A distinct form of mathematical understanding is associated with each of the 3 levels of Band 5.
Now even before we can consider the primes we must already admit the existence of the two original numbers i.e. 1 and 0.
And the importance of these two numbers i.e. the binary digits cannot be overstated.
As is well-known, the IT revolution which is now having such an impact on so many aspects of modern living is intimately related to the binary digits 1 and 0 as a means of potentially encoding all data. Thus all information can be converted in a binary digital manner.
And in this this context we are referring to the quantitative (analytic) use of these two digits where 1 is clearly separated from 0.
However what is not at all well known is that these same two digits can equally be given a qualitative (holistic) meaning, where ultimately they become entirely interdependent with each other.
This latter meaning is reflected very well in some mystical traditions where (phenomenal) form becomes ultimately identical with (spiritual) emptiness.
Now if we rephrase this slightly it entails that ultimately the unity of all form (1) is indistinguishable from nothingness (0) as the creative void from which phenomena potentially emerge.
And just as all information can be encoded in an (analytic) binary digital, likewise all transformation processes can be likewise potentially encoded in a (holistic) binary digital manner.
So the main task that absorbed my attention during Level 1 (Band 5) was the ambitious attempt to encode the full spectrum with respect to the stages of development (with complementary physical and psychological aspects) in a holistic binary digital fashion.
Now basically 1, which geometrically resembles the line can be used to represent conscious linear type development and 0 which resembles (with small variations) the circle can be used to represent corresponding circular development (as the indirect paradoxical expression of unconscious type appreciation).
And whereas differentiation in development is related directly to the linear, integration by contrast is directly related to the circular aspect.
So in using this binary digital approach, all stages can be shown to represent a certain unique configuration with respect to both 1 (differentiation) and 0 (integration) respectively.
Thus when applied to the 7 major bands (with 21 major levels), development starts from a state of total confusion with respect to both 1 (differentiation) and 0 (integration) respectively.
The first major task (Band 1) is with respect to the successful differentiation of the various structures (i.e. where 1 is separated from 0). Then the specialisation of these differentiated structures (1) takes place at Band 2. Once again it is important to recognise that conventional interpretation of Mathematics is exclusively based on this band!
Then Band 3 is now concerned with the gradual integration of states and structures (earlier differentiated). So this represents the journey from 1 to 0. Band 4 then relates to the corresponding specialisation of these same states and structures. (0).
Band 5 now represents the attempt to gradually combine 0 with 1 in a mature fashion. Finally Band 6 and Band 7 relate to the mature specialisation of both differentiated and integrated structures (1 and 0).
This represented therefore my first clear statement that reality is dynamically encoded in mathematical manner with the binary digital approach representing the most universal holistic means of satisfactorily expressing all development.
This task chiefly occupied me from the years 2001 to 2008 (approx.). Some of the fruits of this labour can be found at "The Stages of Development".
Now if we move on a little, the final breakthrough comes at Band 6, where - what I refer to as - the twin binary system can be jointly used as both a means of encoding information and transformation.
Indeed from one very important perspective, all reality can then be viewed as representing in fact such a dynamic twin-based binary system.
Now the great concern I would have regarding our present world is that it is becoming increasingly subject to changes directly or indirectly brought about through the rapid pace of IT developments.
However at root this IT revolution solely relates to information. Though it is certainly bringing about the consequent need for rapid transformation in social, political and economic terms, it runs directly counter to the requirements for authentic transformation.
So what we are seeing is adaptation to many changes that is increasingly of a superficial nature. Indeed we are now reaching the stage in developed societies where democracy itself is slowly being undermined due to an inability to achieve the required transformation in an authentic manner.
This is why I believe that a radical change is now urgently required in our perspective on Mathematics, with the hidden holistic aspect of all its symbols at last properly realised.
For in truth though the present approach is certainly highly specialised in a quantitative manner, it is likewise extremely unbalanced. This in turn fosters a mind-set that is deeply inimical with respect to providing a truly integral approach to dealing with so many of our problems.
However before fully completing my binary approach, I became completely absorbed by a new problem, which I already suspected reached into the deepest recesses of unconscious development. This was the famed Riemann Hypothesis and the profound issues it poses with respect to the nature of the primes and their relationship with the natural numbers.
This constituted Level 2 (Band 5). In fact most of the blog entries I have contributed since 2009, which approach 1 million words, relate directly to this issue.
Now, because I have dealt with it at considerable length - at least with respect to the issues that I felt especially important to pursue - I will not repeat it all again.
Suffice it to say that my initial expectations were vindicated to a remarkable degree in that proper understanding of the nature of the number system reaches into the deepest level of unconscious understanding both with respect to "higher" super-conscious and "lower" sub-conscious development.
So what was eventually revealed was the hidden qualitative basis of both the ordinal and cardinal number systems respectively.
Properly understood, we cannot hope to understood either system in an absolute quantitative type manner (based on mere rational notions). Rather the number system, both with respect to its ordinal and cardinal aspects must be conceived in a dynamic relative fashion, representing the interaction of quantitative (analytic) and holistic (qualitative) aspects. Thus absolute conventional interpretation represents just one special limiting case.
And whereas the Zeta 2 zeros represent the indirect quantitative means of expressing the qualitative basis of the ordinal aspect, the Zeta 1 (Riemann) zeros represent the indirect quantitative means of expressing the corresponding qualitative basis of the cardinal number system.
Perhaps the biggest discovery here was the realisation that the mathematical task of understanding the true nature of the zeros (Zeta 1 and Zeta 2) and the psychological task of attempting to achieve full integration of conscious with unconscious are really one and the same.
So the zeros (representing different energy states) in both cases, express - again in an indirect quantitative manner - the very manner through which the unconscious achieves integration with the conscious mind. And this in turn - as already discovered with respect to the Zeta 1 zeros - has complementary links with the physical world with respect to the different energy levels of atoms.
Thus when one properly understands the dynamic interaction of the primes with the natural numbers, one recognises that they are ultimately completely interdependent with each other in an ineffable manner. And this likewise applies to the zeta zeros which dynamically mediate as between both aspects. So ultimately the number system - with respect to its cardinal and ordinal aspects - behaves in a fully synchronistic dynamic manner.
Further development then takes place at Level 3 (Band 5).
One of the tasks here is to achieve better integration of the earlier binary approach with the new prime/natural number based approach.
For it has to be acknowledged that 1 and 0 in a very important sense precede the primes.
Though the binary system does indeed provide for the most basic way of encoding all quantitative and qualitative type interactions, it does not enable the same diversity that then follows from the prime/natural number approach.
So ultimately all the rich variety in nature with respect to both quantitative and qualitative characteristics are rooted in the latter number system. However even here the binary numbers fully underlie this diversity. So just as the binary numbers need to be decoded to reveal the rich diversity of information data, likewise the double binary system needs to be decoded, using the two-way relationship as between the primes and the natural numbers, to reveal the rich diversity of all phenomenal reality (in quantitative and qualitative terms).
Even with respect to the Zeta 1 and Zeta 2 zeros, we can see clearly the influence of 1 and 0.
First of all it is the zero solutions to the equations (= 0) that are especially relevant in both cases.
Then with respect to the Zeta 2, the collective sum of quantitative roots of 1 (as the indirect quantitative expression of ordinal members of a number group) = 0. Then, in complementary fashion, each quantitative part of the corresponding cardinal number = 1.
Also whereas the sum of roots = 0, the product of roots = 1 (+ 1 for an odd number and – 1 for an even number)
Then in relation to the Zeta 2 solutions, these come in conjugate pairs as .5 + it and .5 – it respectively. So when one adds the two, the result = 1.
Another advance at Level 3 is in relation to one's understanding of how the cardinal and ordinal systems interact.
At Level 2, one tends still to think of both in a relatively separate manner. However two-way dynamic interaction between both aspects takes place at Level 3 so that one clearly realises that the ordinal system has no strict meaning in the absence of the cardinal (and that likewise the cardinal has no meaning in the absence of the ordinal).
And in psychological terms this further entails the ability to appreciate both sets of Zeros (Zeta 1 and Zeta 2) and the primes and natural numbers in cognitive and affective terms.
Now even before we can consider the primes we must already admit the existence of the two original numbers i.e. 1 and 0.
And the importance of these two numbers i.e. the binary digits cannot be overstated.
As is well-known, the IT revolution which is now having such an impact on so many aspects of modern living is intimately related to the binary digits 1 and 0 as a means of potentially encoding all data. Thus all information can be converted in a binary digital manner.
And in this this context we are referring to the quantitative (analytic) use of these two digits where 1 is clearly separated from 0.
However what is not at all well known is that these same two digits can equally be given a qualitative (holistic) meaning, where ultimately they become entirely interdependent with each other.
This latter meaning is reflected very well in some mystical traditions where (phenomenal) form becomes ultimately identical with (spiritual) emptiness.
Now if we rephrase this slightly it entails that ultimately the unity of all form (1) is indistinguishable from nothingness (0) as the creative void from which phenomena potentially emerge.
And just as all information can be encoded in an (analytic) binary digital, likewise all transformation processes can be likewise potentially encoded in a (holistic) binary digital manner.
So the main task that absorbed my attention during Level 1 (Band 5) was the ambitious attempt to encode the full spectrum with respect to the stages of development (with complementary physical and psychological aspects) in a holistic binary digital fashion.
Now basically 1, which geometrically resembles the line can be used to represent conscious linear type development and 0 which resembles (with small variations) the circle can be used to represent corresponding circular development (as the indirect paradoxical expression of unconscious type appreciation).
And whereas differentiation in development is related directly to the linear, integration by contrast is directly related to the circular aspect.
So in using this binary digital approach, all stages can be shown to represent a certain unique configuration with respect to both 1 (differentiation) and 0 (integration) respectively.
Thus when applied to the 7 major bands (with 21 major levels), development starts from a state of total confusion with respect to both 1 (differentiation) and 0 (integration) respectively.
The first major task (Band 1) is with respect to the successful differentiation of the various structures (i.e. where 1 is separated from 0). Then the specialisation of these differentiated structures (1) takes place at Band 2. Once again it is important to recognise that conventional interpretation of Mathematics is exclusively based on this band!
Then Band 3 is now concerned with the gradual integration of states and structures (earlier differentiated). So this represents the journey from 1 to 0. Band 4 then relates to the corresponding specialisation of these same states and structures. (0).
Band 5 now represents the attempt to gradually combine 0 with 1 in a mature fashion. Finally Band 6 and Band 7 relate to the mature specialisation of both differentiated and integrated structures (1 and 0).
This represented therefore my first clear statement that reality is dynamically encoded in mathematical manner with the binary digital approach representing the most universal holistic means of satisfactorily expressing all development.
This task chiefly occupied me from the years 2001 to 2008 (approx.). Some of the fruits of this labour can be found at "The Stages of Development".
Now if we move on a little, the final breakthrough comes at Band 6, where - what I refer to as - the twin binary system can be jointly used as both a means of encoding information and transformation.
Indeed from one very important perspective, all reality can then be viewed as representing in fact such a dynamic twin-based binary system.
Now the great concern I would have regarding our present world is that it is becoming increasingly subject to changes directly or indirectly brought about through the rapid pace of IT developments.
However at root this IT revolution solely relates to information. Though it is certainly bringing about the consequent need for rapid transformation in social, political and economic terms, it runs directly counter to the requirements for authentic transformation.
So what we are seeing is adaptation to many changes that is increasingly of a superficial nature. Indeed we are now reaching the stage in developed societies where democracy itself is slowly being undermined due to an inability to achieve the required transformation in an authentic manner.
This is why I believe that a radical change is now urgently required in our perspective on Mathematics, with the hidden holistic aspect of all its symbols at last properly realised.
For in truth though the present approach is certainly highly specialised in a quantitative manner, it is likewise extremely unbalanced. This in turn fosters a mind-set that is deeply inimical with respect to providing a truly integral approach to dealing with so many of our problems.
However before fully completing my binary approach, I became completely absorbed by a new problem, which I already suspected reached into the deepest recesses of unconscious development. This was the famed Riemann Hypothesis and the profound issues it poses with respect to the nature of the primes and their relationship with the natural numbers.
This constituted Level 2 (Band 5). In fact most of the blog entries I have contributed since 2009, which approach 1 million words, relate directly to this issue.
Now, because I have dealt with it at considerable length - at least with respect to the issues that I felt especially important to pursue - I will not repeat it all again.
Suffice it to say that my initial expectations were vindicated to a remarkable degree in that proper understanding of the nature of the number system reaches into the deepest level of unconscious understanding both with respect to "higher" super-conscious and "lower" sub-conscious development.
So what was eventually revealed was the hidden qualitative basis of both the ordinal and cardinal number systems respectively.
Properly understood, we cannot hope to understood either system in an absolute quantitative type manner (based on mere rational notions). Rather the number system, both with respect to its ordinal and cardinal aspects must be conceived in a dynamic relative fashion, representing the interaction of quantitative (analytic) and holistic (qualitative) aspects. Thus absolute conventional interpretation represents just one special limiting case.
And whereas the Zeta 2 zeros represent the indirect quantitative means of expressing the qualitative basis of the ordinal aspect, the Zeta 1 (Riemann) zeros represent the indirect quantitative means of expressing the corresponding qualitative basis of the cardinal number system.
Perhaps the biggest discovery here was the realisation that the mathematical task of understanding the true nature of the zeros (Zeta 1 and Zeta 2) and the psychological task of attempting to achieve full integration of conscious with unconscious are really one and the same.
So the zeros (representing different energy states) in both cases, express - again in an indirect quantitative manner - the very manner through which the unconscious achieves integration with the conscious mind. And this in turn - as already discovered with respect to the Zeta 1 zeros - has complementary links with the physical world with respect to the different energy levels of atoms.
Thus when one properly understands the dynamic interaction of the primes with the natural numbers, one recognises that they are ultimately completely interdependent with each other in an ineffable manner. And this likewise applies to the zeta zeros which dynamically mediate as between both aspects. So ultimately the number system - with respect to its cardinal and ordinal aspects - behaves in a fully synchronistic dynamic manner.
Further development then takes place at Level 3 (Band 5).
One of the tasks here is to achieve better integration of the earlier binary approach with the new prime/natural number based approach.
For it has to be acknowledged that 1 and 0 in a very important sense precede the primes.
Though the binary system does indeed provide for the most basic way of encoding all quantitative and qualitative type interactions, it does not enable the same diversity that then follows from the prime/natural number approach.
So ultimately all the rich variety in nature with respect to both quantitative and qualitative characteristics are rooted in the latter number system. However even here the binary numbers fully underlie this diversity. So just as the binary numbers need to be decoded to reveal the rich diversity of information data, likewise the double binary system needs to be decoded, using the two-way relationship as between the primes and the natural numbers, to reveal the rich diversity of all phenomenal reality (in quantitative and qualitative terms).
Even with respect to the Zeta 1 and Zeta 2 zeros, we can see clearly the influence of 1 and 0.
First of all it is the zero solutions to the equations (= 0) that are especially relevant in both cases.
Then with respect to the Zeta 2, the collective sum of quantitative roots of 1 (as the indirect quantitative expression of ordinal members of a number group) = 0. Then, in complementary fashion, each quantitative part of the corresponding cardinal number = 1.
Also whereas the sum of roots = 0, the product of roots = 1 (+ 1 for an odd number and – 1 for an even number)
Then in relation to the Zeta 2 solutions, these come in conjugate pairs as .5 + it and .5 – it respectively. So when one adds the two, the result = 1.
Another advance at Level 3 is in relation to one's understanding of how the cardinal and ordinal systems interact.
At Level 2, one tends still to think of both in a relatively separate manner. However two-way dynamic interaction between both aspects takes place at Level 3 so that one clearly realises that the ordinal system has no strict meaning in the absence of the cardinal (and that likewise the cardinal has no meaning in the absence of the ordinal).
And in psychological terms this further entails the ability to appreciate both sets of Zeros (Zeta 1 and Zeta 2) and the primes and natural numbers in cognitive and affective terms.
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