaginary parts of the 6th and 12th roots of 1 = 1/2 respectively).
This would also strongly indicate (as I have long suspected) that the Twin Prime Hypothesis is intimately tied to the Riemann Hypothesis.
Therefore as the Riemann Hypothesis can have no solution (in conventional mathematical terms) this would imply that likewise the Twin Prime Hypothesis can have no solution!
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