We will finally, in this blog entry, further probe the precise holistic significance of the Riemann Hypothesis, where all non-trivial zeros of the zeta function are postulated to lie on the imaginary line through 1/2.
The significance of 1/2 is fascinating from a holistic context. What it in fact entails is that the notion of mathematical truth in objective terms is dynamically inseparable from the corresponding notion of this same truth as (subjective) interpretation!
Though from the conventional mathematical perspective, we are accustomed to interpreting number objects in an absolute objective fashion, this makes no sense in experiential terms. And remember all mathematical understanding is necessarily experiential in nature!
So for example when one becomes objectively aware of a number - say "2" - this necessarily corresponds with the mental perception of "2". So strictly speaking in experiential terms, "2" represents a dynamic interaction governed by two reference poles that are - relatively - external and internal with respect to each other.
So, if we refer to the external (objective) pole in quantitative terms, then the corresponding internal (subjective) pole as perception is - relatively - qualitative in nature.
So once again the understanding of "2", in our lived experience, properly entails both quantitative and qualitative aspects in dynamic interaction with each other.
Now the very nature of conventional mathematical interpretation is the attempt to freeze this interaction in an absolute manner (through recognition of just one polar reference frame).
This then leads to the mistaken notion that mathematical objects enjoy an absolute existence in an objective space (which can be - literally - abstracted from experience).
So "2" is now defined absolutely in a merely static objective manner.
If pressed, a mathematician may reluctantly agree that one cannot form knowledge of a number object, without a corresponding mental construct!
However any consideration of the necessary dynamic interaction as between object and construct will then be quickly dismissed as irrelevant in a mathematical context.
Thus in effect the assumption is made that the mental constructs absolutely correspond with their objective counterparts.
In other words the qualitative aspect of appreciation is thereby reduced in a mere quantitative manner!
Though it clearly makes no sense to insist that these two polarities of understanding (external and internal) can be absolutely separated in an independent manner from each other, this is the unwarranted key assumption that currently underlies all accepted mathematical interpretation.
For even when this assumption (of the abstract nature of mathematical objects) is insisted on in formal terms, the actual experience of understanding implicitly requires that some dynamic relationship (however small) be maintained as between external and internal poles.
So properly understood, the standard analytic approach to Mathematics really represents just one limiting extreme, where one attempts to approach as close as possible to an absolute type appreciation of mathematical objects (which however always remains at the level of relative approximation)!
So again even though one may interpret "2" as representing a number with an absolute identity, this can never be strictly true, for the very nature of mathematical communication represents a form of social consensus that can only be approximate.
Indeed it should be very clear that in this discussion, I am directly challenging the very nature of this consensus. This of course does not mean that satisfactory communication still cannot take place (within well-defined contexts). However once again, mathematical truth remains always of a relative - rather than absolute - nature.
So the analytic extreme requires the attempt to keep that opposite poles that underlie all phenomenal experience (such as external/internal and whole/part) as independent as possible.
Thus once again, in any context, just one pole of reference is used in an unambiguous manner, with relationships now literally conforming to analysis through linear (i.e. 1-dimensional) reason .
However, the holistic extreme represents the complete opposite;
Here, rather than concentrating on the relative independence of opposite poles (such as external and internal) one rather focuses on their complementary identity (through interdependence). Then in extremes - which conforms directly to highly refined intuitive understanding - one approaches a pure energy state (largely free of any phenomenal notion of number).
Attaining this nondual state - which again can only be approximated in relative manner in experience - requires achieving the golden mean as between the two poles (external and internal) so that both are harmonised as close as possible with each other.
As we have seen, 1-dimensional entails sole recognition of just the pole (i.e. the external pole of "objective" quantitative recognition).
Now in a wonderful holistic manner, 1/2 represents the equal splitting of this dimension as between both the external (quantitative) and internal (qualitative) poles.
So the true holistic significance of the fact that the non-trivial zeros are postulated to all lie on the line through 1/2, is that this condition is necessary to ensure the identity of both the quantitative and qualitative aspects of number recognition.
And what is vital to appreciate is that this requirement can only be properly appreciated in a holistic intuitive manner (where both quantitative and qualitative aspects of recognition are directly harmonised in experience).
As I explained in an earlier blog entry, we can then indirectly translate this intuitive recognition in a circular logical fashion, through paradox.
So once again, the zeros represent points on an imaginary number line, where both the primes and natural numbers are identical, where randomness and order with respect to the number system are identical, where the cardinal and ordinal interpretation of number is identical and finally where addition is identical with multiplication. And all of this results from being simultaneously able to "see" number relationships with respect to two complementary poles of reference!
However I must stress once again that conventional mathematical ability (of an analytic nature) is of no avail in this regard. Indeed as it is based in extreme fashion on just one independent pole of recognition it is only likely to hinder such ability.
So again, Holistic Mathematics represents an utterly distinctive form of mathematical recognition, whose very development initially requires substantial initial withdrawal from the unfortunate "indoctrination" imposed through conventional mathematical procedures.
I know very well from experience how deeply ingrained is the conventional mathematical "ideology" in our culture.
But for the fact that I formed a strong conviction at the age of 10, regarding a fundamental unaddressed problem with respect to multiplication, I doubt if my "conversion" to Holistic Mathematics could have readily taken place.
However though this has led to many wonderful discoveries, that have greatly enriched my life, it has always remained very much a solo voyage. In fact, remarkably few are yet prepared to seriously question the key construction faults that deeply underlie the mathematical edifice.
I will address now briefly the holistic reason why the zeros lie on an "imaginary" line.
I have already mentioned that one can indirectly express the holistic intuitive appreciation of the nature of the zeros indirectly in a circular logical fashion (through rational paradox).
Now in holistic terms, the "imaginary" line is the appropriate means to express notions of an inherently circular nature in an acceptable linear manner.
When I was addressing the numerical values of the zeros I showed that they were closely related to the divisors (natural factors) of each number.
However this required converting from a circular scale (represented by the circumference of a circle) to a corresponding linear scale (represented by the radius).
In similar holistic fashion, zeros that lie on an "imaginary" line represent a means through rational discourse of expressing the fact that these points conform to a different logical system (i.e. that is holistic - rather than analytic - in nature).
In normal discourse "imaginary" - as opposed to "real" - is generally used to indicate what is of unconscious origin. It is very similar in holistic mathematical terms.
So the postulate that all the zeros lie on an "imaginary" line implies that the holistic unconscious basis of the (cardinal) number system is fully consistent with its corresponding analytic interpretation. Again this is an assumption that is already implicit in the use of the standard axioms of Conventional Mathematics! (In other words, the truth of the Riemann Hypothesis is assumed in the these axioms, which means of course that it supersedes them and cannot be thereby proved or disproved through their use). Thus once again, the Riemann Hypothesis points to the key requirement that both quantitative and qualitative aspects of interpretation can be fully reconciled with respect to the number system.
Finally I will briefly address the fact that the non-trivial zeros can be given negative as well as positive values.
This once again points to the pure holistic nature of these zeros.
In holistic terms, the fact that each positive zero has a matching negative value, implies that in their very understanding the momentary positing of a zero (as consciously independent) must be immediately negated (in an unconscious manner). It is only then that its pure holistic nature (as a psychological energy state) can be continually maintained. And of course this has a complementary interpretation in terms of a physical energy state (within the atomic structure of matter).
So once again we are here at the holistic extreme to number (which is based on the intuitive recognition of the pure interdependence of number) which complements - in relative fashion - the analytic extreme (of its pure independence).
Friday, November 13, 2015
Thursday, November 12, 2015
Clarifying Analytic and Holistic Interpretations of "2" (19)
To put it simply, the Riemann (Zeta 1) zeros represent the (hidden) holistic basis of the cardinal natural number system (as explicitly understood in the standard analytic fashion).
From a corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same natural number system (as explicitly understood in a conscious rational manner).
Then in a parallel fashion, the alternative (Zeta 2) zeros represent the (hidden) holistic basis of the ordinal natural number system (as again explicitly understood in the standard analytic fashion).
And from the corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same ordinal natural number system (as explicitly understood in a conscious rational manner).
However because these two sets of zeros (Zeta 1 and Zeta 2) are dynamically complementary with the analytic aspects in a two-way interactive fashion, they are in truth both simultaneously involved as the hidden holistic basis with respect to the natural number system (in cardinal and ordinal terms).
All of this leads to the need for the most fundamental change possible with respect to the customary interpretation of number.
Rather than the number system existing in an unchanging absolute fashion in some universal "mathematical heaven", it has an inherently dynamic interactive nature, entailing both quantitative and qualitative aspects in relative fashion.
The ultimate nature of this system is synchronistic in a holistic manner, approaching a pure ineffable identity. It is then only at the opposite extreme of analytic type interpretation (within isolated polar reference frames) that it approaches the appearance of being composed of rigid number objects of form!
Crucially, as the true interpretation of number, with respect to its dynamic interactive nature, entails the balanced interaction of both conscious (rational) and unconscious (intuitive) aspects of understanding, it is simply not possible to appreciate its nature within the accepted confines of present Mathematics. As such Mathematics is formally interpreted within a merely (conscious) rational type framework, it inevitably reduces the qualitative aspect in a quantitative manner.
It thereby reduces the inherent dynamic nature of the number system in a limited static manner!
As the nature of number is so fundamental, underlying everything else in phenomenal creation, the greatest revolution yet in our intellectual history now awaits, where the unconscious aspect of understanding needs to be explicitly united with its conscious counterpart.
This revolution, which will require a great deal of time to unfold, will then not have only profound implications for Mathematics but also most importantly for all related sciences and for society generally, gradually leading to a much more comprehensive world view that at present would be impossible to envisage.
One may validly inquire as to the status of the Riemann Hypothesis from this new dynamic perspective on number.
In fact, its very nature in now transformed in a manner that cannot be appreciated within the conventional mathematical framework.
I have commented before many times on the important fact that from the analytic quantitative perspective, the Riemann zeta function is undefined at just one point, i.e. where s (representing a power or dimensional number) = 1.
The holistic counterpart of this is that the Riemann zeta function likewise is undefined for s = 1, where it now defines in qualitative terms the 1-dimensional approach to mathematical understanding.
And as Conventional Mathematics is formally defined in a 1-dimensional manner (i.e. within isolated polar reference frames) then the true significance of the Riemann zeta function simply cannot be appreciated from this standpoint.
In other words, properly understood, the Riemann zeta function maps out numerical values that correspond to both analytic (quantitative) and holistic (qualitative) interpretations of number respectively.
Basically values for the function on the RHS for values of s > 1 correspond to analytic interpretation that are mapped with corresponding holistic values on the LHS for values of s (< 0).
So through the Riemann functional equation, values of ζ(s) can be mapped with corresponding values of ζ(1 – s).
This means for example that ζ(2) is mapped with ζ(– 1).
Now ζ(2) = 1 + 1/4 + 1/9 + 1/16 + ..... which in the standard logic of numerical interpretation, does indeed converge to a finite value (π2/6).
Put another way, this entails that both the analytic (quantitative) and holistic (qualitative) interpretations of number coincide.
In fact, properly understood, the true significance of the Riemann Hypothesis is much greater than what is presently realised, for it in fact starkly reveals the enormous limitations imposed by the present conventional mathematical approach (which attempt to formally exclude qualitative considerations from all its procedures).
Thus the Riemann Hypothesis - when properly interpreted - expresses the mysterious requirement for the coincidence of quantitative with qualitative type interpretation of mathematical symbols.
From a psychological perspective, this relates to the requirement that both conscious (rational) and unconscious (intuitive) aspects of understanding can be synchronised in a consistent manner with respect to all mathematical interpretation.
This cannot be proved for conventional proof is already implicitly based on the assumption (that such synchronisation of meaning is already assured).
Thus correctly understood, all mathematical interpretation is implicitly based on an initial massive act of faith in the consistent correspondence of both the quantitative and qualitative meaning of its symbols !
From a corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same natural number system (as explicitly understood in a conscious rational manner).
Then in a parallel fashion, the alternative (Zeta 2) zeros represent the (hidden) holistic basis of the ordinal natural number system (as again explicitly understood in the standard analytic fashion).
And from the corresponding psychological perspective, these zeros represent the (hidden) unconscious basis of the same ordinal natural number system (as explicitly understood in a conscious rational manner).
However because these two sets of zeros (Zeta 1 and Zeta 2) are dynamically complementary with the analytic aspects in a two-way interactive fashion, they are in truth both simultaneously involved as the hidden holistic basis with respect to the natural number system (in cardinal and ordinal terms).
All of this leads to the need for the most fundamental change possible with respect to the customary interpretation of number.
Rather than the number system existing in an unchanging absolute fashion in some universal "mathematical heaven", it has an inherently dynamic interactive nature, entailing both quantitative and qualitative aspects in relative fashion.
The ultimate nature of this system is synchronistic in a holistic manner, approaching a pure ineffable identity. It is then only at the opposite extreme of analytic type interpretation (within isolated polar reference frames) that it approaches the appearance of being composed of rigid number objects of form!
Crucially, as the true interpretation of number, with respect to its dynamic interactive nature, entails the balanced interaction of both conscious (rational) and unconscious (intuitive) aspects of understanding, it is simply not possible to appreciate its nature within the accepted confines of present Mathematics. As such Mathematics is formally interpreted within a merely (conscious) rational type framework, it inevitably reduces the qualitative aspect in a quantitative manner.
It thereby reduces the inherent dynamic nature of the number system in a limited static manner!
As the nature of number is so fundamental, underlying everything else in phenomenal creation, the greatest revolution yet in our intellectual history now awaits, where the unconscious aspect of understanding needs to be explicitly united with its conscious counterpart.
This revolution, which will require a great deal of time to unfold, will then not have only profound implications for Mathematics but also most importantly for all related sciences and for society generally, gradually leading to a much more comprehensive world view that at present would be impossible to envisage.
One may validly inquire as to the status of the Riemann Hypothesis from this new dynamic perspective on number.
In fact, its very nature in now transformed in a manner that cannot be appreciated within the conventional mathematical framework.
I have commented before many times on the important fact that from the analytic quantitative perspective, the Riemann zeta function is undefined at just one point, i.e. where s (representing a power or dimensional number) = 1.
The holistic counterpart of this is that the Riemann zeta function likewise is undefined for s = 1, where it now defines in qualitative terms the 1-dimensional approach to mathematical understanding.
And as Conventional Mathematics is formally defined in a 1-dimensional manner (i.e. within isolated polar reference frames) then the true significance of the Riemann zeta function simply cannot be appreciated from this standpoint.
In other words, properly understood, the Riemann zeta function maps out numerical values that correspond to both analytic (quantitative) and holistic (qualitative) interpretations of number respectively.
Basically values for the function on the RHS for values of s > 1 correspond to analytic interpretation that are mapped with corresponding holistic values on the LHS for values of s (< 0).
So through the Riemann functional equation, values of ζ(s) can be mapped with corresponding values of ζ(1 – s).
This means for example that ζ(2) is mapped with ζ(– 1).
Now ζ(2) = 1 + 1/4 + 1/9 + 1/16 + ..... which in the standard logic of numerical interpretation, does indeed converge to a finite value (π2/6).
Then ζ(– 1) = 1 + 2 + 3 + 4 +.... which again in the standard logic of numerical interpretation diverges (to an infinite value).
However in terms of the Riemann zeta function, ζ(– 1) = – 1/12!
So clearly, a different form of interpretation is required to give meaning to this value, which in fact now corresponds to a holistic - rather than analytic - value!
And because analytic and holistic are complementary opposites, this entails that where a series diverges (in the standard analytic manner) it converges (in a corresponding holistic manner) and vice versa.
Now the value for s = .5, assumes a central importance here, for it entails that both ζ(s) and ζ(1 – s) are now identical.
Put another way, this entails that both the analytic (quantitative) and holistic (qualitative) interpretations of number coincide.
The additional requirement that both ζ(s) and ζ(1 – s) = 0 requires that all of the non-trivial zeros lie on an imaginary line through .5.
Therefore we can now reinterpret the Riemann Hypothesis as the central requirement for ensuring the coincidence of both the quantitative and qualitative aspects of number. In other words it serves as the fundamental requirement for ensuring the consistency of the natural number system with respect to both cardinal and ordinal interpretation!
Now, when one reflects on this for a moment, it should be obvious that there is no way that this proposition can be proved (or disproved) in the conventional mathematical manner.
Because the qualitative aspect is reduced (in every context) to quantitative interpretation, within conventional mathematical axioms, this means that the consistency of quantitative with qualitative aspects is in effect already assumed through the use of these axioms.
Therefore one cannot hope to prove (or disprove) a prior assumption, using an axiomatic approach that already implicitly contains that prior assumption (literally as an act of faith).
Thus the Riemann Hypothesis - when properly interpreted - expresses the mysterious requirement for the coincidence of quantitative with qualitative type interpretation of mathematical symbols.
From a psychological perspective, this relates to the requirement that both conscious (rational) and unconscious (intuitive) aspects of understanding can be synchronised in a consistent manner with respect to all mathematical interpretation.
This cannot be proved for conventional proof is already implicitly based on the assumption (that such synchronisation of meaning is already assured).
Thus correctly understood, all mathematical interpretation is implicitly based on an initial massive act of faith in the consistent correspondence of both the quantitative and qualitative meaning of its symbols !
Wednesday, November 11, 2015
Clarifying Analytic and Holistic Interpretations of "2" (18)
I mentioned yesterday, how the appreciation of the true nature of the non-trivial zeros is directly of a holistic intuitive nature, which appears deeply paradoxical when expressed in the standard analytic rational manner.
So once again in a rational manner, we can identify two distinct reference frames for investigating the relationship of the primes to the natural numbers.
In the first (Type 1) frame we attempt to understand the natural numbers as derived from independent prime "building blocks".
In the second (Type 2) frame, we attempt to understand the natural number factors (within each number) as likewise composed of prime "building blocks".
In the first frame, the numbers in their independent linear state, represent the (base) quantitative aspect; however in the second frame, in their interdependent state as factors, they - relatively - represent the complementary (dimensional) qualitative aspect.
And we also saw yesterday that the Riemann (Zeta 1) zeros entail the simultaneous recognition with respect to both frames.
In a direct sense, this comes from refined holistic mathematical insight (of a pure intuitive nature);
However indirectly this can be then expressed in a circular rational manner, which appears deeply paradoxical from the standard dualistic rational perspective.
So once again, the Riemann zeros represent - on an imaginary scale - the points where the coincidence of both the prime and natural numbers is dynamically obtained in a necessarily approximate relative manner; alternatively we could express this as the two-way coincidence of both the quantitative and qualitative aspects of number. Equally, we could say that zeros represent the points where randomness and order with respect to the number system are fully reconciled or finally, the points where multiplication is identical with addition.
The clear implication of this is that the zeros directly represent psychological energy states, which ultimately approach pure formlessness.
From a complementary perspective, the zeros equally represent physical energy states, which again ultimately approach pure formlessness.
The deeper implication of all this is that number itself can no longer be properly understood in abstraction from phenomena of form (physical and psychological) but must be understood rather as embedded in these forms as their inherent encoded nature.
This would explain for example why the Riemann zeros bear such a close relationship at a quantum level with the physical energy systems of atomic structures!
However what has not at all been stressed, is the complementary fact that the zeros equally relate intimately to psychological energy states (of a highly intuitive spiritual nature).
Some time in the future, it will be clearly realised how the zeros will play a profoundly important role in the consolidated attainment of the most spiritually advanced contemplative states (that can only be dimly realised at the current level of human evolution)!
Once again when appropriately understood in a dynamic interactive manner, the number system operates as between two limiting extremes.
From the standard analytic perspective, number is identified in a somewhat rigid absolute manner with unchanging objects of form.
This is especially the case with respect to the conventional appreciation of the primes that are given a universally fixed identity with respect to mathematical interpretation.
However this properly represents but the (reduced) analytic interpretation of number (where the qualitative aspect of interpretation is inevitably reduced in a quantitative manner).
However number can also be given an equally valid holistic interpretation. So once again whereas the quantitative nature of a number such as "2" can be given an actual finite identity, the corresponding qualitative aspect of "2" (as "twoness") relates properly to a potential infinite identity.
And this latter aspect of number (relating to number interdependence) cannot be grasped directly by (linear) reason but rather by refined mathematical intuition of a holistic kind. However indirectly, it can then be expressed through (circular) reason, where interpretation seems deeply paradoxical (from the conventional analytic perspective).
Now in experiential terms, neither the analytic (quantitative) nor holistic (qualitative) aspects of number interpretation can be viewed in isolation.
Rather both are dynamically related in complementary fashion with each other in a relative (approximate) manner.
This means that - properly understood - both the analytic and holistic aspects of number interpretation necessarily enjoy a relative - rather than absolute - validity.
Thus at one extreme, we have the analytic aspect which approaches towards absolute forms.
Now one might seek to maintain that the primes and natural numbers are indeed of an absolute nature!
However, strictly this is not the case. For example, there is an inevitable uncertainty with respect to communication. One, for example might use the number "2". However I would be pretty sure that my interpretation of what the number "2" represents, differs in crucial respects from the conventional viewpoint. Now this this does not mean that successful communication cannot take place within well-defined contexts, but it clearly must always remain of an approximate uncertain nature.
At the other extreme we have the holistic aspect (of number as representing energy states) which approaches towards pure formlessness.
So - relative to the analytic interpretation of the number system - the holistic appreciation of the zeta zeros (Zeta and Zeta 2) approaches as close as is possible the opposite extreme of the formless identity of number.
Now we still do identify these zeros in terms of form. However properly understood the true appreciation of a Zeta 1 zero for example - a complex number with an imaginary part that is transcendental - requires understanding that is so dynamic and interactive in nature that its identity as a fixed form is so fleeting that it is continually erased from conscious understanding.
In other words the zeta zeros form the final partition - as it were - as between phenomena of form and pure ineffable reality.
The first reality to be encountered with respect to the physical world are the zeta zeros (Zeta 1 and Zeta 2) as the inherent basis of all number notions.
The final reality to be encountered in a fully evolved psychological manner are again the same zeta zeros, now fully unearthed from the unconscious and explicitly understood as the fundamental basis - not only of number but - ultimately of all phenomenal creation!
So once again in a rational manner, we can identify two distinct reference frames for investigating the relationship of the primes to the natural numbers.
In the first (Type 1) frame we attempt to understand the natural numbers as derived from independent prime "building blocks".
In the second (Type 2) frame, we attempt to understand the natural number factors (within each number) as likewise composed of prime "building blocks".
In the first frame, the numbers in their independent linear state, represent the (base) quantitative aspect; however in the second frame, in their interdependent state as factors, they - relatively - represent the complementary (dimensional) qualitative aspect.
And we also saw yesterday that the Riemann (Zeta 1) zeros entail the simultaneous recognition with respect to both frames.
In a direct sense, this comes from refined holistic mathematical insight (of a pure intuitive nature);
However indirectly this can be then expressed in a circular rational manner, which appears deeply paradoxical from the standard dualistic rational perspective.
So once again, the Riemann zeros represent - on an imaginary scale - the points where the coincidence of both the prime and natural numbers is dynamically obtained in a necessarily approximate relative manner; alternatively we could express this as the two-way coincidence of both the quantitative and qualitative aspects of number. Equally, we could say that zeros represent the points where randomness and order with respect to the number system are fully reconciled or finally, the points where multiplication is identical with addition.
The clear implication of this is that the zeros directly represent psychological energy states, which ultimately approach pure formlessness.
From a complementary perspective, the zeros equally represent physical energy states, which again ultimately approach pure formlessness.
The deeper implication of all this is that number itself can no longer be properly understood in abstraction from phenomena of form (physical and psychological) but must be understood rather as embedded in these forms as their inherent encoded nature.
This would explain for example why the Riemann zeros bear such a close relationship at a quantum level with the physical energy systems of atomic structures!
However what has not at all been stressed, is the complementary fact that the zeros equally relate intimately to psychological energy states (of a highly intuitive spiritual nature).
Some time in the future, it will be clearly realised how the zeros will play a profoundly important role in the consolidated attainment of the most spiritually advanced contemplative states (that can only be dimly realised at the current level of human evolution)!
Once again when appropriately understood in a dynamic interactive manner, the number system operates as between two limiting extremes.
From the standard analytic perspective, number is identified in a somewhat rigid absolute manner with unchanging objects of form.
This is especially the case with respect to the conventional appreciation of the primes that are given a universally fixed identity with respect to mathematical interpretation.
However this properly represents but the (reduced) analytic interpretation of number (where the qualitative aspect of interpretation is inevitably reduced in a quantitative manner).
However number can also be given an equally valid holistic interpretation. So once again whereas the quantitative nature of a number such as "2" can be given an actual finite identity, the corresponding qualitative aspect of "2" (as "twoness") relates properly to a potential infinite identity.
And this latter aspect of number (relating to number interdependence) cannot be grasped directly by (linear) reason but rather by refined mathematical intuition of a holistic kind. However indirectly, it can then be expressed through (circular) reason, where interpretation seems deeply paradoxical (from the conventional analytic perspective).
Now in experiential terms, neither the analytic (quantitative) nor holistic (qualitative) aspects of number interpretation can be viewed in isolation.
Rather both are dynamically related in complementary fashion with each other in a relative (approximate) manner.
This means that - properly understood - both the analytic and holistic aspects of number interpretation necessarily enjoy a relative - rather than absolute - validity.
Thus at one extreme, we have the analytic aspect which approaches towards absolute forms.
Now one might seek to maintain that the primes and natural numbers are indeed of an absolute nature!
However, strictly this is not the case. For example, there is an inevitable uncertainty with respect to communication. One, for example might use the number "2". However I would be pretty sure that my interpretation of what the number "2" represents, differs in crucial respects from the conventional viewpoint. Now this this does not mean that successful communication cannot take place within well-defined contexts, but it clearly must always remain of an approximate uncertain nature.
At the other extreme we have the holistic aspect (of number as representing energy states) which approaches towards pure formlessness.
So - relative to the analytic interpretation of the number system - the holistic appreciation of the zeta zeros (Zeta and Zeta 2) approaches as close as is possible the opposite extreme of the formless identity of number.
Now we still do identify these zeros in terms of form. However properly understood the true appreciation of a Zeta 1 zero for example - a complex number with an imaginary part that is transcendental - requires understanding that is so dynamic and interactive in nature that its identity as a fixed form is so fleeting that it is continually erased from conscious understanding.
In other words the zeta zeros form the final partition - as it were - as between phenomena of form and pure ineffable reality.
The first reality to be encountered with respect to the physical world are the zeta zeros (Zeta 1 and Zeta 2) as the inherent basis of all number notions.
The final reality to be encountered in a fully evolved psychological manner are again the same zeta zeros, now fully unearthed from the unconscious and explicitly understood as the fundamental basis - not only of number but - ultimately of all phenomenal creation!
Tuesday, November 10, 2015
Clarifying Analytic and Holistic Interpretations of "2" (17)
We saw yesterday how there is a direct relationship as between the accumulated frequency of the natural factors of each number (i.e divisors) and the corresponding frequency of the the non-trivial (Zeta 1) zeros.
Therefore to estimate the frequency of natural factors to n, we calculate the corresponding frequency of non-trivial zeros to t, on an imaginary scale (where n = t/2π)
The well-known formula for calculation of non-trivial zeros is given as:
(i) t/2π(log t/2π – 1)
Now, I have suggested a simple amendment through the addition of 1 i.e. (ii) t/2π(log t/2π – 1) + 1.
This gives an amazingly accurate estimate of the frequency of non-trivial zeros, which when rounded, even at the highest numbers known (for existence of the zeros) gives estimates that are either exactly correct (in absolute terms) or in error by no more than 1!
Therefore, with respect to formula (i) where n = t/2π, the corresponding formula for calculation of accumulated frequency of (Type 2) natural number factors is given as
n(log n – 1).
This of course then bears a complementary type relationship with the corresponding formula for calculation of the frequency of (Type 1) primes i.e. n/(log n – 1).
In even simpler terms we can see the dual importance of log n in both Type 1 and Type 2 terms.
From the well known Type 1 perspective, log n approximates the average spread or gap as between primes among the natural numbers.
However from the corresponding Type 2 perspective, log n approximates the average frequency of natural number factors (within each number).
However, though as we have seen the frequency of non-trivial zeros (to t) bears a direct relationship to the corresponding frequency of natural number factors (to n), it is not an exact relationship.
When we look at the frequency of natural number factors, they certainly do not occur in a random fashion.
So each prime (with no non-trivial factors) is followed by a natural number (or numbers) where an accumulation of factors occurs.
Thus again illustrating up to n = 10, we have 0, 0, 0, 2, 0, 3, 0, 3, 2, 3 factors for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 respectively.
So the non-trivial zeros can best be seen as the attempt to balance the independent behaviour of the primes (with 0 factors) and the corresponding interdependent behaviour of the (composite) natural numbers (with 2 or more factors).
Now, if one thinks back to the example of a crossroads, the recognition that a turn can be both left and right (depending on context) is paradoxical in terms of normal conventional logic, where a turn must be unambiguously either left or right (separately).
In other words, the paradoxical recognition of the two-way interdependence of left and right - which is directly of an intuitive rather than rational nature - comes from the ability to simultaneously "see" the situation from two opposing polar reference frames.
So once again, when one approaches the crossroads in terms of just one such reference frame (i.e. from a S or N direction) unambiguous identifications of left and right turns can be made. However when one simultaneously then tries to view these turns from both N and S directions, left and right turns have a merely paradoxical interpretation.
In principle the appreciation of the nature of the non-trivial zeros is exactly similar.
As we have seen, there are two distinct reference frames (Type 1 and Type 2 respectively) through which we can view the relationship of the primes to the natural numbers.
From the Type 1 perspective, we concentrate on the primes as the building blocks of the natural numbers in their distinct separate identities.
However from the Type 2 perspective, we concentrate on the natural factors within each number with respect to their interdependent identity.
So again from the Type 1 perspective, we are viewing the number system in a quantitative manner composed of prime building blocks.
Then - relatively - from the Type 2 perspective we are viewing each number in a qualitative manner in its relationships expressed through a variety of natural number factors!
And again in particular we have the dual significance of log n, which in Type 1 terms approximates the average gap as between individual primes and then in corresponding Type 2 terms approximates the average frequency of the natural factors of each number.
Therefore the key to appreciation of the very nature of the non-trivial (Zeta 1) zeros is that they represent the simultaneous recognition of the two-way relationship as between the primes and natural numbers (and natural numbers and primes) in both Type 1 and Type 2 terms.
Thus the essential "seeing" of their nature is directly of an intuitive - rather than rational - nature.
In other words, when we attempt to translate the nature of the non-trivial zeros in a rational manner, they appear paradoxical.
Thus we could validly maintain that the non-trivial zeros represent points on an imaginary line where both the primes and natural numbers are dynamically identical with each other (which of course makes no sense in Type 1 or Type 2 terms, as considered separately).
We could equally maintain that these zeros represent points where both the quantitative and qualitative aspects of number are identical (again in a dynamic approximate manner).
Put in an equivalent fashion, we could say that they represent the dynamic identity of both the cardinal and ordinal aspects of number.
From another important perspective, we could say that the zeros represent the dynamic identity of both notions of randomness and order with respect to the number system.
Also from yet another important perspective, we could say that they represent the points, where both addition and multiplication with respect to the number system are mutually reconciled.
Therefore to estimate the frequency of natural factors to n, we calculate the corresponding frequency of non-trivial zeros to t, on an imaginary scale (where n = t/2π)
The well-known formula for calculation of non-trivial zeros is given as:
(i) t/2π(log t/2π – 1)
Now, I have suggested a simple amendment through the addition of 1 i.e. (ii) t/2π(log t/2π – 1) + 1.
This gives an amazingly accurate estimate of the frequency of non-trivial zeros, which when rounded, even at the highest numbers known (for existence of the zeros) gives estimates that are either exactly correct (in absolute terms) or in error by no more than 1!
Therefore, with respect to formula (i) where n = t/2π, the corresponding formula for calculation of accumulated frequency of (Type 2) natural number factors is given as
n(log n – 1).
This of course then bears a complementary type relationship with the corresponding formula for calculation of the frequency of (Type 1) primes i.e. n/(log n – 1).
In even simpler terms we can see the dual importance of log n in both Type 1 and Type 2 terms.
From the well known Type 1 perspective, log n approximates the average spread or gap as between primes among the natural numbers.
However from the corresponding Type 2 perspective, log n approximates the average frequency of natural number factors (within each number).
However, though as we have seen the frequency of non-trivial zeros (to t) bears a direct relationship to the corresponding frequency of natural number factors (to n), it is not an exact relationship.
When we look at the frequency of natural number factors, they certainly do not occur in a random fashion.
So each prime (with no non-trivial factors) is followed by a natural number (or numbers) where an accumulation of factors occurs.
Thus again illustrating up to n = 10, we have 0, 0, 0, 2, 0, 3, 0, 3, 2, 3 factors for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 respectively.
So the non-trivial zeros can best be seen as the attempt to balance the independent behaviour of the primes (with 0 factors) and the corresponding interdependent behaviour of the (composite) natural numbers (with 2 or more factors).
Now, if one thinks back to the example of a crossroads, the recognition that a turn can be both left and right (depending on context) is paradoxical in terms of normal conventional logic, where a turn must be unambiguously either left or right (separately).
In other words, the paradoxical recognition of the two-way interdependence of left and right - which is directly of an intuitive rather than rational nature - comes from the ability to simultaneously "see" the situation from two opposing polar reference frames.
So once again, when one approaches the crossroads in terms of just one such reference frame (i.e. from a S or N direction) unambiguous identifications of left and right turns can be made. However when one simultaneously then tries to view these turns from both N and S directions, left and right turns have a merely paradoxical interpretation.
In principle the appreciation of the nature of the non-trivial zeros is exactly similar.
As we have seen, there are two distinct reference frames (Type 1 and Type 2 respectively) through which we can view the relationship of the primes to the natural numbers.
From the Type 1 perspective, we concentrate on the primes as the building blocks of the natural numbers in their distinct separate identities.
However from the Type 2 perspective, we concentrate on the natural factors within each number with respect to their interdependent identity.
So again from the Type 1 perspective, we are viewing the number system in a quantitative manner composed of prime building blocks.
Then - relatively - from the Type 2 perspective we are viewing each number in a qualitative manner in its relationships expressed through a variety of natural number factors!
And again in particular we have the dual significance of log n, which in Type 1 terms approximates the average gap as between individual primes and then in corresponding Type 2 terms approximates the average frequency of the natural factors of each number.
Therefore the key to appreciation of the very nature of the non-trivial (Zeta 1) zeros is that they represent the simultaneous recognition of the two-way relationship as between the primes and natural numbers (and natural numbers and primes) in both Type 1 and Type 2 terms.
Thus the essential "seeing" of their nature is directly of an intuitive - rather than rational - nature.
In other words, when we attempt to translate the nature of the non-trivial zeros in a rational manner, they appear paradoxical.
Thus we could validly maintain that the non-trivial zeros represent points on an imaginary line where both the primes and natural numbers are dynamically identical with each other (which of course makes no sense in Type 1 or Type 2 terms, as considered separately).
We could equally maintain that these zeros represent points where both the quantitative and qualitative aspects of number are identical (again in a dynamic approximate manner).
Put in an equivalent fashion, we could say that they represent the dynamic identity of both the cardinal and ordinal aspects of number.
From another important perspective, we could say that the zeros represent the dynamic identity of both notions of randomness and order with respect to the number system.
Also from yet another important perspective, we could say that they represent the points, where both addition and multiplication with respect to the number system are mutually reconciled.
Monday, November 9, 2015
Clarifying Analytic and Holistic Interpretations of "2" (16)
We have seen in previous blog entries how the multiplication of primes, with respect to each composite number, is associated with a unique qualitative resonance for each prime (and indeed combination of primes as natural factors of the number involved).
And remarkably we can show a very close relationship as between the qualitative nature of these number combinations (as factors) and the notion of the Riemann (i.e. Zeta 1) zeros.
So put more simply the frequency of the Riemann (Zeta 1) zeros bears a very close relationship with the corresponding cumulative frequency of the natural factors of numbers.
However whereas the (cumulative) frequency of factors of the natural numbers is expressed on a linear scale, the corresponding frequency of Riemann (Zeta 1) zeros is expressed with respect to a circular scale (where the circumference = linear radius * 2π).
So, to convert from the circular notion of the frequency of non-trivial zeros up to a given number (t) to the linear notion of frequency of factors up to given number (n), we set n = t/2π
Therefore, to illustrate with a simple example the cumulative frequency of factors up to n = 10, it should thereby equate with the corresponding frequency of non-trivial zeros to t = 62.83 (approx).
Now using the method I have already suggested, we can manually calculate the cumulative frequency of factors to 10.
So associated with 1, 2 and 3, we have no factors. Remember the rule with primes is to disregard the two factors of 1 and the prime in question as trivial factors!
Then with 4, we have 2 factors i.e. 2 and 4. (Again because 4 is now a composite number representing the combination of primes 2 * 2, we include it also as a factor).
So the important notion to grasp here is that both 2 and 4 (as factors) acquire a unique qualitative resonance through relationship with the no. 4.
Then 5 as a prime again has no factors.
However 6 (as composite) has 3 i.e. 2, 3 and 6. So once again 2, 3 and 6 acquire a unique qualitative resonance through relationship (as factors) to 6.
7 (as prime) has no factors. However 8 (as composite) has 3, i.e. 2, 4 and 8.
Then 9 (as composite) has 2, i.e. 3 and 9.
Finally 10 (as composite) has 3, i.e. 2, 5 and 10.
So if we now accumulate over the 10 natural numbers involved (i.e. n = 10), we get 0 + 0 + 0 + 2 + 0 + 3 + 0 + 3 + 2 + 3 = 13.
Now according to what I have stated, this should then equate well with the corresponding frequency of non-trivial zeros (to t = 62.83).
Now (correct to 2 decimal places) the non-trivial zeros to 62.83 are 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01, 49.77, 52.97, 56.45, 59.35 and 60.83.
So this gives a total of 14 zeros (to t = 62.83), which already equates very well with the corresponding frequency of factors (to n = t/2π = 10)) of 13!
In fact, I manually calculated the cumulative frequency of factors to n = 100 which gave 357.
The corresponding frequency of non-trivial zeros to t = 628.3 = 362.
So in fact we can already see a very close relationship as between the two sets of measurements.
Thus we have the extremely interesting phenomenon regarding the non-trivial zeros, which seems to me greatly missing from conventional mathematical appreciation.
Therefore, from a dynamic complementary perspective, though the Riemann (Zeta 1) zeros bear a complementary (opposite) relationship to the primes (without factors), they also bear a direct relationship to the natural numbers (with respect to the cumulative nature of their factors).
One further issue that need to be explained at this point relates to the fact that the non-trivial zeros are measured on an imaginary - as opposed to a real - scale.
Now, the holistic mathematical reason for this is very simple (though also very revealing) in that it points to the fact that these zeros relate to the qualitative - as opposed to quantitative - aspect of number.
Therefore from one perspective, we can attempt to look at the natural numbers as merely quantitative measurements on a linear (1-dimensional) scale.
We can then equally attempt to look at numbers (representing the frequency of factors) equally as quantitative measurements on a linear scale.
However like the directions at a crossroads, the frame of reference has now switched with multiple dimensions (as factors) involved.
Put another way the two measurements are quantitative as to qualitative with respect to each other.
And the basic way of indirectly expressing - what refers to - a qualitative type measurement in quantitative terms, is to use imaginary rather than real units.
So the Riemann zeros - in being directly related to the factor composition of the natural numbers - relate inherently to a qualitative, rather than quantitative notion of number.
In other words, when we view the primes in quantitative terms, the Riemann zeros are thereby of a complementary qualitative nature.
In psychological terms, if we view the primes in a conscious (rational) manner, therefore the Riemann zeros are thereby of an unconscious (intuitive) nature.
In fact this is all deeply relevant, for in very true experiential manner - corresponding well with Jungian type notions - the Riemann zeros represent the (hidden) shadow system of the conventional natural number system (understood in a merely quantitative rational manner).
Put another way, the Riemann zeros - correctly appreciated - represent the (unrecognised) unconscious basis of the natural number system (that is conventionally understood in a merely conscious manner).
However to properly understood the relationship between both the (recognised) conscious and (unrecognised) unconscious aspects, we must view both in a dynamic interactive manner.
And remarkably we can show a very close relationship as between the qualitative nature of these number combinations (as factors) and the notion of the Riemann (i.e. Zeta 1) zeros.
So put more simply the frequency of the Riemann (Zeta 1) zeros bears a very close relationship with the corresponding cumulative frequency of the natural factors of numbers.
However whereas the (cumulative) frequency of factors of the natural numbers is expressed on a linear scale, the corresponding frequency of Riemann (Zeta 1) zeros is expressed with respect to a circular scale (where the circumference = linear radius * 2π).
So, to convert from the circular notion of the frequency of non-trivial zeros up to a given number (t) to the linear notion of frequency of factors up to given number (n), we set n = t/2π
Therefore, to illustrate with a simple example the cumulative frequency of factors up to n = 10, it should thereby equate with the corresponding frequency of non-trivial zeros to t = 62.83 (approx).
Now using the method I have already suggested, we can manually calculate the cumulative frequency of factors to 10.
So associated with 1, 2 and 3, we have no factors. Remember the rule with primes is to disregard the two factors of 1 and the prime in question as trivial factors!
Then with 4, we have 2 factors i.e. 2 and 4. (Again because 4 is now a composite number representing the combination of primes 2 * 2, we include it also as a factor).
So the important notion to grasp here is that both 2 and 4 (as factors) acquire a unique qualitative resonance through relationship with the no. 4.
Then 5 as a prime again has no factors.
However 6 (as composite) has 3 i.e. 2, 3 and 6. So once again 2, 3 and 6 acquire a unique qualitative resonance through relationship (as factors) to 6.
7 (as prime) has no factors. However 8 (as composite) has 3, i.e. 2, 4 and 8.
Then 9 (as composite) has 2, i.e. 3 and 9.
Finally 10 (as composite) has 3, i.e. 2, 5 and 10.
So if we now accumulate over the 10 natural numbers involved (i.e. n = 10), we get 0 + 0 + 0 + 2 + 0 + 3 + 0 + 3 + 2 + 3 = 13.
Now according to what I have stated, this should then equate well with the corresponding frequency of non-trivial zeros (to t = 62.83).
Now (correct to 2 decimal places) the non-trivial zeros to 62.83 are 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01, 49.77, 52.97, 56.45, 59.35 and 60.83.
So this gives a total of 14 zeros (to t = 62.83), which already equates very well with the corresponding frequency of factors (to n = t/2π = 10)) of 13!
In fact, I manually calculated the cumulative frequency of factors to n = 100 which gave 357.
The corresponding frequency of non-trivial zeros to t = 628.3 = 362.
So in fact we can already see a very close relationship as between the two sets of measurements.
Thus we have the extremely interesting phenomenon regarding the non-trivial zeros, which seems to me greatly missing from conventional mathematical appreciation.
Therefore, from a dynamic complementary perspective, though the Riemann (Zeta 1) zeros bear a complementary (opposite) relationship to the primes (without factors), they also bear a direct relationship to the natural numbers (with respect to the cumulative nature of their factors).
One further issue that need to be explained at this point relates to the fact that the non-trivial zeros are measured on an imaginary - as opposed to a real - scale.
Now, the holistic mathematical reason for this is very simple (though also very revealing) in that it points to the fact that these zeros relate to the qualitative - as opposed to quantitative - aspect of number.
Therefore from one perspective, we can attempt to look at the natural numbers as merely quantitative measurements on a linear (1-dimensional) scale.
We can then equally attempt to look at numbers (representing the frequency of factors) equally as quantitative measurements on a linear scale.
However like the directions at a crossroads, the frame of reference has now switched with multiple dimensions (as factors) involved.
Put another way the two measurements are quantitative as to qualitative with respect to each other.
And the basic way of indirectly expressing - what refers to - a qualitative type measurement in quantitative terms, is to use imaginary rather than real units.
So the Riemann zeros - in being directly related to the factor composition of the natural numbers - relate inherently to a qualitative, rather than quantitative notion of number.
In other words, when we view the primes in quantitative terms, the Riemann zeros are thereby of a complementary qualitative nature.
In psychological terms, if we view the primes in a conscious (rational) manner, therefore the Riemann zeros are thereby of an unconscious (intuitive) nature.
In fact this is all deeply relevant, for in very true experiential manner - corresponding well with Jungian type notions - the Riemann zeros represent the (hidden) shadow system of the conventional natural number system (understood in a merely quantitative rational manner).
Put another way, the Riemann zeros - correctly appreciated - represent the (unrecognised) unconscious basis of the natural number system (that is conventionally understood in a merely conscious manner).
However to properly understood the relationship between both the (recognised) conscious and (unrecognised) unconscious aspects, we must view both in a dynamic interactive manner.
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