I am attempting in these blogs to use a variety of related insights to convey the true meaning of the famed zeta zeros and their relationship to the primes and natural number system.
My key contention all along is however that we cannot hope to properly understand the nature of these zeros from within the restricted current paradigm of Mathematics, geared as it is to the formal interpretation of meaning that is of a reduced – merely quantitative – nature.
So it is the interaction of polar aspects of understanding that are (1) internal and external with and (2) quantitative and qualitative with respect to each other that the true meaning of number (and the corresponding nature of prime and natural numbers) properly resides.
Once again, appropriately interpreted in a dynamic relative manner, all mathematical meaning entails the relationship as between what is objectively known (as external) with corresponding subjective constructs of mental interpretation (as – relatively – internal).
Numbers from this perspective inevitably reflect the manner by which they are interpreted.
Therefore when we interpret them in a reduced manner in 1-dimensional terms (as solely objective) they appear to have a fixed absolute identity.
However when we interpret them in a more balanced manner (as reflecting the inevitable interaction of objective data with subjective interpretation) they now appear in dynamic terms as possessing a merely relative identity.
Likewise all mathematical meaning entails the interaction of what is identified as independent in quantitative terms with an interdependent aspect through relationship that is strictly of a qualitative nature.
Once again when we seek to interpret numbers is a reduced manner as solely independent they appear to possess a merely quantitative identity.
However in a more balanced understanding, an inevitable interaction necessarily exists as between two polar aspects that are – relatively - quantitative and qualitative with respect to each other.
Expressed in an equivalent manner all numbers possess both analytic (quantitative) and holistic (qualitative) aspects in a dynamic relative manner.
So the physical identity of numbers (as objective) possesses a psychological aspect (as interpretation) with analytic and holistic aspects respectively.
Whereas the psychological counterpart of the analytic aspect relates to conscious (rational), the psychological aspect of the holistic aspect relates to unconscious (intuitive) understanding respectively.
Indirectly however this holistic aspect can be expressed in rational terms in a circular (paradoxical) fashion.
What is even more remarkable is that this (real) circular aspect can then be further indirectly conveyed in a linear rational fashion that is imaginary!
So once again not only can real and imaginary notions be given a merely quantitative (analytic) interpretation in mathematical terms; equally they can be given a qualitative (holistic) interpretation as imaginary.
Therefore to understand number relationships – and indeed all mathematical relationships – in a more comprehensive manner (with interacting analytic and holistic aspects) we require a complex rational paradigm (in qualitative terms) with both real and imaginary aspects.
Those familiar with Jungian notions – which lends itself very well to holistic mathematical interpretation – will readily recognise the notion of the shadow.
So when a conscious function of understanding is unduly dominant in experience, the shadow unconscious aspect is projected outwards as blind projection.
Therefore though the meaning of the projection is properly of an (unconscious) holistic nature, it is misleadingly confused with specific conscious object phenomena.
We see this for example in relation to spiritual beliefs which can be identified in an unduly narrow fashion with the symbols and rituals of the various religious traditions. Too often the projection of such limited interpretations has served to justify war and persecution on a grand scale.
Therefore, though projections are necessarily embodied in conscious symbols their deeper significance is of a holistic (unconscious) nature.
As Conventional Mathematics rigidly identifies itself in a merely conscious rational manner, this thereby betrays a deep unrecognised shadow whereby it remains steadfastly blind to holistic interpretation at all levels.
This is of paramount significance in relation to appreciation of the true nature of the famed non-trivial zeros of the number system.
Rightly understood these can fruitfully be interpreted as representing the perfect shadow counterpart to the prime numbers.
In other words when we view the prime numbers as separate (in an analytic rational manner) as the independent building blocks of the natural number system, the non-trivial zeros exist in relative terms as the perfect holistic counterpart to this system.
As we have seen this holistic aspect properly relates to the unconscious aspect of understanding that directly manifests itself in an intuitive manner. This can then indirectly be expressed in a circular rational fashion which in turn can be converted in a linear imaginary fashion.
Therefore right away here we have the simple qualitative explanation of why all the non-trivial zeros line up on the same imaginary line!
So the prime numbers as separate independent entities – and this can only properly be viewed in a relative manner – have their perfect shadow number counterpart in the non-trivial zeros, which through their overall collective behaviour represent the interdependent extreme of the relationship of the primes to the natural numbers.
Thus the one extreme of the prime numbers viewed - in relative terms - as analytic and independent, is necessarily counterbalanced by the opposite extreme of the non-trivial zeros as holistic and interdependent!
We could equally say that the discrete particle nature of the prime numbers is necessarily counterbalanced by the continual wave nature of the non-trivial zeros (which are – relatively – analytic and holistic with respect to each other).
However from the equally valid opposite perspective each non-trivial zero has a discrete independent identity in imaginary terms (i.e. as a point on the same imaginary line).
Therefore from this imaginary perspective the discrete independent identity of each non-trivial zero (in analytic terms) is perfectly counterbalanced by the collective interdependent identity of the prime numbers.
So just as we can explain the actual deviations of individual prime numbers from their overall general behaviour through the collective behaviour of the non-trivial zeros (in holistic terms), equally we can explain the actual deviations of individual trivial zeros from their overall general behaviour through the collective behaviour of the prime numbers (again in holistic terms).
So properly understood in a dynamic interactive perspective, both the primes and the non-trivial zeros can be given, in their relevant contexts, both extreme analytic and holistic interpretations respectively.
This points to the fact that ultimately the relationship between both is of a purely relative nature (in phenomenal terms) which equally implies their ineffable origin in an absolute manner.
Once again the great (unrecognised) limitation of Conventional Mathematics is that it cannot – by its very interpretations – view the relationship between the primes and the non-trivial zeta zeros (and the non-trivial zeros and the primes) in a satisfactory manner.
Though the relationship between both is inherently dynamic and interactive with twin analytic and holistic aspects, Conventional Mathematics can only attempt to view both in a fixed analytic fashion. This therefore ultimately only serves to misrepresent their very nature!
Thursday, February 28, 2013
Wednesday, February 20, 2013
The Truly Remarkable Nature of the Number System
As readers of this blog will know, I have been posting now for some time on the Riemann Hypothesis.
I had already become convinced before the first posting that the Riemann Hypothesis pointed to an - as yet - important unaddressed issue with relation to the true nature of number.
Indeed so fundamental is this issue that it had already become very clear to me then that not only is the Riemann Hypothesis incapable of proof within the standard form of accepted Mathematics but that its very nature cannot be properly interpreted from this perspective!
So for me the Riemann Hypothesis has served as an invaluable pathway towards a deeper understanding of the true nature of the number system. And at last perhaps the basic explanation of this nature can now be given.
The crucial starting point is that - properly understood - all mathematical notions can be given distinctive analytic and holistic interpretations. As I have repeatedly stated this is really what the Riemann Hypothesis is about i.e. the ultimate reconciliation of both the quantitative (analytic) and qualitative (holistic) aspects of number!
Therefore this distinction as between quantitative and qualitative aspects intimately applies to the very nature of number. So rather that viewing the number system in a somewhat fixed absolute manner in merely quantitative terms, we must dramatically change our perspective so as to view it in as inherently dynamic, representing the interaction of twin complementary analytic (quantitative) and holistic (qualitative) aspects.
The quantitative (analytic) aspect basically relates to the understanding of number as separate and independent, Because of the dominance of this aspect we have come to view numbers - misleadingly - as absolute fixed entities. However though this perspective is so deeply ingrained in our culture and accepted largely without question, in truth it represents but a reduced and ultimately distorted viewpoint.
In order to be related, numbers must enjoy a certain interdependence with other numbers. So numbers necessarily enjoy both an independent aspect (as separate from) and interdependent (as shared with) other members. So both aspects are of a relative - rather than absolute - nature! And it is in the recognition of this interdependence that the holistic (qualitative) aspect relates. Therefore to equally recognise both the independent and interdependent aspects of number, we must employ two distinctive means of mathematical interpretation.
Thus in the language I customarily use, we have therefore both Type 1 (analytic) and Type 2 (holistic) aspects to all mathematical interpretation.
Initially both aspects can be developed in relative isolation from each other. However, comprehensive mathematical understanding requires the combined dynamic interaction of the two aspects (in what I refer to as the Type 3 approach).
Though Type 1 and Type 2 aspects are - relatively - analytic and holistic with respect to each other, both can be given (in their isolated separate contexts) an analytic presentation which readily concurs with common sense understanding.
Because from a conventional mathematical perspective, no clear distinction is made as between analytic and holistic aspects, the natural number system is uniformly defined for example as 1, 2, 3, 4,......
However properly understood this number system has two distinct aspects (which highlights the distinction as between addition and multiplication).
So from the Type 1 perspective all natural numbers are defined with respect to a (default) dimensional value of 1
i.e. 1^1, 2^1, 3^1, 4^1,......
Thus the natural number 3 from this aspect implies 3^1. Then when we attempt to define it with respect to its individual units we get 3 = 1 + 1 + 1.
From the Type 2 perspective all natural numbers are defined - in an inverse manner - as dimensional powers with respect to a (default) base number quantity of 1.
i.e. 1^1, 1^2, 1^3, 1^4,......
Then the natural 3 from this alternative aspect imples 1^3. So when we attempt to define it with respect to its individual units we get 3 = 1 * 1 * 1.
So right away we see that the fundamental distinction as between addition and multiplication relates respectively to the two differing aspects of the number system (Type 1 and Type 2).
However as Conventional Mathematics does not recognise in formal terms the distinction as between analytic (quantitative) and holistic (qualitative) meaning it has no means of adequately dealing with this key issue.
Indeed I have come to firmly hold what might seem an outrageous position i.e. that Conventional Mathematics is simply not fit for purpose as - by its very definitions - it can provide no satisfactory means of reconciling the key notion of (qualitative) interdependence, with respect to any relationship, with that of (quantitative) independence.
The fact that mathematicians in practice do not recognise this as the no. 1 issue again indicates how deeply ingrained the reduced (i.e. merely quantitative) approach to Mathematics has now become!
Of course it is well recognised - in the context of the Riemann Hypothesis - that a deep problem exists in terms of reconciling the nature of addition with multiplication. However mathematicians attempt to view this in a merely quantitative manner whereas - correctly understood - it fundamentally relates to a prior distinction as between the quantitative and qualitative means of mathematical interpretation.
However what is truly remarkable is the existence for both the Type 1 and Type 2 analytic aspects of the number system of two hidden interpenetrating holistic systems (the full nature of which is just mindboggling in its implications).
These two systems derive from the non-trivial solutions for both the Zeta 1 and Zeta 2 equations.
Now again because Conventional Mathematics does not formally recognise the Type 2 number system, not surprisingly it can provide no real appreciation of - what I refer to as - the Zeta 2 non-trivial solutions.
However recently I have come to realise that our very ability to make consistent ordinal distinctions as between numbers, intimately depends on these Type 2 solutions (which manifest themselves as successive roots of 1). And as we have seen the prime numbered roots of 1 lead to a unique form of circular interdependence, in relational terms, that is the exact opposite of the Type 1 appreciation of primes as comprising the unique (independent) building blocks of the number system.
So the holistic (circular) complex number system as the non-trivial unique solutions (except 1) for prime numbered roots of 1, is vitally necessary in order to make consistent ordinal distinctions as between numbers. Thus the logical sequence of 1st, 2nd, 3rd, 4th etc. would have no meaning in the absence of this unique number system.
In complementary fashion the non-trivial zeros to the Zeta 1 function as an infinite set of special complex numbers is necessary to enable the very identification of natural numbers in finite terms. Without this equally unique holistic system, it would not be possible to use numbers consistently in a cardinal sense.
So putting it simply there is one holistic system of zeta zeros for the Zeta 1 Function that underlies the cardinal number system ensuring its consistency; there is an alternative holistic system of zeta zeros - whose function is yet completely unrecognised - that likewise underlies the ordinal system ensuring its consistency. Using quantum mechanical terminology, the number system therefore comprises two distinct aspects with particle (analytic) and wave (holistic) manifestations in both cases.
However in Type 3 terms - representing the most comprehensive type of mathematical appreciation - these two systems are simultaneously co-determined with respect to both their analytic and holistic aspects. So implicit in the emergence of starting cardinal notions of order are corresponding ordinal notions; and implicit in the starting emergence of ordinal notions are corresponding cardinal notions. So underlying our customary cardinal and ordinal notions of order (in any context) are two hidden holistic number systems (of complex form) that ultimately are mutually co-determined in an ineffable manner.
And herein lies the great secret of life underlying all phenomenal meaning in both quantitative and qualitative terms. However this will never be properly appreciated while Mathematics remains deeply rooted in its distorted (i.e. merely one-sided) quantitative perspective.
I had already become convinced before the first posting that the Riemann Hypothesis pointed to an - as yet - important unaddressed issue with relation to the true nature of number.
Indeed so fundamental is this issue that it had already become very clear to me then that not only is the Riemann Hypothesis incapable of proof within the standard form of accepted Mathematics but that its very nature cannot be properly interpreted from this perspective!
So for me the Riemann Hypothesis has served as an invaluable pathway towards a deeper understanding of the true nature of the number system. And at last perhaps the basic explanation of this nature can now be given.
The crucial starting point is that - properly understood - all mathematical notions can be given distinctive analytic and holistic interpretations. As I have repeatedly stated this is really what the Riemann Hypothesis is about i.e. the ultimate reconciliation of both the quantitative (analytic) and qualitative (holistic) aspects of number!
Therefore this distinction as between quantitative and qualitative aspects intimately applies to the very nature of number. So rather that viewing the number system in a somewhat fixed absolute manner in merely quantitative terms, we must dramatically change our perspective so as to view it in as inherently dynamic, representing the interaction of twin complementary analytic (quantitative) and holistic (qualitative) aspects.
The quantitative (analytic) aspect basically relates to the understanding of number as separate and independent, Because of the dominance of this aspect we have come to view numbers - misleadingly - as absolute fixed entities. However though this perspective is so deeply ingrained in our culture and accepted largely without question, in truth it represents but a reduced and ultimately distorted viewpoint.
In order to be related, numbers must enjoy a certain interdependence with other numbers. So numbers necessarily enjoy both an independent aspect (as separate from) and interdependent (as shared with) other members. So both aspects are of a relative - rather than absolute - nature! And it is in the recognition of this interdependence that the holistic (qualitative) aspect relates. Therefore to equally recognise both the independent and interdependent aspects of number, we must employ two distinctive means of mathematical interpretation.
Thus in the language I customarily use, we have therefore both Type 1 (analytic) and Type 2 (holistic) aspects to all mathematical interpretation.
Initially both aspects can be developed in relative isolation from each other. However, comprehensive mathematical understanding requires the combined dynamic interaction of the two aspects (in what I refer to as the Type 3 approach).
Though Type 1 and Type 2 aspects are - relatively - analytic and holistic with respect to each other, both can be given (in their isolated separate contexts) an analytic presentation which readily concurs with common sense understanding.
Because from a conventional mathematical perspective, no clear distinction is made as between analytic and holistic aspects, the natural number system is uniformly defined for example as 1, 2, 3, 4,......
However properly understood this number system has two distinct aspects (which highlights the distinction as between addition and multiplication).
So from the Type 1 perspective all natural numbers are defined with respect to a (default) dimensional value of 1
i.e. 1^1, 2^1, 3^1, 4^1,......
Thus the natural number 3 from this aspect implies 3^1. Then when we attempt to define it with respect to its individual units we get 3 = 1 + 1 + 1.
From the Type 2 perspective all natural numbers are defined - in an inverse manner - as dimensional powers with respect to a (default) base number quantity of 1.
i.e. 1^1, 1^2, 1^3, 1^4,......
Then the natural 3 from this alternative aspect imples 1^3. So when we attempt to define it with respect to its individual units we get 3 = 1 * 1 * 1.
So right away we see that the fundamental distinction as between addition and multiplication relates respectively to the two differing aspects of the number system (Type 1 and Type 2).
However as Conventional Mathematics does not recognise in formal terms the distinction as between analytic (quantitative) and holistic (qualitative) meaning it has no means of adequately dealing with this key issue.
Indeed I have come to firmly hold what might seem an outrageous position i.e. that Conventional Mathematics is simply not fit for purpose as - by its very definitions - it can provide no satisfactory means of reconciling the key notion of (qualitative) interdependence, with respect to any relationship, with that of (quantitative) independence.
The fact that mathematicians in practice do not recognise this as the no. 1 issue again indicates how deeply ingrained the reduced (i.e. merely quantitative) approach to Mathematics has now become!
Of course it is well recognised - in the context of the Riemann Hypothesis - that a deep problem exists in terms of reconciling the nature of addition with multiplication. However mathematicians attempt to view this in a merely quantitative manner whereas - correctly understood - it fundamentally relates to a prior distinction as between the quantitative and qualitative means of mathematical interpretation.
However what is truly remarkable is the existence for both the Type 1 and Type 2 analytic aspects of the number system of two hidden interpenetrating holistic systems (the full nature of which is just mindboggling in its implications).
These two systems derive from the non-trivial solutions for both the Zeta 1 and Zeta 2 equations.
Now again because Conventional Mathematics does not formally recognise the Type 2 number system, not surprisingly it can provide no real appreciation of - what I refer to as - the Zeta 2 non-trivial solutions.
However recently I have come to realise that our very ability to make consistent ordinal distinctions as between numbers, intimately depends on these Type 2 solutions (which manifest themselves as successive roots of 1). And as we have seen the prime numbered roots of 1 lead to a unique form of circular interdependence, in relational terms, that is the exact opposite of the Type 1 appreciation of primes as comprising the unique (independent) building blocks of the number system.
So the holistic (circular) complex number system as the non-trivial unique solutions (except 1) for prime numbered roots of 1, is vitally necessary in order to make consistent ordinal distinctions as between numbers. Thus the logical sequence of 1st, 2nd, 3rd, 4th etc. would have no meaning in the absence of this unique number system.
In complementary fashion the non-trivial zeros to the Zeta 1 function as an infinite set of special complex numbers is necessary to enable the very identification of natural numbers in finite terms. Without this equally unique holistic system, it would not be possible to use numbers consistently in a cardinal sense.
So putting it simply there is one holistic system of zeta zeros for the Zeta 1 Function that underlies the cardinal number system ensuring its consistency; there is an alternative holistic system of zeta zeros - whose function is yet completely unrecognised - that likewise underlies the ordinal system ensuring its consistency. Using quantum mechanical terminology, the number system therefore comprises two distinct aspects with particle (analytic) and wave (holistic) manifestations in both cases.
However in Type 3 terms - representing the most comprehensive type of mathematical appreciation - these two systems are simultaneously co-determined with respect to both their analytic and holistic aspects. So implicit in the emergence of starting cardinal notions of order are corresponding ordinal notions; and implicit in the starting emergence of ordinal notions are corresponding cardinal notions. So underlying our customary cardinal and ordinal notions of order (in any context) are two hidden holistic number systems (of complex form) that ultimately are mutually co-determined in an ineffable manner.
And herein lies the great secret of life underlying all phenomenal meaning in both quantitative and qualitative terms. However this will never be properly appreciated while Mathematics remains deeply rooted in its distorted (i.e. merely one-sided) quantitative perspective.
Monday, February 18, 2013
Calculating the Shadow Deviation
I have mentioned many times before that proper appreciation of the nature of the Riemann Hypothesis requires complementary (Type 1) analytic and (Type 2) holistic methods of mathematical interpretation.
Then Type 3 interpretation - which alone can reveal the true ultimate significance of the Hypothesis - entails the mutual dynamic synthesis of both Type 1 and Type 2 aspects.
In the same manner as in quantum physics where the wave interpretation of matter equally possesses a particle aspect (and the particle a wave aspect) the Type 1 analytic equally possesses a holistic aspect and the Type 2 holistic equally an analytic aspect of interpretation respectively.
Quite remarkably - and this is the most crucially important point of all - Conventional Mathematics gives no recognition whatsoever in formal terms to the Type 2 aspect of interpretation. Thus it inevitably attempts to interpret meaning that is properly of a holistic qualitative nature in a merely reduced analytic fashion.
The implications of this could not be more far reaching! For not alone is the Riemann Hypothesis not capable of proof in the standard Type 1 context; its very nature cannot be properly appreciated in this manner.
In other words the fundamental nature of the primes (to which the Riemann Hypothesis points) relates in number terms to the ultimate reconciliation of quantitative notions of independence with qualitative notions of interdependence.
While readily admitting its great achievements, the futility of present (Type 1) Mathematics resides in the continued attempt to deal with qualitative holistic notions (that properly relate to interdependence) in a merely reduced quantitative manner (where numbers are treated in an independent fashion). And nowhere is this problem of interpretation more exposed than with respect to the fundamental nature of prime numbers!
From the Type 1 perspective the natural numbers are viewed in cardinal terms (where integers literally reflect collective whole units). Here the primes are viewed as the (independent) building blocks of the natural numbers with every such number (except 1) representing a unique combination of prime factors.
However I have repeatedly pointed to the key limitation of this approach in that viewing numbers in this cardinal manner leaves us with no means of making ordinal distinctions.
Though this issue is repeatedly glossed over in conventional presentation, the making of ordinal distinctions as between numbers strictly reflects qualitative appreciation which does not come from their cardinal nature.
If we attempt to break down a cardinal number such as 3 in terms of its individual units we would represent it as 1 + 1 + 1. However because these units are both independent and homogeneous (i.e. lacking qualitative distinction) it would not be possible to rank them in ordinal fashion i.e. as 1st, 2nd and 3rd members of their collective group (cardinally represented as 3).
In other words the very ability to rank numbers in an ordinal fashion relates to qualitative notions of relational interdependence. And strictly as Type 1 Mathematics views numbers as separate independent units (in merely quantitative terms) it thereby can offer no satisfactory way of establishing relationships between numbers (which necessarily implies qualitative notions of interdependence). So quite simply once again Conventional (Type 1) Mathematics can only deal with such qualitative notions in a reduced (i.e. merely quantitative) manner.
The Type 2 approach as to the relationship as between the primes and the natural numbers completely inverts the nature of Type 1 appreciation.
Here each prime number - except 1 - is viewed in ordinal terms as representing a unique circle of interdependence in natural number terms.
Thus, if we again take the number 3 this is now viewed in ordinal terms as comprising a 1st, 2nd and 3rd member.
When we reflect on the matter ordinal rankings have a merely relative nature. Thus for example what is 2nd (with respect to a group of 2) clearly has a different meaning than what is 2nd (with respect to a group of 3). So in fact for every ordinal number, an unlimited set of possible interpretations exist (depending on the size of the number group).
Now to give an unambiguous meaning to these ordinal numbers we use the Type 2 number system to obtain the corresponding roots of 1.
So the 3 roots of 1 are 1, - .5 + .866i and - .5 - .866i. 1 here corresponds to the 3rd root of 1 (with exponent 3/3 = 1); - .5 + .866i corresponds to the 1st root with exponent 1/3 and - .5 - .866i corresponds to the 2nd root of 1 with exponent 2/3.
The circular interdependence of these roots is demonstrated by the fact that the sum of the 3 values = 0. In more general terms the circular interdependence of the n root of 1 is demonstrated by the fact that the sum of the n roots = 0 (except where n = 1). Once again this clearly indicates that qualitative notions of interdependence can be given no meaning in a merely linear interpretation (i.e. 1-dimensional). And once again it is such linear rational interpretation that characterises the very nature of Conventional Mathematics!
The uniqueness of the prime numbers with respect to the number of roots of unity is that with the exception of 1 (which is common to all roots) the other members are uniquely defined i.e. cannot occur in any smaller number sequence of roots. I refer to these roots (other than 1) as the non-trivial zeros of the Zeta 2 Function.
So where n = p roots are involved the non-trivial zeros arise as unique solutions for the equation,
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0
The non-trivial zeros for the Zeta 1 Functions arise as solutions for the infinite sequence of terms:
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
The non-trivial zeros for the Zeta 2 Function correspondingly arise for the complementary reverse equation (i.e. where the exponent becomes the base quantity and the base quantity the exponent) with a finite no. of terms.
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
Thus from the Type 2 perspective the very nature of the prime numbers is completely inverted. Whereas in Type 1 cardinal terms they are considered as the most independent of numbers (serving as the building blocks of the natural number system), in Type 2 terms they are now seen in complementary fashion as the most interdependent of all numbers (that are defined in circular terms by a unique set of natural numbers now given an ordinal interpretation).
So from the Type 1 perspective we use the primes in a cardinal manner to generate the natural numbers in a unique fashion; from the corresponding Type 2 perspective we use the natural numbers in ordinal terms to define uniquely each prime number!
Now when one provides both the Type 1 and Type 2 perspectives, it becomes readily apparent that the primes and natural numbers are ultimately fully interdependent with each other in an ineffable manner that both transcends all phenomenal attempts at interpretation and yet is already immanent (i.e. inherent) in physical phenomena as soon as they arise in experience.
The Type 2 approach to number is fully complementary with the Type 1 perspective.
Whereas in the Type 1, number is viewed in linear terms as a separate independent entity, by contrast in the type 2 it is viewed in circular terms as a holistic interdependent relationship.
So once again in the Type 1, the number 3 for example is viewed as an independent whole integer on the number line in cardinal terms; by contrast in the Type 2, the number 3 is viewed in holistic terms as entailing the interdependence of its 3 individual ordinal members - expressed as the 3 roots of 1 - in a circular manner.
Of course the sum of these 3 roots = 0 entailing that this qualitative notion of interdependence strictly has no quantitative significance.
However as stated earlier whereas the Type 1 analytic aspect of number has a (hidden) holistic interpretation, equally the Type 2 holistic aspect of number can be given a (hidden) analytic expression.
Thus again, the Type 1 approach in qualitative terms is based on a linear (1-dimensional) method of interpretation. The (hidden) holistic aspect requires reflecting “higher” dimensional interpretations (where numbers representing dimensions are given a corresponding qualitative meaning).
By contrast the Type 2 approach is based directly on such higher dimensional interpretation. The (hidden) analytic aspect requires reducing all numerical values in a linear quantitative manner.
This simply implies with respect to the various roots of 1, treating all values as positive (and imaginary values as real).
We can thereby give a positive real expression to all cos and sin parts of the roots of 1.
I have expressed before that such a treatment leads to a complementary (Type 2) Prime Number Theorem and equally a complementary (Type 2) Riemann Hypothesis.
What we do here is to initially sum up the cos and sin components of roots separately.
For example the sum of the cos parts of the 3 roots of 1 (in this real positive manner) = 1 + ½ + ½ = 2. We then get the average of the sum of these 3 parts = .66666.
We then do the same for the 3 sin parts. The value for one of these parts = 0 with each of the other parts = .866.
Therefore the sum of the 3 roots = 0 + .866 + .866 = 1.732 (approx.)
The average of the sum of 3 parts = .577 (approx.)
Now what I found many years ago is that when the number of n roots increases (initially sticking to prime numbers for n) that the average of the n roots quickly zones in on the constant value 2/π.
What is remarkable is that 2/π = i/ln i.
Though I did not realise the connection immediately, this latter formulation points to the Type 2 (imaginary) equivalent of the Type 1 (real) Prime Number Theorem that the average number of primes up to n approximates n/ln n.
I further began to see an important connection in the ratio of the deviations of the cos part from i/ln i to the corresponding deviation (in absolute terms) of the sin part from this value.
In this example illustrating 3 roots the ratio = . 0300…/.0596… = .5033…
Already in this really example the value is very close to .5 and this value is approximated ever more closely as the value of n increases.
Again, though I did not make the connection initially, I gradually began to see this relationship as in fact offering the complementary (Type 2) formulation to the Riemann Hypothesis.
Once again all the non-trivial solutions (i.e. except 1) arise through solving the equation
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
As we have seen this in fact represents the complementary formulation of the Type 1
equation,
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
So in fact when properly viewed from both the Type 1 and Type 2 perspectives, we have two complementary sets of non-trivial zeros arising with two corresponding formulations of the Riemann Hypothesis!
As we know the non-trivial zeros can be used to zone in on the precise location of the prime numbers with respect to the Type 1 formulation.
In other words we can gradually eliminate all unexplained deviations with respect to the general prime number frequency among the natural numbers in this manner.
What is fascinating is that in reverse manner the prime numbers themselves can be used to explain the deviations with respect to the average sin and cos values for a natural number (n) of roots.
Indeed after some investigation I have found this remarkably simple expression to explain to a high level of accuracy such deviations.
This simple expression for the cos part is π/12(n^2) where n is prime.
I had stated before on these blogs that some years ago I manually worked out the average values of the sum of all prime numbered roots up to 127 (calculating also their deviation from 2/π (= i/ln i).
For example the average of the cos part (in this reduced linear quantitative manner) for the 53 roots of 1 = .636712981….
The corresponding deviation from 2/π (= i/ln i) = .000093209698..
Using the expression π/12(n^2) where n = 53, we obtain .000093202…
So we have already predicted the deviation correctly for the first 8 decimal places and this accuracy increases with higher n values!
To get an even more accurate prediction the following expression can be used
π/12(n^2)*{1 + [π/12(n^2)]}.
For n = 53 this yields .0000932087… which now rounded to the 5th significant figure would be correct for the first 9 decimal places.
As the deviation for the average sin value from quickly approaches twice that of the cos value from 2/π (= i/ln i),
The corresponding simple expression to predict this deviation = π/6(n^2)
The actual deviation for n = 53 is .000186411434…
The predicted deviation, i.e. π/6(n^2) = .0001864004…
When we combine cos and sin values the deviation from 4/π (= 2i/ln i)
= π/12(n^2)
So where n is prime the change in the deviation is seen to be based very closely on the square of the prime value!
Then Type 3 interpretation - which alone can reveal the true ultimate significance of the Hypothesis - entails the mutual dynamic synthesis of both Type 1 and Type 2 aspects.
In the same manner as in quantum physics where the wave interpretation of matter equally possesses a particle aspect (and the particle a wave aspect) the Type 1 analytic equally possesses a holistic aspect and the Type 2 holistic equally an analytic aspect of interpretation respectively.
Quite remarkably - and this is the most crucially important point of all - Conventional Mathematics gives no recognition whatsoever in formal terms to the Type 2 aspect of interpretation. Thus it inevitably attempts to interpret meaning that is properly of a holistic qualitative nature in a merely reduced analytic fashion.
The implications of this could not be more far reaching! For not alone is the Riemann Hypothesis not capable of proof in the standard Type 1 context; its very nature cannot be properly appreciated in this manner.
In other words the fundamental nature of the primes (to which the Riemann Hypothesis points) relates in number terms to the ultimate reconciliation of quantitative notions of independence with qualitative notions of interdependence.
While readily admitting its great achievements, the futility of present (Type 1) Mathematics resides in the continued attempt to deal with qualitative holistic notions (that properly relate to interdependence) in a merely reduced quantitative manner (where numbers are treated in an independent fashion). And nowhere is this problem of interpretation more exposed than with respect to the fundamental nature of prime numbers!
From the Type 1 perspective the natural numbers are viewed in cardinal terms (where integers literally reflect collective whole units). Here the primes are viewed as the (independent) building blocks of the natural numbers with every such number (except 1) representing a unique combination of prime factors.
However I have repeatedly pointed to the key limitation of this approach in that viewing numbers in this cardinal manner leaves us with no means of making ordinal distinctions.
Though this issue is repeatedly glossed over in conventional presentation, the making of ordinal distinctions as between numbers strictly reflects qualitative appreciation which does not come from their cardinal nature.
If we attempt to break down a cardinal number such as 3 in terms of its individual units we would represent it as 1 + 1 + 1. However because these units are both independent and homogeneous (i.e. lacking qualitative distinction) it would not be possible to rank them in ordinal fashion i.e. as 1st, 2nd and 3rd members of their collective group (cardinally represented as 3).
In other words the very ability to rank numbers in an ordinal fashion relates to qualitative notions of relational interdependence. And strictly as Type 1 Mathematics views numbers as separate independent units (in merely quantitative terms) it thereby can offer no satisfactory way of establishing relationships between numbers (which necessarily implies qualitative notions of interdependence). So quite simply once again Conventional (Type 1) Mathematics can only deal with such qualitative notions in a reduced (i.e. merely quantitative) manner.
The Type 2 approach as to the relationship as between the primes and the natural numbers completely inverts the nature of Type 1 appreciation.
Here each prime number - except 1 - is viewed in ordinal terms as representing a unique circle of interdependence in natural number terms.
Thus, if we again take the number 3 this is now viewed in ordinal terms as comprising a 1st, 2nd and 3rd member.
When we reflect on the matter ordinal rankings have a merely relative nature. Thus for example what is 2nd (with respect to a group of 2) clearly has a different meaning than what is 2nd (with respect to a group of 3). So in fact for every ordinal number, an unlimited set of possible interpretations exist (depending on the size of the number group).
Now to give an unambiguous meaning to these ordinal numbers we use the Type 2 number system to obtain the corresponding roots of 1.
So the 3 roots of 1 are 1, - .5 + .866i and - .5 - .866i. 1 here corresponds to the 3rd root of 1 (with exponent 3/3 = 1); - .5 + .866i corresponds to the 1st root with exponent 1/3 and - .5 - .866i corresponds to the 2nd root of 1 with exponent 2/3.
The circular interdependence of these roots is demonstrated by the fact that the sum of the 3 values = 0. In more general terms the circular interdependence of the n root of 1 is demonstrated by the fact that the sum of the n roots = 0 (except where n = 1). Once again this clearly indicates that qualitative notions of interdependence can be given no meaning in a merely linear interpretation (i.e. 1-dimensional). And once again it is such linear rational interpretation that characterises the very nature of Conventional Mathematics!
The uniqueness of the prime numbers with respect to the number of roots of unity is that with the exception of 1 (which is common to all roots) the other members are uniquely defined i.e. cannot occur in any smaller number sequence of roots. I refer to these roots (other than 1) as the non-trivial zeros of the Zeta 2 Function.
So where n = p roots are involved the non-trivial zeros arise as unique solutions for the equation,
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0
The non-trivial zeros for the Zeta 1 Functions arise as solutions for the infinite sequence of terms:
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
The non-trivial zeros for the Zeta 2 Function correspondingly arise for the complementary reverse equation (i.e. where the exponent becomes the base quantity and the base quantity the exponent) with a finite no. of terms.
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
Thus from the Type 2 perspective the very nature of the prime numbers is completely inverted. Whereas in Type 1 cardinal terms they are considered as the most independent of numbers (serving as the building blocks of the natural number system), in Type 2 terms they are now seen in complementary fashion as the most interdependent of all numbers (that are defined in circular terms by a unique set of natural numbers now given an ordinal interpretation).
So from the Type 1 perspective we use the primes in a cardinal manner to generate the natural numbers in a unique fashion; from the corresponding Type 2 perspective we use the natural numbers in ordinal terms to define uniquely each prime number!
Now when one provides both the Type 1 and Type 2 perspectives, it becomes readily apparent that the primes and natural numbers are ultimately fully interdependent with each other in an ineffable manner that both transcends all phenomenal attempts at interpretation and yet is already immanent (i.e. inherent) in physical phenomena as soon as they arise in experience.
The Type 2 approach to number is fully complementary with the Type 1 perspective.
Whereas in the Type 1, number is viewed in linear terms as a separate independent entity, by contrast in the type 2 it is viewed in circular terms as a holistic interdependent relationship.
So once again in the Type 1, the number 3 for example is viewed as an independent whole integer on the number line in cardinal terms; by contrast in the Type 2, the number 3 is viewed in holistic terms as entailing the interdependence of its 3 individual ordinal members - expressed as the 3 roots of 1 - in a circular manner.
Of course the sum of these 3 roots = 0 entailing that this qualitative notion of interdependence strictly has no quantitative significance.
However as stated earlier whereas the Type 1 analytic aspect of number has a (hidden) holistic interpretation, equally the Type 2 holistic aspect of number can be given a (hidden) analytic expression.
Thus again, the Type 1 approach in qualitative terms is based on a linear (1-dimensional) method of interpretation. The (hidden) holistic aspect requires reflecting “higher” dimensional interpretations (where numbers representing dimensions are given a corresponding qualitative meaning).
By contrast the Type 2 approach is based directly on such higher dimensional interpretation. The (hidden) analytic aspect requires reducing all numerical values in a linear quantitative manner.
This simply implies with respect to the various roots of 1, treating all values as positive (and imaginary values as real).
We can thereby give a positive real expression to all cos and sin parts of the roots of 1.
I have expressed before that such a treatment leads to a complementary (Type 2) Prime Number Theorem and equally a complementary (Type 2) Riemann Hypothesis.
What we do here is to initially sum up the cos and sin components of roots separately.
For example the sum of the cos parts of the 3 roots of 1 (in this real positive manner) = 1 + ½ + ½ = 2. We then get the average of the sum of these 3 parts = .66666.
We then do the same for the 3 sin parts. The value for one of these parts = 0 with each of the other parts = .866.
Therefore the sum of the 3 roots = 0 + .866 + .866 = 1.732 (approx.)
The average of the sum of 3 parts = .577 (approx.)
Now what I found many years ago is that when the number of n roots increases (initially sticking to prime numbers for n) that the average of the n roots quickly zones in on the constant value 2/π.
What is remarkable is that 2/π = i/ln i.
Though I did not realise the connection immediately, this latter formulation points to the Type 2 (imaginary) equivalent of the Type 1 (real) Prime Number Theorem that the average number of primes up to n approximates n/ln n.
I further began to see an important connection in the ratio of the deviations of the cos part from i/ln i to the corresponding deviation (in absolute terms) of the sin part from this value.
In this example illustrating 3 roots the ratio = . 0300…/.0596… = .5033…
Already in this really example the value is very close to .5 and this value is approximated ever more closely as the value of n increases.
Again, though I did not make the connection initially, I gradually began to see this relationship as in fact offering the complementary (Type 2) formulation to the Riemann Hypothesis.
Once again all the non-trivial solutions (i.e. except 1) arise through solving the equation
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
As we have seen this in fact represents the complementary formulation of the Type 1
equation,
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
So in fact when properly viewed from both the Type 1 and Type 2 perspectives, we have two complementary sets of non-trivial zeros arising with two corresponding formulations of the Riemann Hypothesis!
As we know the non-trivial zeros can be used to zone in on the precise location of the prime numbers with respect to the Type 1 formulation.
In other words we can gradually eliminate all unexplained deviations with respect to the general prime number frequency among the natural numbers in this manner.
What is fascinating is that in reverse manner the prime numbers themselves can be used to explain the deviations with respect to the average sin and cos values for a natural number (n) of roots.
Indeed after some investigation I have found this remarkably simple expression to explain to a high level of accuracy such deviations.
This simple expression for the cos part is π/12(n^2) where n is prime.
I had stated before on these blogs that some years ago I manually worked out the average values of the sum of all prime numbered roots up to 127 (calculating also their deviation from 2/π (= i/ln i).
For example the average of the cos part (in this reduced linear quantitative manner) for the 53 roots of 1 = .636712981….
The corresponding deviation from 2/π (= i/ln i) = .000093209698..
Using the expression π/12(n^2) where n = 53, we obtain .000093202…
So we have already predicted the deviation correctly for the first 8 decimal places and this accuracy increases with higher n values!
To get an even more accurate prediction the following expression can be used
π/12(n^2)*{1 + [π/12(n^2)]}.
For n = 53 this yields .0000932087… which now rounded to the 5th significant figure would be correct for the first 9 decimal places.
As the deviation for the average sin value from quickly approaches twice that of the cos value from 2/π (= i/ln i),
The corresponding simple expression to predict this deviation = π/6(n^2)
The actual deviation for n = 53 is .000186411434…
The predicted deviation, i.e. π/6(n^2) = .0001864004…
When we combine cos and sin values the deviation from 4/π (= 2i/ln i)
= π/12(n^2)
So where n is prime the change in the deviation is seen to be based very closely on the square of the prime value!
Subscribe to:
Posts (Atom)