Wednesday, August 31, 2016

Riemann Hypothesis: New Perspective (15)

We have already looked at the bi-directional relationship internally (within each prime) of both its quantitative and qualitative aspects. Thus the unit members of the prime from the cardinal (quantitative) perspective are balanced by a set of natural number members from the corresponding ordinal (qualitative) perspective.

Put another way the cardinal notion of a prime (in quantitative terms) can have no strict meaning in the absence of its natural number ordinal members (from a qualitative perspective).

Equally, in reverse terms, the natural number ordinal members can have no strict meaning in the absence of the cardinal notion of the prime.

So for example 3 as a prime has no strict meaning in the absence of its 1st, 2nd and 3rd members; likewise these 1st, 2nd and 3rd members in qualitative terms, have no strict meaning in the absence of the quantitative notion of 3.

Therefore, properly understood, each prime must be understood in a dynamic bi-directional manner entailing both cardinal and ordinal aspects (which are, relatively, quantitative and qualitative with respect to each other).  
We saw in yesterday’s blog entry that if s is a prime, that the sth position (with respect to s) always reduces to standard cardinal interpretation and that this represents the default treatment of ordinal numbers in conventional mathematical terms. This in turn indirectly equates in quantitative terms, with the fact that in every case, one of the s roots of 1 = 1.

We then saw that there exists a Zeta 2 function that complements the well known Zeta 1 (i.e. Riemann) zeta function, where its zeros provide an indirect quantitative means of giving unique expression to all the non-trivial ordinal positions associated with each prime.

So the crucial function of the zeros - when appreciated from this dynamic (Type 3) perspective - is that they provide the ready means of indirectly converting from a qualitative to quantitative type interpretation.  
So for example in the case of the prime number 3, the Zeta 2 function is given as,
1 + s + s= 0.

Therefore the two solutions to this equation provide unique quantitative conversion of the qualitative notion of 1st and 2nd respectively (in the context of 3).

And by extension the Zeta 2 function can thereby be used to provide unique quantitative conversions for (non-trivial) ordinal positions associated with every prime!

All this provides an important basis for appreciating the corresponding external bi-directional relationship as between the primes and the number system as a whole. 

Now, when we look externally at this relationship we find that all natural numbers in quantitative terms are uniquely composed of a combination of one or more primes.

So for the primes themselves only one factor is involved (excluding the “trivial” factor of 1 from consideration).

Then for a composite number such as “6” two factors are involved.

So 6 = 2 * 3 (which is the unique prime factor combination for this number).

And of course prime factors can be repeated.

So 12 for example has 3 prime factors where 2 is repeated twice (i.e. 12 = 2 * 2 * 3).

Now this is all well and good insofar as it goes, but unfortunately as we shall see, completely one-sided.

So in standard mathematical interpretation, the primes represent the unique independent “building blocks” of the natural numbers in a merely quantitative manner.

However what is crucially overlooked in such conventional mathematical interpretation is that - quite literally - a corresponding qualitative dimension arises whenever the multiplication of numbers takes place.

This fact - which I have often recited - hit me forcibly at the age of 10 when studying simple concrete problems involving the areas of fields.

So, for example, if one imagines a large field with length 3 km and width 2 km, the corresponding area will be given in square (i.e. 2-dimensional) units.

Though from a quantitative perspective the answer is indeed 6, a qualitative transformation in the nature of the units has thereby taken place through the very process of multiplication.

There is another simple way also of coming to appreciate this qualitative connection.

Imagine there are 2 rows - say of cars - with 3 cars in each row.

Now from the perspective of addition, one would treat all the items as independent.

Therefore one could count up the 3 items in one row (= 3) and then proceed to the second row again counting up the 3 items (= 3) and then add the two rows.

So we are here treating the items in each row as independent (and indeed the rows themselves as independent).
In this way the total no. of cars = 3 + 3 = 6.

However what is vital to carry out multiplication, is the corresponding recognition of the interdependence of each row (which thereby assumes a common similarity as between the 3 items in each row).

So the very reason one can now relate the operator 2 with 3, though multiplication, is because of the recognition of the common similarity (with respect to the cars in each row).

However such similarity applies to the interdependence of the items with each other (which is qualitative in nature).

Therefore, properly understood, multiplication necessarily entails notions of both quantitative independence and qualitative interdependence respectively, which can then only be properly appreciated in a dynamic relative manner.

So the conventional interpretation of multiplication reduces its nature to that of addition (where the quantitative independence of unit members is solely maintained).

Therefore in conventional terms, in our example, we start with the quantitative “building blocks” i.e. 2 and 3 which are defined in linear (1-dimensional terms) i.e. on the number line.

However when we multiply these two numbers, even though their dimensional nature has changed, the result 6 is still represented on the same number line!

So the crucial necessary qualitative nature of multiplication is thereby missed.

Thus if one is to appreciate the true significance of the Zeta 1 (i.e. Riemann) zeros, it is vital to recognise clearly the qualitative nature of multiplication.

Whereas from a quantitative perspective, the identity of the composite natural numbers is indeed based on the primes, strictly speaking this all reversed in qualitative terms, whereby the identity of the primes is now based on their unique relationship with the natural numbers!

In other words with respect to the composites, the primes have a new interdependent relationship as factors. Therefore though the primes are indeed independent in a quantitative sense (as separate “building blocks”), through relationship with each other as the factors of composite numbers, they likewise share a qualitative interdependence with each other.  

Now I have made this simple point before though in truth it is extremely subtle to understanding, that the very nature of number keeps switching, as it were, between both a particle and wave identity (without ever being noticed in conventional terms).

Thus once again the standard quantitative definition of number is based on the independence of each of its unit members.
So 3 = 1 + 1 + 1 (with each homogeneous unit independent in a quantitative sense).

However if we now say for example that a number - say 30 - has 3 factors the very nature of 3 has now changed (from a dynamic interactive perspective).

The very point about each factor is that its identtity implies that it is related to a composite number. So in this context, 2, 3 and 5 assume a new qualitative identity as factors through their common relationship to the number 30!

And likewise when we now say that 30 has 3 factors, 3 = 1 + 1 + 1 (but now - relatively - in an interdependent rather than independent sense).

So once again this is akin to nature of left and right turns at a crossroads, where what is deemed left and right is merely relative depending on a reference frame based on the direction of approach (which can keep switching).

So just as we saw how the Zeta 2 zeros can be used to deal with the very important internal question of how ordinal qualitative notions (with respect to the members of each prime) can be indirectly converted in a consistent quantitative manner, we now have the parallel problem of how the Zeta 1 (Riemann) zeros can equally be used externally to indirectly convert the qualitative nature of the primes (as factors of natural numbers) in a quantitative manner.

And I have shown before how the frequency of the Riemann zeros is intimately linked to the factor frequency of the natural numbers.

Therefore once again if one accumulates the frequency of the proper factors of the natural numbers (up to n), the total will bear a remarkably close relationship with the corresponding frequency of the non-trivial Riemann zeros (up to t) where n = t/2π.

Therefore the key importance of the Riemann zeros (from this perspective) is that they provide an indirect means of converting the qualitative nature of the primes (through their relationship with the natural numbers as factors) in a quantitative manner.

And both the Zeta 1 and Zeta 2 functions are themselves complementary, relating to the bi-directional relationship of the primes and natural numbers (externally and internally) in both quantitative and qualitative terms. 

So the Zeta 1 (Riemann) function can be expressed:

1– s + 2– s + 3– s + 4– s + …… = 0.

The corresponding Zeta 2 function can then be expressed as

1 + s1 + s2 + s 3 + …… = 0 (initially where s is prime).

Notice the complementarity! Whereas the natural numbers represent the base values with respect to the Zeta 1, they represent the dimensional values with respect to the Zeta 2; and whereas the unknown (s) represents the dimensional values with respect to the Zeta 1, they represent the base values with respect to the Zeta 2.

Also whereas the dimensional values are negative in Zeta 1, they are positive in Zeta 2; finally whereas Zeta 1 represents an infinite, Zeta 1 represents a finite series respectively.

Of course ultimately both internal and external aspects of this bi-directional relationship between the primes and natural numbers are themselves fully interdependent.

From the quantitative perspective, it does indeed appear that the natural numbers are derived from the primes; however equally from the qualitative perspective the primes appear to now obtain their positions (expressing their relationship with each other) through their identity as unique factors of the natural numbers! 

And if one thinks about this for a moment, without knowledge of the gaps between the primes, it would not be possible to list the primes (as quantitative "building blocks" of the natural number system); likewise without knowledge of their quantitative value it would not be possible to establish the gaps between the primes (that express their qualitative relationship with each other).
So rather than an absolute quantitative relationship connecting the primes with the natural numbers in a one-way static manner, rather we have a dynamic two-way interactive relationship that operates relatively in both quantitative and qualitative terms.

This implies that an incredible dynamic synchronicity characterises the relationship between the primes and natural numbers , where they ultimately approach total identity with each other in an ineffable and utterly mysterious manner.

The very ability to - literally - "see" in a pure intuitive manner this remarkable synchronicity, where number as form becomes dynamically inseparable from number as energy, represents the holistic extreme of mathematical understanding (where quantitative can no longer be separated from qualitative appreciation).

At the other extreme we have the totally abstract rational understanding of number representing absolutely fixed forms where quantitative is totally separated (in formal terms) from qualitative appreciation.

The fundamental requirement for Mathematics is then the consistent integration of both types of understanding. However this will require that equal emphasis is given to both analytic and holistic aspects.

And without any hint of exaggeration, Mathematics at present, despite its enormous advances, is hugely unbalanced (through a complete neglect of its vitally important holistic aspect).

Nothing less than a total revolution in perspective will rectify this problem!

Indeed this may be slowly initiated through the inevitable continued failure to prove the Riemann Hypothesis, which not alone cannot be proven (or disproven) but much more importantly cannot be properly appreciated in conventional mathematical terms.   

Sunday, August 28, 2016

Riemann Hypothesis: New Perspective (14)

There are two ways in which we can look at the bi-directional relationship between the primes and natural numbers, which we can usefully refer to as the internal and external approaches respectively.
We will look here in this blog entry at the internal approach.

In accordance with this approach, when we look within each prime, we find that it can be expressed in two ways, which in dynamic interactive terms are complementary.

First we have the cardinal definition where each prime is defined as composed of homogeneous units in a merely quantitative manner.

Therefore in our much used example 3 (as prime) = 1 + 1 + 1.

Then secondly, we have the alternative ordinal definition, where each prime is defined by a unique set of ordinal members in a qualitative manner.

Therefore in this context 3 = 1st + 2nd + 3rd.

Both of these, considered as separate, constitute analytic interpretation that appears unambiguous representing in fact equivalent statements..

However when considered holistically in dynamic interactive terms as complementary, direct paradox arises.
(This again parallels the example of a crossroads where both left and right turns at the crossroads appear unambiguous when approached from just one direction (N or S); however when when both N and S directions of approach are viewed simultaneously, left and right turns are now rendered paradoxical). 

In the first (cardinal) case each prime is considered as an independent "building block" of the natural number system; however in the second (ordinal) case, each prime is already composed of an interdependent set of ordinal natural number members! Thus from this dynamic perspective, each prime and its ordinal set of natural numbers are mutually interdependent with each other in a manner ultimately approaching total synchronicity

Now I have stated that the two complementary perspectives - with respect to the cardinal and ordinal nature of the primes - entail the corresponding complementarity of the cognitive and affective aspects of understanding.

Thus when the cardinal numbers are understood in conventional cognitive terms, true ordinal recognition (i.e. that is not of a reduced nature) entails corresponding affective appreciation

And then when reference frames are reversed, when a cardinal number is understood in affective terms (where the individual uniqueness of the number is recognised) then ordinal appreciation entails corresponding cognitive appreciation.

The conventional quantitative notion of 3 is properly expressed (in Type 1 terms) as 31.

The corresponding qualitative notion of 3 is then expressed (in Type 2 terms) as 13.

Holistic appreciation then entails bringing these two aspects together whereby the qualitative dimensional notion of 3 indirectly finds its conventional (1-dimensional) expression. This then entails obtaining the 3 roots of 1 (i.e  11/3, 12/3 and 13/3) respectively.

In this way we can indirectly convert qualitative Type 2 notions in a quantitative manner.

In other words the ordinal notion of 1st (of 3) is given as – .5 + .866i; the ordinal notion of 2nd (of 3) is then given as – .5 .866i and the ordinal notion of 3rd (of 3) as given as 1. And these three positions are fully interchangeable with each other depending on the criterion chosen for (relative) ordinal rankings..

We can now perhaps appreciate precisely how ordinal notions are reduced in conventional mathematical terms. When we speak of 1st, its meaning is fixed as the last of a group of 1 item
(i.e.  11/1). Then when we speak of 2nd, its meaning is fixed as the last of a group of two items
 (i.e. 12/2). and then when we speak of 3rd, its meaning is fixed as the last member of 3 items (i.e. 13/3). 

So here 1st, 2nd and 3rd in turn are reduced to 1 (as the last fixed unit in a series).

Therefore, in this context 1st + 2nd + 3rd in ordinal terms becomes indistinguishable from 1 + 1 + 1 (= 3) in a cardinal manner. 

However the true holistic meaning of 1st, 2nd and 3rd have a potential (interchangeable) rather than actual (fixed) meaning.

Therefore, when attempting to rank number members, each in turn can be 1st, 2nd or 3rd (depending on the relative context). So what might for example be ranked 1st (according to one criterion, might be ranked 2nd according to another and then 3rd with respect to yet some other criterion). And this likewise applies to the other two members.

So the true relative interdependence of the 3 members in ordinal terms is reflected by the fact that each can be ranked 1st, 2nd or 3rd (depending on arbitrary context).

And this reflects the holistic meaning of the number 3, where each unit member potentially shares all three possible positions (1st, 2nd and 3rd). And this qualitative nature of "threeness" is directly appreciated in an intuitive manner that then indirectly can be expressed in a circular rational fashion through the 3 roots of 1.

So the quality of 3 (i.e. as "threeness") arises when one intuitively appreciates the interdependence - rather than independence - of the three unit members of a number  This entails - as we have seen - that each member shares the potential ordinal quality of 1st, 2nd and 3rd respectively.   

With such ordinal rankings, there is always one less degree of freedom that the number of units involved as when s – 1 positions are ranked the remaining position is automatically determined.
We can refer to this then as a trivial, with the other positions constituting non-trivial solutions!

So if we refer to the case where the sth root = 1, this concurs with analytic rather than holistic appreciation. Therefore to isolate the roots - indirectly corresponding to holistic understanding - we divide sn – 1 = 0 by s – 1 = 0 or in preferred form,

1 – sn = 0 by 1 – s = 0, to obtain

1 + s + s2 + ...... + sn – 1 = 0 (where initially s is prime).

Again it can be easily seen that as the s – 1 roots (other than 1) are unique for each prime, this means that the non-trivial natural number ordinal positions are thereby likewise unique for every prime.  

This is what I refer to as the Zeta 2 function which complements in an internal fashion the better known Zeta 1 function (Riemann zeta function) that expresses the external relationship as between the primes and zeros.

In other words the very purpose of the Zeta 2 zeros (as solutions to this finite equation) is to express the holistic notion of the qualitative interdependence of the unique set of natural number ordinal members of each prime, indirectly in a quantitative manner.

In this way the Zeta 2 zeros give quantitative expression to the qualitative notion of the uniqueness of all non-trivial ordinal positions  (1st, 2nd, 3rd,.........,  s – 1th) with respect to each prime.

So in the case of 3 (which of course is prime),

1 + s + s = 0.

Therefore the two solutions for s, i.e. – .5 + .866i and – .5 .866i give unique quantitative expression to the qualitative notions of 1st and 2nd (in the context of 3 members).
In this manner, therefore unique quantitative expression can be indirectly provided for the qualitative nature of all the (non-trivial) natural number ordinal members of each prime.

Thursday, August 11, 2016

Riemann Hypothesis: New Perspective (13)

I have mentioned that number recognition entails - relatively - both cognitive and affective modes of understanding.

Through the former aspect, one comes to appreciation of the quantitative (impersonal) nature of number in cardinal terms; through the latter one comes to corresponding appreciation of its qualitative (personal) nature in an ordinal manner.
So number is thereby inherently dynamic, entailing both collective whole and individual part aspects.

So again the collective whole aspect of a natural number (as cardinal) entails the homogeneous similarity of its independent part units (which thereby lack a qualitative identity); however the individual part aspects of the number (as ordinal) entail a unique distinction with respect to its unit parts (with the collective sum thereby lacking a quantitative identity).

Therefore, when properly understood, in a dynamic interactive manner, these two aspects of number are revealed as fully complementary with each other!  

However in conventional mathematical terms due to its rigid absolute framework, the qualitative aspect of number is thereby reduced to mere quantitative interpretation

Now, I am aware that I have stated these points repeatedly. However I believe it is necessary so as to fully convince you that the present accepted mathematical framework - which is rarely ever questioned - is simply not fit for purpose.

Therefore an enormous revolution in understanding now awaits, which promises to be the greatest yet to occur in our intellectual history. This will intimately affect every possible notion in mathematics and the sciences with dramatic consequences for the future evolution of our world.

So far in the present discussion, I have concentrated on the analytic understanding of number.

Once again, the analytic has two aspects relating properly to cognitive and affective type appreciation respectively.

It is interesting how in accepted understanding, the highest form of reason requires the ability to abstract from more limited concrete information provided by the senses.

Therefore in earlier childhood, one only can come to a knowledge of number with reference to simple concrete type examples (where counting is still associated with the concrete objects of counting).

So both cognitive and affective aspects are here naturally involved in number experience in a somewhat immature manner. 

Then later one becomes able to continually abstract from mere concrete understanding to obtain a purer mathematical appreciation of number (based on specialised reason).

However the reverse also is the case. Therefore one develops the pure affective appreciation of number through the corresponding ability to detach the senses from reason.

In this way, one becomes able as it were to properly distinguish the intuitive aspect of sense understanding from reason.
As we have seen however, because this specialised ability is not even recognised in conventional mathematical terms, genuine intuition, where it arises, is simply reduced in rational terms.

So from a dynamic interactive perspective, both cognitive and affective dimensions are necessarily involved in number experience, which are - initially - associated with reason and intuition respectively.

Thus in the past few entries I have been attempting to clarify the proper role of  both the cognitive aspect (refined reason) and the affective aspect ( refined intuition) in the analytical understanding of number.

Thus it requires a very developed form of intuition (where the affective can be properly differentiated from the cognitive aspect) to recognise that the ordinal recognition of number is not strictly provided by reason. Once again when one attempts such recognition through reason, ordinal notions lose their qualitative uniqueness and become thereby reduced in a mere quantitative manner!

However there is also the holistic recognition of number (entailing the simultaneous juxtaposition of complementary opposite reference frames of understanding).

And this also has both cognitive and affective aspects, which I will return to in the next entry.

Wednesday, August 10, 2016

Riemann Hypothesis: New Perspective (12)

In our present culture, a huge split separates the domain of reason from that of emotion exemplified by a corresponding clear split as between the sciences and the arts.

And nowhere is that split more clearly evident than in the conventional interpretation of Mathematics, which formally is based exclusively on a reduced form of rational interpretation.

I have already identified in earlier blog entries with respect to the basic operations of addition and multiplication (and subtraction and division) that two distinct types are involved in each case (which are not properly distinguished in conventional terms).

So once again for example when we say that 3 (as a cardinal number)  = 1 + 1 + 1, implicit in this definition is the assumption that these units are of a homogeneous independent quantitative nature (thereby lacking any qualitative identity).

However when we express 3 in the alternative ordinal fashion, 3 = 1st + 2nd + 3rd, implicit in this definition is the assumption that each of the units is now unique in an individual qualitative manner where, strictly, the sum of the units lacks any quantitative identity!

So properly understood, we have now switched from the quantitative to the qualitative notion of 3 (as "threeness").

Therefore once we accept that both cardinal and ordinal notions of number must be coherently related, we have to abandon completely the conventional assumption of natural numbers as possessing an absolute independent identity.

Rather, number is now understood in a dynamic interactive manner, which entails notions of both relative independence and relative interdependence respectively.

So now number is clearly seen as composed of two complementary aspects that interact with each other in a relative - rather than absolute - manner.

Therefore with respect to the cardinal (quantitative) aspect, we can now say that addition of the component units of 3 (i.e. 1 + 1 + 1) relates to their independent status (in relative terms).

However, the corresponding addition with respect to the three ordinal units (1st + 2nd + 3rd) relates to their interdependent status (again in relative terms).

So we have now properly recognised the two distinct aspects of addition in relative terms (as relating to independent and interdependent units respectively).

Once more - and it is vital to properly grasp this key point - the abstract stance of conventional mathematical interpretation is fatally flawed in that it attempts to view number reality in an absolute rigid manner. This thereby reduces in every context meaning that is truly qualitative in a merely quantitative manner. Alternatively expressed this implies the corresponding reduction of ordinal notions in cardinal terms.

Properly understood however, number reality - which is inherently dynamic and interactive - is always relates in experience to both its complementary physical and psychological expressions.

The deeper implication of what I am stating here is that we can no longer hope, from this dynamic perspective, to view Mathematics in merely cognitive terms as relating to rational truth!

We have already seen that Mathematics contains both positive (conscious) and negative (unconscious) aspects.

Whereas one may indeed initially identify the positive aspect with rational understanding, the corresponding negative aspect relates direct to intuitive type appreciation (that indirectly can be conveyed in a circular rational manner).

So once more, mathematicians may indeed informally accept the importance of intuition (especially for creative work). However, because of the rigid framework from which interpretation is attempted, they have no way of properly distinguishing intuition from reason. In effect intuition is simply reduced to reason, with all proof formally expressed in rational terms.

Likewise however - perhaps more surprisingly - one eventually discovers  that Mathematics also contains cognitive (real) and affective (imaginary) aspects.

And in a comprehensive mathematical understanding, both cognitive and affective aspects must be properly harmonised with each other

So again with respect to our simple example, when one recognises in quantitative terms that 3 = 1 + 1 + 1, this directly entails cognitive type understanding (of a rational impersonal nature).

However implicitly when one recognises in qualitative terms that 3 (as "threeness" =  1st + 2nd + 3rd) this entails affective type understanding (where the senses are involved in a personal manner).

Unfortunately, appreciation of this latter type of affective recognition has all but been completely lost in the modern understanding of Mathematics.
Though implicitly it still requires at least some small degree of affective sense recognition to successfully make ordinal type distinctions, at an explicit level this recognition is then reduced in a merely rational impersonal manner.

So one of the truly damaging effects of the great cultural influence of mathematical - and by extension all scientific - understanding is that this is exercising an enormous influence in reducing qualitative to quantitative meaning in so many areas of our lives.

And the root of this very problem lies at the very heart of Mathematics itself in the attempted preservation of  but a limited - and ultimately hugely distorted - rigid understanding of the true nature of all its relationships.

Tuesday, August 9, 2016

Riemann Hypothesis: New Perspective (11)

I have mentioned before how a future golden age of Mathematics will contain at least three distinctive ways of interpreting mathematical symbols.

1) The conventional rational approach based on the quantitative interpretation of mathematical symbols in a conscious manner.

2) The - largely - unrecognised intuitive approach based on the qualitative interpretation of mathematical symbols in an unconscious manner. Though this approach is indeed directly based on refined intuitive type recognition that cannot be successfully reduced in standard rational terms, indirectly it can be intellectually translated in a (circular) rational manner (entailing paradox from a dualistic perspective).

3) the comprehensive radial approach based on the mutual interpenetration, in a coherent integrated fashion, of mathematical symbols in both a conscious and unconscious manner.

However the great surprise that awaits entails the additional recognition that all mathematical symbols have both cognitive rational and affective sense interpretations.

So comprehensive mathematical appreciation of symbols entails the emotional as well as rational domain!

Indeed ultimately it entails also the volitional domain as the very means for successfully harmonising - relatively - both conscious and unconscious aspects with respect to cognitive and affective aspects is through the volitional aspect (i.e. will).

Now in this context it would be helpful to carefully distinguish both these two aspects of the psychological recognition of mathematical symbols through a simple example.

Imagine one is looking at 3 cars (say parked in a driveway)!

The cognitive recognition here relates to the common collective identity of the cars (as belonging to the same class).

This directly concurs with the cardinal notion of number (where each unit of the number in question enjoys an impersonal homogeneous identity in quantitative terms.
So 3 = 1 + 1 + 1.

However affective (sense) recognition is quite distinct in relating to the unique individual identity of each car (arising from their ordinal relationship with each other.

.This then directly concurs with the ordinal notion of number where each unit now enjoys a distinct personal identity in a qualitative manner.

So from this perspective 3 = 1st + 2nd + 3rd!

In conventional mathematical terms, the latter each interpretation is simply reduced in a cardinal manner. So the personal unique identity of each item - corresponding initially with sense recognition of an affective kind - is thereby lost.

However properly understood - when we recognise the true complementary nature of both aspects of number recognition - cognitive and affective aspects are necessarily involved in the dynamics of all number recognition.

Indeed these dynamics relate directly to the true relation of whole and part (and part and whole).
Once more each prime, from the conventional mathematical perspective is considered in a quantitative whole manner (where all units are considered as homogenous and thereby lacking any qualitative distinction).

So again to illustrate, 3 (as a prime) = 1 + 1 + 1 (where the quantitative units lack any qualitative identity).

This concurs with the standard rational (i.e. cognitive) interpretation of number where each prime is considered as a "building block" of the cardinal natural number system.

However, from the complementary (unrecognised) perspective 3 (as a prime) is uniquely defined by its ordinal members in natural number terms.

So 3 = 1st + 2nd + 3rd.

Now here, in reverse, 3 (as the unique combination of individual ordinal units )  strictly lacks a quantitative identity. So 3, in this context, properly relates to "threeness" (as the qualitative nature of 3).

However this latter qualitative recognition i.e. that number units bear a necessary relationship with each other, pertains directly to sense recognition (of an affective kind).

So when one fails to recognise the necessary interaction of both cognitive and affective recognition with respect to each prime, a fundamentally distorted interpretation of the relationship of whole and parts results.

So in conventional mathematical terms - reflecting the dominance of the merely cognitive (rational) aspect of understanding - the number system is interpreted in a merely reduced quantitative manner (where primes are unambiguously viewed as the "building blocks" of the natural numbers).

However when one properly allows for the corresponding affective (sense) aspect of understanding, the number system is likewise seen in a true qualitative manner (where each prime is defined by its natural number members in an ordinal manner).

So from the customary analytic perspective, the relationship between the primes and natural numbers (and natural numbers and primes) is considered in a one-way unambiguous manner.
So in standard (Type 1) terms each prime serves as a quantitative "building block" of the natural number system a cardinal manner.

Then in corresponding (Type 2) terms each prime is already uniquely defined by its natural number mebers in an ordinal manner.

Then the simultaneous recognition of both Type 1 and Type 2 aspects (i.e. Type 3) requires true holistic understanding in the inherent understanding of the number system in a dynamic interactive manner where both aspects - that appear unambiguous from within each reference frame considered in isolation - now appear as deeply paradoxical (when indirectly conveyed in a circular rational manner).

Friday, August 5, 2016

Riemann Hypothesis: New Perspective (10)

For many years, I have been aware of the remarkable similarity - in holistic mathematical terms - as between the notion of prime numbers on the one hand, and primitive instinctive behaviour on the other.

Gradually I have come to realise that they in fact constitute two sides of the same coin, as it were, with respect to their complementary relationship with each other in physical and psychological terms.

Therefore the same dynamic experiential process of coming to understand the true nature of the primes with respect to both their analytic and holistic aspects is the same process as coming to unravel the confused nature of primitive instinctive behaviour in conscious and unconscious fashion.

The very nature of primitive instinctive response is that a direct confusion pertains with respect to the relationship as between the conscious and unconscious aspects of behaviour. Thus the holistic nature of the unconscious in the quest for overall meaning is thereby directly identified with the analytic (localised) nature of conscious phenomena. This then leads to the reduction of holistic meaning (which is ultimately formless in nature) in a merely localised phenomenal manner.

Indeed in earliest infant development when the conscious aspect of personality is still largely merged with the unconscious, primitive instinctive response is so immediate that phenomena of form cannot be properly placed in a holistic dimensional environment of space and time. So these dimensions quickly collapse in infant behaviour with recall of events of the most short-lived nature.

In psychological terms, this parallels the very nature of sub-atomic particles whole existence again is of such a fleeting nature. These particles are so close to the holistic ground of nature, from which they instantaneously emerge that they are yet unable to consolidate a stable phenomenal existence in space and time.

So in the deepest holistic sense the world of sub-atomic particles is characterised by the holistic nature of prime dimensions with respect to space and time. So in a very true sense this behaviour therefore charcterises the primitive (i.e prime) nature of matter.
So when I read some years ago that mathematicians were now discovering that the Riemann zeros (which represent the hidden shadow aspect of the primes) bear an uncanny relationship to energy levels of particles (documented in quantum physics), I for one was not in the least surprised.

Indeed because of the holistic perspective with which I had long looked at these matters I had already suspected that something very similar would in fact be the case!

Now going back to child development, in Western culture, the early stages are given over to the gradual differentiation of the conscious aspect from the unconscious.

Then all going well in teen and adult life, the conscious aspect undergoes increasing rational specialisation.

So from an intellectual perspective, exemplified especially by Conventional Mathematics, understanding is - mistakenly - viewed as standing alone from the unconscious.

Thus the formal interpretation of mathematical relationships is of an absolute static nature that is viewed in a merely rational manner.

Now when once accepts that the true nature of prime numbers entails both conscious and unconscious aspects in a dynamic interactive manner, then the present mathematical approach - despite all its wonderful analytic developments - can be clearly seen as extremely reduced and limited.

So the requirement in terms of full appreciation of the primes - which in a dynamic context is necessarily of a relative approximate nature - is the continuation in later development of corresponding growth with respect to the unconscious aspect leading to the eventual specialisation of holistic intuitive type understanding.

In former times such development was largely associated with the spiritual contemplative traditions, with detailed accounts of the various states that unfold provided.

However what has been greatly overlooked so far in our cultural history is the great relevance that these intuitive states possess for the holistic appreciation of mathematical relationships.

In brief, associated with the accepted analytic (Type 1) interpretation of mathematical symbols in a quantitative manner, is a corresponding - largely unrecognised - holistic (Type 2) interpretation, whereby their corresponding qualitative significance is realised.

For example I made some years ago what I considered at the time the startling discovery that all the stages of psychological (and physical) development were precisely encoded in holistic mathematical manner (based on the qualitative interpretation of the various number types). And just as the analytic interpretation has immense significance for the encoding of information, likewise the holistic interpretation of the same binary digits has immense significance for the potential encoding of all transformation processes!  

Then by far the most comprehensive appreciation of mathematical symbols unfolds when both analytic (rational) and holistic (intuitive) aspects are integrated in a dynamic interactive manner. I frequently refer to this final fullest understanding as radial (Type 3) interpretation.

And the basic contention of these blog entries is that the relationship of the primes to the natural numbers (and natural numbers to the primes) clearly requires radial (Type 3) interpretation for its proper comprehension.

And this physical mathematical quest to understand the relationship of the primes to the natural numbers cannot be divorced from the corresponding psychological quest to finally unravel the unconscious nature of primitive instinctive behaviour so that it can be properly integrated with  rational type understanding.

In other words the two-way relationship of the primes to the natural numbers in a physical mathematical sense, precisely complements in psychological experience the same relationship as between both the conscious and unconscious aspects of personality.

So it is only when both conscious and unconscious are fully integrated in personality that the experiential appreciation of the ultimate synchronous identity of primes and natural numbers (as perfect interacting mirrors of each other) can take place. And this state of union in both physical and psychological terms - which ultimately is ineffable - can only be approximated ever more closely in a dynamic relative manner.

It is now being slowly realised that very intimate connections exist as between the Riemann zeros and atomic energy states in high energy physics.

Some time in the future, it will be equally recognised that the same Riemann zeros bear an intimate connection with the "high energy" states in psychological terms associated with advanced spiritual contemplation.

So from a physical perspective, the Riemann zeros are the numbers - representing energy states - that enable perfect conversion as between the quantitative (Type 1) and qualitative (Type 2) aspects of the number system; equally from a psychological perspective, the Riemann zeros are likewise  those same numbers - now representing (intuitive) energy states - that enable perfect conversion as between both the conscious and unconscious aspects of experience.

And as both physical and psychological aspects are in truth ultimately fully complementary, neither aspect can be properly understood in isolation but rather in intimate relationship with each other.  

Tuesday, August 2, 2016

Riemann Hypothesis: New Perspective (9)

We address here the holistic mathematical reason as to why the Riemann zeros are postulated to lie on the imaginary line through .5!

So far we have concentrated on the dynamic relationship between whole and part i.e. in the manner in which number keeps switching - relatively - as between its quantitative and qualitative (and qualitative and quantitative)  aspects.

However there is another dynamic interaction that necessarily applies to all numbers with respect to both external and internal aspects.

Therefore when one refers to a number as existing "out there" in an objective manner, the - relatively - external aspect of number is involved.

However when one then refers to a number "in here" as a subjective mental perception, then the corresponding internal aspect is involved.

Now in conventional mathematical terms, the internal aspect is effectively reduced in terms of the external in an absolute static manner.

So a mathematician - if pressed - might reluctantly concede that one's actual experience of number entails both the objective notion of number together with its subjective (mental) interpretation.

However, such dynamics will then be quickly forgotten with number in formal terms treated as enjoying an absolute objective existence in an external mathematical environment!

Therefore it is vital to appreciate that - properly understood - number represents a dynamic interaction as between both its external and internal aspects, which in holistic mathematical terms are positive and negative with respect to each other.

Now the key to such balanced dynamic understanding of number is the realisation that both conscious and unconscious aspect of understanding are equally involved.

Thus the actual positing of number with respect to its external direction takes place in a (rational) conscious manner.
The corresponding dynamic negation of this aspect then occurs in an (intuitive) unconscious manner, which causes a switch in the direction of experience.

So now, the actual positing of number takes place in a corresponding internal direction as mental interpretation (again in a conscious manner).

And then once more, dynamic negation with respect to this aspect occurs in an unconscious manner causing another switch in direction.

In this way, one keeps alternating in a balanced manner as between both external and internal aspects of number appreciation (with both understood in a relative manner).

So this refined rational appreciation of number - where neither external nor internal aspect attains dominance - coincides with a corresponding highly refined intuitive appreciation.

In other words, both the conscious and unconscious aspects of mathematical appreciation are then brought fully into harmony with each other.

Now once again with respect to the Riemann Hypothesis the non-trivial zeros are all postulated to lie on the line 1/2 + it (and 1/2 – it) respectively.

These complex numbers refer to dimensional powers with respect to the natural numbers!

Now when we raise the first number i.e. 1 to 1/2 i.e. 11/2, we get – 1.

The clue here then in holistic mathematical terms is that the proper appreciation of the Riemann Hypothesis (i.e. in a dynamic interactive manner) requires that both unconscious (holistic) and conscious (analytic) interpretation be equally involved.

The requirement that all the zeros lie on the line through .5, points directly to the fact that in dynamic interactive terms, ultimately number is equally both physical and psychological in nature i.e. as representing both (objective) fact and (subjective) mental interpretation respectively.  

In fact when one is able to appreciate this necessary dynamic balance as between the external and internal aspects of all numbers then the pure intuitive nature of numbers, as energy states, can be directly appreciated in a psycho spiritual manner. This then complements the corresponding realisation coming from comparison of the zeros with quantum chaotic states in physics that the zeros represent (physical) energy states.

The deeper implications of all this is that properly understood, number is already always inherent in phenomena (with respect to their most fundamental encoding) in both physical and psychological terms. Thus the supposed abstract existence of numbers – that has dominated mathematical thinking now for millennia – represents but a fundamentally distorted perspective!

Therefore for one to approximate experientially as close as possible to the truth enshrined in the Riemann Hypothesis, then it is necessary that both conscious (analytic)) and unconscious (holistic) modes of understanding be employed in the most refined and balanced manner.

This relates firstly to both the external and internal aspects of all number phenomena; secondly it also relates to the quantitative and qualitative aspects (with respect to whole/part interactions) of the same number phenomena.

Such understanding - at its most refined level - then forms the thinnest partition possible as between the (manifest) world of form and the (unmanifest) ineffable origins of all creation.

So number - when appropriately understood - most perfectly bridges this mysterious gap that divides (phenomenal) form from (spiritual) emptiness.

Monday, August 1, 2016

Riemann Hypothesis: New Perspective (8)

We have looked at the apparent fact that all the Riemann zeros lie on an imaginary line (drawn through .5 on the real axis).

Therefore in dynamic interactive terms, the "real" nature of the natural number line i.e. where all real numbers are viewed in linear rational terms (as lying on the 1-dimensional line) is complemented by the "imaginary" nature of a corresponding number line on which all the Riemann zeros are postulated to lie.

An as we have seen - again in dynamic interactive terms - when the interpretation of the "real" number line takes place in the conventional analytic manner (based on the assumed independence of number), then the interpretation of the "imaginary" number line (containing the Riemann zeros) should then rightly take place in a holistic manner (where the corresponding interdependence of number - which appears paradoxical in analytic terms - is now equally emphasised).

Looked at from a psychological experiential perspective, this implies that both conscious (analytic) and (unconscious) holistic appreciation of number be brought into a dynamic equilibrium with each other (the ultimate nature of which is truly ineffable).
Expressed more simply, this implies the balanced recognition of both reason and intuition with respect to all number relationships.

Because of the reduced nature of accepted mathematical interpretation, number is treated solely with respect to its quantitative (independent) nature that is viewed in an absolute manner.

However there is always - inescapably - an unrecognised qualitative aspect to recognition, where one accepts that numbers can be consistently related with each other (i.e. as interdependent with each other).

In conventional mathematical terms this qualitative aspect is blindly assumed to be consistent with the quantitative aspect (in a static absolute manner) which strictly speaking is a completely untenable position.

So once we recognise the equal importance of both independent and interdependent aspects, we must then treat number in a dynamic interactive fashion with complementary aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.

So in this dynamic context, the key issue for Mathematics is that consistency can be maintained as between both quantitative and qualitative aspects.

And this consistency requires than a complementary holistic linear formulation of number exists that  complements the accepted analytic linear interpretation.

And this again is the statement of the Riemann Hypothesis with the additional requirement that the imaginary line - containing the Riemann zeros - passes through .5 on the real axis!

Once again this clearly cannot be proven through conventional mathematical methods (as its axioms already blindly assume consistency).

So its truth depends on acceptance of the twin complementary nature of the number system, which ultimately reflects an initial act of faith in the subsequent consistency of the whole mathematical enterprise.

We also know that for every "positive" expression of a Riemann zero i.e. a + it, a corresponding "negative" expression equally exists i.e. a - it and again the assumption of the Riemann Hypothesis is that a = .5.

Now one might query as to what the second "negative" version of each zero refers!

To briefly recap, I have suggested that the the frequency of the Riemann zeros coincides very closely to the manner in which the natural factors of numbers  accumulate as we move up the number scale. And just as these factors progressively increase (as we move higher up the linear scale to n), likewise it is similar with the trivial zeros moving on a corresponding "circular" scale up to t where n = t/2π.

However, we are referring solely here to the "positive" zeros!

One must remember however that in dynamic interactive terms, switching of reference frames continually takes place.

Therefore when we associate the "positive" zeros with the holistic interpretation of the zeros (that complement the analytic interpretation of the natural number system) then the "negative zeros coincide with the corresponding analytic interpretation of the zeros (that complement the corresponding holistic interpretation of the natural number system).

And of course what is "positive" and "negative" in this dynamic interpretation is merely relative depending on context.

Therefore the fact that all Riemann zeros are postulated to have both a positive and negative equal identity on the imaginary line simply reflects the fact that all these zeros can be given - in dynamic interactive terms - both a holistic and analytic interpretation respectively.