Wednesday, February 29, 2012

Riemann Zeta Function (6)

(Though this extract is very lengthy, it is perhaps the most important I have yet written. It illustrates with respect to the new Riemann Zeta Function for s = 4), the enlarged three strand approach outlined in the last blog (indicating how Type 1, Type 2 and Type 3 Mathematics are applied in this context).

We will continue on now with our examination of the Zeta Function for real dimensional values of s > 1. And for the moment, we are concentrating on the even integer values of s.

In a previous blog we looked at the Function where s = 2.

So we will know move on to examine the Function with respect to the next even number i.e. 4.

Now in attempting this task, I will deliberately illustrate the three strand approach - outlined in yesterday's blog - that is necessary to give a comprehensive interpretation of what is involved.

So we will look first at the Function with respect to s = 4, from the standard Type 1 perspective based merely on quantitative type interpretation.

We will then look at Function with respect to the same value of s = 4, from the holistic Type 2 perspective based on corresponding qualitative interpretation.

Finally we will look at the Function again with respect to the value s = 4, from the more dynamically interactive Type 3 perspective (where both quantitative and qualitative aspects are inherently combined as interdependent).

The first task is the easiest, as extensive work has already been done with respect to the quantitative aspects of the Function.

So when 4 is used as dimensional number (or exponent) in this context the standard linear notion - in qualitative terms - applies. This then enables the dimensional transformation of the terms of the Zeta Function to be interpreted merely in a (reduced) quantitative fashion i.e. in the way that a power of a number is normally understood in Mathematics.

So as Euler discovered the quantitative value of the Function (when s = 4) is (pi^4)/90.

Of course Euler discovered a lot more and was able to prove that for all (positive) even integer values of s, the quantitative result of the Function can be expressed in terms of an expression of the form k*(pi^s) where k always in the form of a rational number.

I have also drawn attention in this context - though I imagine it is already well known - to the fact that the denominator of k bears a special relationship to the prime numbers. So in brief, when s represents a power of 2, the denominator seemingly will always entail an expression representing the product of all prime numbers up to and including s + 1 (and only these primes).

In all other cases where s is even (but not a power of 2), the denominator will entail an expression representing the product of all prime numbers from 3 up to and including s + 1 (and only these primes).

Now I am using these features here to illustrate the - already accepted - Type 1 strand.

And in conclusion, a key important fact to bear in mind (especially when considering the alternative Type 2 interpretation), is that in qualitative terms, Conventional Type 1 Mathematics represents a linear (1-dimensional) method of interpretation i.e. where s (in qualitative terms) = 1.

We now move on to the more difficult task of providing a corresponding Type 2 interpretation in holistic qualitative terms for each relationship (already dealt with in a Type 1 quantitative manner).

Now, as we have seen in Type 1 terms, when 4 is used as a dimensional number, it is from a merely reduced quantitative perspective (where in qualitative terms 1 as dimension applies).

This can be easily illustrated. To make it simple, let's look at the expression 2^4 (in quantitative terms. Now the answer here is of course 16 (which written more completely is 16^1). This must necessarily be the case with any such quantity, as clearly a different value would result if the dimensional number ≠ 1. So numbers in quantitative terms are ultimately expressed with respect to a default dimension of 1.

So in this quantitative system 1, 2, 3, 4,... are more fully expressed as

1^1, 2^1, 3^1, 4^1....

However when we move on to the true qualitative circular notion of dimension, in reverse terms, the dimensional number is defined with respect to a fixed base quantity.

So in the qualitative system (where numbers represent dimensional qualities),

1, 2, 3, 4.... are more fully expressed as:

1^1, 1^2, 1^3, 1^4,...

As I have stated previously, a complementary inverse relationship exists as between the dimensional numbers and corresponding roots.

I have already explained the key reason for this elsewhere in explaining the fascinating transition from a whole to a part number.

In quantitative terms 4 and 1/4 for example bear an inverse relationship to each other. Now 1/4 can be written as 4^(- 1).

So, in corresponding qualitative terms 4 and 1/4 bear an inverse relationship as dimensional numbers where 4 can be equally written as 4 ^(- 1). Now - 1 in this context relates - literally - to the dynamic negation of (conscious) rational understanding, which results in (unconscious) intuitive appreciation. This then can indirectly be given a circular rational interpretation.

Then in the second system 1/4 is more fully expressed as 1^(1/4) which then gives us a quantitative value, as one of the 4 roots of 1, lying on the circle of unit radius (in the complex plane). So all 4 roots (corresponding to 1^1, 1^2, 1^3 and 1^4), each raised to the power of 1/4, lie as equidistant points on the same circle.

Likewise the four dimensions of 1 in inverse terms lie on the same circle in the complex plane, but now interpreted in a qualitative manner.

Structurally, these four dimensions are therefore identical with their corresponding quantitative roots i.e. 1, - 1, i and - i (but now given a qualitative rather than quantitative interpretation).

So in circular qualitative terms, we express their significance in the Zeta Function for s = 4, by interpreting the qualitative nature of 1, -1, i and - i.

Whereas in the case where s = 2, we could confine ourselves to one complementary pairing with positive (+) and negative (-) polarities, here we have two such complementary pairings, with one relating to real and the other to imaginary poles respectively.

And may I say right away, that interpretation of 4 as dimension is extraordinarily important. After all we are accustomed to defining our world in 4-dimensional terms. In Physics such dimensions are given a mere quantitative explanation! However the qualitative correspondent in truth is equally - if not more - important.

Basically this 4-dimensional perspective, in qualitative holistic terms, relates to the fact that all reality (including of course mathematical reality) is conditioned by two fundamental pairs of polar opposites.

One of these relates to the inevitable fact that all external objects in experience necessarily relate to an internal interpretation (that - relatively - is subjective in nature). So strictly speaking from this perspective, the notion of "abstract objects" in Mathematics is meaningless.

The reason why they do indeed appear abstract is itself very interesting in that it reflects the attempt to view them in absolute terms as if independent of interpretation. Or to put in another way, as repeatedly stated in these blogs, it represents the attempt to view the quantitative as totally separate from the qualitative (which is ultimately a completely untenable position).

The other equally important pairing relates to that as between whole and part.

Again strictly speaking, it is impossible in dynamic experiential terms to have wholes (without parts) or parts (without wholes). Again because of the great lack of this dynamic perspective, the very nature of interdependence is thereby fundamentally misrepresented within Mathematics.

Now true interdependence necessarily requires - at a minimum - two complementary poles. So from this perspective, obviously we cannot have quantitative without qualitative (or qualitative without quantitative). Yet when it comes to the key issue of trying to understand the two-way relationship as between individual primes and their collective behaviour, conventionally in Mathematics, this is merely attempted from within a quantitative framework.

You know, it truly amazes me that such an enormous amount of energy, requiring ever more specialised techniques, has gone into so many attempts to prove the Riemann Hypothesis. However the basic application of a little bit of holistic interpretation would quickly indicate why the Hypothesis is not capable of proof in the first place!

Sadly however, a continued attempt has been made to exclude completely the qualitative dimension - in formal terms - from mathematical research. This unfortunately has led to a situation where specialists continually miss the obvious, due simply to not thinking in a holistic (i.e. qualitative) manner!

So interpretation of 4 (as dimensional number) in wany ways can be seen as an extension of that for 2. We now attempt to relate complementary opposites with respect to both external and internal (and internal and external) polarities and also whole and part (and part and whole) polarities.

It then remains to be explained as to what the imaginary polarities relate.

Well, we have mentioned before that inherent is all conscious recognition is an - often unrecognised - unconscious aspect. Therefore actual experience is always working at two levels, where local objects in many ways serve an unconscious holistic purpose. And of course this is likewise true of mathematical objects when understood in a dynamic manner.

Ultimately this points to the fact that both conscious and unconscious necessarily interact with each other in all mathematical experience.

The conscious rational aspect of such understanding is "real", whereas the unconscious aspect is - relatively - "imaginary". I have indicated before the precise nature of this imaginary notion. So - 1 in qualitative terms, relates to the (unconscious) negation of unitary form (and in dynamic terms combines both positive and negative poles). So to express this indirectly in a unipolar manner, we
need the qualitative equivalent of a square root. So as this corresponds to the square root of - 1, i.e. i, the result is imaginary (in a precise holistic mathematical fashion).

The imaginary in this sense bears a very close relationship with circular understanding, where it is indirectly conveyed through linear type symbols.

And this is precisely what I am doing here. So I am illustrating the second strand of the Riemann Zeta function as an attempt to convey the holistic qualitative significance of mathematical symbols (whhich are customarily used in a linear manner).

So again in very precise holistic mathematical sense Type 2 represents the "imaginary" aspect of mathematical understanding (in qualitative terms).

Now, as we know, the Riemann Zeta Function is defined with respect to the complex plane in quantitative terms (entailing both real and imaginary aspects).

Then Type 1 Mathematics attempts to understand the Function employing merely the real aspect (of qualitative interpretation).

However it requires a complex approach (in qualitative terms) to properly interpret complex relationships (relating to the quantitative perspective). It another words it needs both Type 1 (real) and Type 2 (imaginary) aspects. And once again the Type 2 aspect is entirely missing in the conventional approach!

So I have spent some time in explaining s = 4 (in qualitative terms) though it would take a long book - indeed several long books - to enlarge properly on its great significance.

We now look at the result of the Function for s = 4 i.e. (pi^4)/90.

What is required here in qualitative terms is to give a holistic mathematical explanation as to why the resulting expression has this precise numerical structure.

Now, once again this represents an extension of where s = 2. We saw then that in qualitative terms pi represents the (pure) relationship as between circular appreciation (where the two way interdependence of opposite poles is recognised) and linear understanding (where initially each pole is interpreted within one isolated pole of reference).

So it is is similar here except that a more refined appreciation is required where linear understanding with respect to four separate poles takes place initially before the combined (4-way interdependece) of all poles together can be holistically appreciated in a circular manner.

Thus, in geometric terms we can imagine this in terms of a circle divided into four quadrants with the horizontal line through the centre repesenting the real and the vertical vertical line representing the imaginary poles! Also there is the added requirement here that we can equally recognise that understanding in relative terms, keeps switching as between the "real" poles as representative of conscious recognition on the one hand, and the corresponding "imaginary" poles as the representative of unconscious appreciation (indirectly expressed through conscious symbols).

Now the capacity for such understanding has not been even remotely developed in our culture. It would be characteristic in practice of some who have attained to genuine contemplative awareness, though even here it is highly unlikely that they would translate such experience in a qualitative mathematical manner.

However my true intention is to show - quite literally - that innumerable higher dimensions of understanding have yet to be attained and also that Type 1 Mathematics in still being confined to 1 as dimension, is thereby operating at the "lowest" possible dimension!

With respect to the second strand of Type 2 understanding, we can even suggest a holistic explanation for the interesting prime number pattern in the denominator (which we mentioned in the Type 1 case).

We have commented before on the two two extreme capacities of primes. The individual primes are unique (in that they have no factors). However the relational (collective) aspect of the primes is due to the fact that every natural number can be uniquely expressed as the product of prime factors.

So we could look at the individual aspect as the linear capacity and the relational aspect as the circular aspect of the primes respectively.

I have also mentioned before that when we switch the linear dimensional number i.e. 1 to - 1, we thereby switch from linear to circular format with respect to the primes.

This can even be illustrated simply in quantitative terms. For example if we raise 7 to (- 1) we obtain .142857 where the 6 digits continually recur. This is perhaps the best known of the cyclic primes and possesses many amazing circular type properties.

Now when we look at the at (pi^4)/90, the rational part is 1/90 i.e. 90^(-1).

This would thereby suggest a circular (i.e relational) pattern to the primes and in this case (and in all subsequent cases corresponding to even values of s > 4 that are likewise powers of 2). So indeed here we have the most ordered arrangement possible i.e. where all the primes are included as factors from 2 to s + 1 (and only these primes).

The final strand represents explanation of how the combined interaction of both quantitative and qualitative aspects takes place (in Type 3 terms).

Though we can only scratch the surface as it were here I will try and convey something of what is involved.

Again we start with s = 4. Now we have already explained the nature of this dimensional number - in a relative separate manner - with respect to both its Type 1 (quantitative) and Type 2 (qualitative) significance.

The implication of Type 3 understanding is that we now simultaneously can recognise the number with respect to both its quantitative and qualitative aspects (thereby enabling ready switching as between both types of meaning). So at one moment, we recognise its quantitative significance (in Type 1 terms) and then where holistic appreciation is required, its alternative qualitative significance (in Type 2 terms).

Now with mental and spiritual capacities well developed, one could indeed understand mathematical relationships in a comprehensive (radial) type manner, thereby combining considerable quantitative rigour with a greatly enhanced quality of holistic intuitive awareness.

And as I have repeatedly stated, Mathematics as a speciality is defined in an extremely narrow manner by the profession. So, I would see its manifest lack of holistic awareness in tackling problems (such as the Riemann Hypothesis) as its greatest limitation.

The result of the Function for s = 4 i.e. (pi^4)/90 would also be interpreted in a distinctive manner from the more advanced Type 3 perspective.

The expression now is not seen in either a quantitative or qualitative manner (as separate) but rather as combining both aspects in a dynamically interactive manner.

And as already suggested this leads to the need for a considerable refinement with respect to both quantitative (and qualitative) interpretation.

When one looks at (pi^4)/90 from a Type 1 perspective, its value is interpreted in a merely quantitative manner.

So, in maintaining that the expression represents a constant value, one understands this to unambiguously mean its quantitative value.

However in the light of Type 3 interpretation, this is now seen to be strictly in error.

The constant value here relates to a combined interaction (of both quantitative and qualitative aspects) which is ineffable. In other words its constant nature cannot be expressed in either a quantitative or qualitative manner (as separate). And as all phenomenal understanding requires a degree of separation of both aspects, this entails that its true (constant) value must necessarily remain ineffable.

Thus when we try to actualise the value of the expression (pi^4)/90 i.e. 1.082323233..., an indeterminacy necessarily operates (with respect to its true unknowable value). So as we keep attempting to achieve greater accuracy in our finite approximation, the number (from this relative quantitative perspective) necessarily keeps changing!

So the Uncertainty Principle here strictly applies with respect to the number expression (pi^4)/90 in an actual finite manner. So to use an analogy that may be helpful, we are here dealing with the quantum mechanical properties of number!

Now we already witnessed a similar situation with respect to (pi/2)/6 where the number likewise keeps changing.

Also, the qualitative aspect of interpretation keeps changing. In general terms, the reason for such quantitative uncertainty still relates to the dynamic interaction of quantitative and qualitative aspects. These can be appreciated as representing in more refined terms both aspects (understood now in both a real and qualitative manner).

Finally even with respect to the denominator of (pi/2)/90, the two aspects can now be seen in a more synchronous manner i.e. with respect to the quantitative nature of prime numbers involved and the qualitative appreciation of why they are related in this manner.

So there is a lot to digest here. It is not necessary to understand every detail. What is important at this stage is the general appreciation that this represents (in embryonic form) the nature of an altogether more comprehensive approach to Mathematics, which has the intention of combining both the quantitative and qualitative aspects of Mathematics in a balanced harmonious fashion.

I have no doubt that such an integrated approach has the capacity in time to completely transform the world we live in.

Tuesday, February 28, 2012

Marvellous Vistas

We will be returning again to further clarification of this new approach to the Riemann Zeta Function.

I am conscious with each blog that I am repeating points already made but as this new formulation represents a radical departure from the conventional approach, I feel that it is valuable to keep reflecting it, as it were, from slightly different angles so as to facilitate a better appreciation of what is on offer.

So, in this slightly more refined formulation of the new Riemann Function, I am emphasising three strands (reflecting the three Mathematical Types (Type 1, Type 2 and Type 3 respectively).

The Type 1 strand reflects the traditional formulation of the Riemann Zeta Function based on a (mere) quantitative approach. So here the Function is defined in the complex plane for every value of s (representing the dimensional power of the Function) except for s = 1 (where it is undefined).

The Type 2 strand reflects the (unrecognised) formulation of the Riemann Zeta function based on the complementary qualitative approach. The significant difference here is that the values of s are now interpreted in qualitative terms (as distinctive multi-dimensional means of holistic interpretation), so that s and the resulting values from the Function are in each case interpreted in a unique manner!

As this holistic strand - despite its equal importance to the quantitative - remains totally undeveloped within Mathematics, my insights here are based very much on my own work carried out intensively over the past 45 years.

So here again the Function is defined in the complex plane (which remember again is now defined in an appropriate qualitative manner) for every value of s, except for s = 1.

The striking relevance of this last point once again is that this excludes the conventional (Type 1) approach entirely as an appropriate means of qualitative interpretation of the Zeta Function. And as the "music of the primes" fundamentally relates to this qualitative aspect, it simply entails that the Function does not properly lend itself to such interpretation in conventional terms. So the qualitative nature of the primes thereby completely eludes the mere quantitative approach!

The Type 3 strand is ultimately the most important by far offering a comprehensive interactive map of the Zeta Function. It combines both Type 1 and Type 2 strands, demonstrating how both quantitative and qualitative aspects (defined in relative isolation in the first two approaches) now interact in a vast multitude of dynamic complementary type relationships.

And at the heart of this understanding stands the Riemann Hypothesis as the central piece of the jigsaw, that seamlessly completes the whole complex tapestry in the ultimate identity of quantitative with qualitative (and qualitative with quantitative) interpretation.

Unfortunately there is further bad news for the Type 1 approach here.

Because of the truly dynamic interactive nature of the approach, both quantitative and qualitative aspects thereby undergo continual transformation.

Now originally in the Type 1 approach all quantitative type relationships are defined in a static (unchanging) manner.

However now, through continual interaction, the very nature of such quantitative type relationships subtly changes.

What this entails in effect is that a limitless number of possible qualitative interpretative models (corresponding to numbers as dimensions) can now be brought to bear on mathematical reality;, through such interpretative interaction, all quantitative relationships now likewise reflect the same limitless possibilities for change.

I illustrated this in a small way in yesterday's blog contribution, when I demonstrated how, through 2-dimensional interpretation (as qualitative) - which is the simplest of the alternative interpretations to demonstrate - the very nature of pi changes (in quantitative terms).

So we saw that pi is defined as a constant in quantitative terms (from a 1-dimensional perspective); however when then viewed in 2-dimensional terms, pi is redefined as having a merely relative approximate value (in quantitative terms) that is subject to uncertainty. So its constant nature is now understood as reflecting the combined interaction of both its quantitative and qualitative aspects (in an ultimately ineffable manner).

The upshot of all this means that when placed in its proper context, all quantitative relationships in Mathematics will ultimately need to be radically redefined. This clearly implies the that the Type 1 formulation - despite its apparent quantitative rigour - represents but an extremely reduced version of mathematical truth.

On a personal note the clear realisation of this last point has brought me now full circle in my own mathematical quest!

This journey had commenced in boyhood when I already sensed significant cracks in the mathematical edifice. This original disillusionment then reached its zenith at University where I realised clearly that the treatment of the all important infinite notion in Conventional Mathematics was totally unsatisfactory. So I was left with no option then but to attempt radical reconstruction of this mathematical world from outside the fold (trusting my own intuitions rather than conventional wisdom).

And finally now I am seeing clearly (at least for myself) the fruits of this long labour.

In other places I have further sub-divided the Type 3 (which I formerly termed radial) into three subsidiary classes.

Type 3 (A) represents the integration of both Type 1 and Type 2 aspects in a dynamic appreciation of the quantitative/qualitative interaction inherent in all mathematical processes.

However here the emphasis - relatively - is more on the qualitative side, with attention to the quantitative aspect primarily - though not exclusively - erepresenting the appropriate interpretation of already established relationships.

And this would accurately describe my own position. So, my mathematical focus started out with concern for such quantitative interpretation (at an embryonic level) and has culminated now in this much more developed position.

Type 3 (B) again requires a dynamic appreciation of the quantitative/qualitative interaction inherent in all mathematical processes. However here the emphasis - relatively - is more on using such appreciation to creatively generate new areas of mathematical enquiry, possibly culminating in radical new quantitative type hypotheses. In this it regard it would reflect a Type 1 approach (that has become considerably refined through deep contemplative vision)

Type 3 (C) would represent the most perfect expression of this approach, where one can display great mastery at a quantitative level in ways both highly productive and immensely creative, while also proving expert with respect to holistic type interpretation.

However this really represents a vision of the future which we have not yet remotely reached!

So with respect to the Riemann Zeta Function and the Type 3 strand, the Function is defined in the complex plane in both a quantitative and qualitative type manner (where now quantitative and qualitative are understood in dynamic interactive terms as inherent in all mathematical symbols, relationships etc.)

Thus, here the Function is understood in both a mutually complementary quantitative and qualitative manner for all values of s and corresponding results, except once again where s = 1 (in both a quantitative and qualitative fashion).

And finally once again the Riemann Hypothesis lies at the centre of it all (as the raison d'etre for the existence of everything).

Thus here the identity of both the quantitative and qualitative aspects of the primes both as numbers and as means of interpreting such numbers become identical in ineffable mystery.

From this enlarged perspective one may get some idea of how incredibly limited is our present conception of Mathematics.

It is as if we have confined ourselves for all our history to a tiny island when in truth vast continents lie completely unexplored close by.

Though it will be extremely difficult to emerge from such self imposed exile, marvellous vistas of completely new mathematical meaning await us in the future in what promises to be the most radical revolution in our intellectual history. And of course by extension because of the pre-eminence of mathematics this revolution will then quickly spread in a completely unprecedented fashion to all of the sciences.

And in this extraordinary new mathematical world, the long untenable quest to prove the Riemann Hypothesis will be remembered as a mere folly.

Monday, February 27, 2012

The Music of Pi

It is significant how the musical analogy is so often invoked in relation to the primes.

I have referred before to the fact that the Chapter in Keith Devlin's "The Millennium Problems" on the Riemann Hypothesis is entitled "The Music of the Primes".

Then the very same title is used for Marcus de Sautoy's popular book on the primes.

And Michael Berry who has done extensive work in recent years with respect to the physical implications of the non-trivial zeros says

“... we can give a one-line nontechnical statement of the Riemann hypothesis: The primes have music in them."

Indeed this musical analogy extends right back to the time of Pythagoras who discovered intimate connections as between terms in the - since titled - harmonic series and musical sounds.

And the harmonic series in turn serve as the basic foundation stone as it were on which the subsequent Euler and Riemann Zeta Functions have been built.

However what I find quite baffling, is why the obvious implications of such links as between music and the primes have been repeatedly missed in a formal mathematical context.

Probably the most universal way in which the qualitative dimension of experience is conveyed is through music. So this should immediately suggest therefore a marked qualitative dimension to the mathematical study of the primes.

There is indeed also a marked quantitative aspect with respect to the structure of music; however which however is not to be confused with its qualitative appreciation. It should perhaps further suggest that with respect to the primes that both quantitative and qualitative aspects are closely linked.

Also it is interesting to note that musical often goes with mathematical ability.

Euler for example, who made such ground breaking discoveries with respect to the primes wrote extensively about music, manufactured instruments and was also a practitioner of some talent.

However the simple fact remains that mathematicians have never really made the important connection that the "music of the primes" implies an important qualitative dimension to the very nature of Mathematics. Rather, with ever more specialised technical developments rapidly taking place in the discipline, it has become increasingly defined within a hermetically-sealed chamber (that allows no intrusion of the all-important qualitative dimension).

For some reason from an early age I was naturally able to make this qualitative connection as an obvious consequence of the multiplication process.

I have related before that in doing multiplication problems as a young schoolboy, as for example in calculating the area of a field, I quickly appreciated that a qualitative as well as quantitative dimension was involved.
So with a rectangular field we start with two sides measured in linear (1-dimensional) units. However after multiplication the area is then correctly expressed in square i.e. (2-dimensional) units.

However in the more abstract use of number in pure Mathematics, this qualitative dimensional aspect is ignored altogether with mathematical results interpreted in a merely reduced quantitative manner.

That such a fundamental issue can be glossed over with such apparent ease, despite the so-called "rigour” of pure Mathematics, therefore led me to be highly sceptical regarding the nature of mathematical interpretation. So I have since operated with an expectation – arising from this early experience - that I would find equally important fundamental issues in Mathematics that are completely overlooked.

Thus in the context of mathematics generally - and most especially the primes – there are in truth are two equally important aspects to Mathematics i.e. quantitative and qualitative with the qualitative aspect entirely censored out - in formal terms - of what conventionally passes as Mathematics.

So having accepted that there is indeed an (unrecognised) qualitative dimension to Mathematics, the next task was to see how this could be successfully incorporated in a manner going beyond mere philosophical conviction regarding its true nature.
And it eventually dawned on me that the qualitative aspect is intimately tied up with the dimensional notion of number (when given an appropriate circular type interpretation).

In short this qualitative notion of a dimensional number fits exactly from with the inverse notion of the corresponding roots of that number (in quantitative terms).

Thus, right away we can pinpoint exactly the nature of Conventional Mathematics. In qualitative terms this is defined by its linear (1-dimensional) nature and the corresponding 1 root of unity is of course equally 1. So – quite literally – from a 1-dimensional perspective, the qualitative is not distinguished from quantitative meaning with the result that the qualitative is necessarily reduced – in any given context – to mere quantitative interpretation!

And it should be already patently obvious – though it still clearly eludes the Mathematics profession - that we are never going to understand “the music of the primes” from this perspective. We may indeed obtain a reduced quantitative interpretation with respect to many features of the primes, but it will always remain like one though having an unrivalled technical mastery – say – of the structure of Beethoven’s music yet has never listened to his work (and worse still refuses to listen when invited to do so!).

So when we next move on to the number 2 (as the qualitative aspect of dimension) its structure is inversely related to the two roots of unity.
Now in quantitative terms this relates to the two separate results + 1 and – 1. (So linear always implies separation of opposites).

However circular interpretation by contrast entails interdependence. So
2-dimensional interpretation is thereby defined as the complementary of polar opposites (i.e. with positive and negative aspects).

This again directly implies that the very appreciation of interdependence requires from a qualitative perspective circular rather linear logical understanding. And once more – in formal terms – this is totally excluded in Conventional Mathematics.

So when mathematicians attempt to deal with interdependence they do so within uni-polar (separate) frames of reference, So like one who understands the two turns at a crossroads as both left, Conventional Mathematics deals with the interdependence of important opposite poles such as the general and particular (as for example the relationship of a theoretical result to individual cases) in merely quantitative terms.

Again from my standpoint it should be patently obvious that a qualitative aspect is also necessarily involved; but because of accepted (unquestioned) orthodoxy going back now more than two millennia this is again missed.

Now in a previous blog we dealt with the nature of pi. It is worth returning here to illustrate a few more significant points.

From a conventional (linear) perspective, pi is treated in a merely quantitative manner. So from this perspective pi represents the ratio of the circular circumference to its line diameter (in quantitative terms).

However as always we can give these symbols a corresponding qualitative (holistic) interpretation. So from this context pi now expresses the relationship as between circular and linear type understanding.

The two roots of unity can be geometrically expressed through drawing a circle with its line diameter with the direction to the right marked positive and that to the left negative.

This diagram likewise serves as a good illustration of 2-dimensional understanding (in qualitative terms).

So initially using linear understanding, we deal with the two opposite poles as separate. Understanding at this level does indeed seem unambiguous (with both interpreted in absolute terms as positive). However once we relate both poles as interdependent, paradox immediately arises (with respect to linear interpretation). Thus this requires a movement to a distinctive type of holistic understanding that can embrace paradox (i.e. that is circular in nature).

Therefore the very interpretation of pi itself has both a coherent qualitative as well as quantitative interpretation.

So when we understand pi correctly from a qualitative perspective, we realise that it entails both linear and circular aspects of interpretation.

And just as in quantitative terms pi is recognised as perhaps the best known example of a transcendental number, we can now likewise perhaps appreciate the qualitative notion of transcendental, which represents the dynamic relationship as between both linear and circular aspects of understanding (necessarily expressed in a reduced rational manner).

Now pi represents the pure relationship here; however all transcendental numbers entail the same basic relationship! By contrast an algebraic irrational number such as the square root of 2 represents merely the circular aspect (interpreted in a linear reduced manner).

It is not crucial at this stage to appreciate the implications of everything stated here. Rather the key requirement is to get some notion that there is indeed a valid alternative aspect to Mathematics (where every symbol that already has a defined quantitative meaning can likewise be given a distinctive qualitative interpretation).

So having defined the nature of pi in both (Type 1) quantitative and (Type 2) qualitative terms (separately as it were), we then bring both aspects together in Type 3 understanding.

And the Riemann Zeta Function requires such combined (Type 3) understanding for its proper comprehension.

Again when we define pi in 1-dimensional terms it has merely a quantitative interpretation.

Pi is referred to as a constant though its numerical value cannot be written down precisely. Though it is recognised that it is not a rational number (but rather a transcendental) this too is defined in a merely reduced quantitative manner (i.e. that cannot be the solution of a polynomial equation with rational coefficients). So though clearly rational and transcendental type numbers possess a profound qualitative dimension – even the very word transcendental would suggest as much – by the very nature of the linear (1-dimensional) approach, this aspect cannot be approached.

However when we interpret pi in a 2-dimensional qualitative context, both quantitative and qualitative aspects dynamically co-exist (in the same number as it were).

Not alone does this imply a new qualitative interpretation entailing both linear and circular elements, but - because quantitative and qualitative are now related - it also implies an important subtle change in the interpretation of the quantitative nature of pi.

Again, the two linear frames of reference come from the more refined recognition that as a particular number, pi is necessarily in relationship with its general number concept. And the circular aspect comes when we recognise both aspects as interdependent.

However, as I say, this then subtly changes the quantitative interpretation of pi.
So from this quantitative perspective, we understand pi in dynamic relative terms, as having a merely approximate value! (So strictly pi cannot now be a constant in quantitative terms).

What this means is that the actual value of pi can never be fully known, with an inherent uncertainty attaching to its sequence of digits. So to keep it simple, if we write down the digits in binary form, with even trillions of these digits calculated already, we only have a 50/50 chance of correctly predicting the next digit!

Now this might seem reminiscent of Quantum Mechanics and indeed it is! In fact the deeper roots of the Uncertainty Principle in Quantum Mechanics, spring from a more fundamental uncertainty in Mathematics (which ultimately relates to the complementarity of both its quantitative and qualitative aspects).

And when we begin to understand Mathematics from all other dimensions (except 1) the same uncertainty applies to everything.

It is indeed like moving from a Newtonian to a Quantum Mechanical Universe except that it is much more fundamental than that!.

For example when interpreted in a 2-dimensional sense (which is the most accessible of the alternative dimensions) all mathematical proof is subject to the Uncertainty Principle.

So in what sense is pi now a constant!

Well! remember at the 2-dimensional level of appreciation, pi represents a dynamic interaction of both quantitative and qualitative aspects. So it is in this combined entity as it were (that the constant value resides).

However there is always something ineffable about this situation, as we can never know what it is exactly, once we separate quantitative and qualitative aspects. And in phenomenal terms quantitative and qualitative aspects must necessarily be separated to a degree.

So its quantitative value – as befits dynamic interaction – can only be known in a relative approximate fashion.

Indeed the deeper root of the ineffable nature of pi lies – ultimately – in the ineffable nature of the Riemann Hypothesis.

Sunday, February 26, 2012

Riemann Zeta Function (5)

It struck me today that it may be misleading to continue referring to the Riemann Zeta Function and what it means, as I am offering an enlarged and - ultimately - very different interpretation than that proposed by Riemann.

Both Euler and Riemann of course operated within the accepted standard quantitative framework of Mathematics (which I refer to as Type 1 Mathematics).

However my basic standpoint is that Mathematics - when appropriately understood - equally possesses an (unrecognised) qualitative aspect. So every number, symbol, sign relationship, hypothesis etc. that can be given a quantitative type interpretation in Type 1 terms, can equally be given a coherent qualitative interpretation from a holistic mathematical perspective (which I term Type 2 Mathematics).

Then, finally my position is that truly comprehensive mathematical understanding requires the harmonious integration of both Type 1 and Type 2 aspects (which I refer to as Type 3 Mathematics).

So my new treatment of the Riemann Zeta Function - which perhaps more accurately could be referred to as the Zeta 3 Function - is intended as representing a preliminary introduction to Type 3 Mathematics.

So Euler first defined the Zeta Function in quantitative (Type 1) with respect to real values of the function (where s > 1).

Riemann then extended the Zeta Function – again in quantitative (Type 1) terms - with respect to all complex values of the function (except where s = 1).

However I am now proposing a radical further extension – incorporating both quantitative (Type 1) and qualitative (Type 2) aspects, with respect to all complex values of the Function (except where s = 1). And as subtle complementary relationships on a variety of different levels connect both quantitative and qualitative interpretations of s, then in effect this thereby represents an exercise in Type 3 Mathematics.

So to put some perspective on where we have travelled with the Zeta Function, we have started on the real axis with the intention of explaining the nature of both quantitative and qualitative type interpretation for values of the Function (where s > 1).

Though from a quantitative perspective we can readily accept the results already obtained by Euler for the Function, even here a more careful subtle interpretation is required, where in actual terms all values obtained correctly represent relative finite approximations (to an unknowable “true” value). This is precisely because the “true” value now is seen to represent the dynamic interaction of both quantitative and qualitative aspects of interpretation).

However our main focus here is – necessarily – on the unrecognised qualitative aspect of interpretation.

So, all quantitative values of s (on the RHS for s > 1) are initially defined in a linear manner i.e. in Type 1 terms, which accords in qualitative terms with a default 1-dimensional interpretation.

However the appropriate qualitative interpretation of s must be taken in accordance with the dimensional value of s that applies in each case to the Function.

So in yesterday’s blog entry I spent some time clarifying the nature for s = 2, of what 2-dimensional interpretation in qualitative terms precisely implies.

One startling conclusion that immediately arises, in the context of the newly defined Zeta Function, is that Type 1 Mathematics is uniquely unsuited to appropriate clarification of what is involved.

Because Type 1 Mathematics – in formal terms – is solely defined with respect to its quantitative aspect, it therefore has no means of correctly representing an interpretation that – by definition – is based on the incorporation of both quantitative and qualitative aspects.

So, just as The Riemann Zeta Function remains undefined for s = 1, in quantitative terms, the new Zeta Function remains undefined for s = 1, in a corresponding qualitative manner.
And what this simply means is that Type 1 Mathematics again is uniquely unsuited to interpretation of this newly defined Function.

This further implies that it is futile to seek a proof of the Riemann Hypothesis using the standard Type 1 approach.

In the new Zeta Function, the Riemann Hypothesis represents the key condition for obtaining the mutual identity of both quantitative and qualitative aspects of interpretation. Obviously an approach that does not recognise the qualitative aspect is thereby not fit for purpose!

Indeed just as we demonstrated yesterday that the key limitation of Type 1 Mathematics ultimately relates to an unsatisfactory interpretation of finite and infinite notions, this same problem is endemic in all conventional (Type 1) mathematical proof.

Correctly understood finite and infinite represent distinctive notions which cannot be reduced in terms of each other. The infinite represents a holistic notion relating to a (universal) potential meaning whereas the finite represents a specific notion relating to a (partial) actual meaning. However though they cannot be properly reduced in terms of each other, they do necessarily continually interact in dynamic fashion.

So when appropriately viewed the infinite is seen as always inherent within actual phenomena (which of course includes numbers). Now, with respect to the new Zeta Function, the only case where this not occur is where s = 1 (in both quantitative and qualitative terms). So here, there is a disassociation of both finite and infinite, where from one perspective the infinite notion is treated as a linear extension of the finite, or from the opposite perspective the finite notion treated as a linear extension of the infinite.

And it is this latter case that intimately applies to the nature of mathematical proof.

If for example I say that the Pythagorean Theorem has been proved, this correctly implies that this has been established in general terms (as potentially applying to “all” cases). So, essentially this relates to a qualitative type conclusion that is infinite in scope.

However application of a proof must necessarily apply in a limited number of actual cases (entailing quantitative meaning of a finite nature).

So right at the heart of mathematical proof (in Type 1 terms) we have a basic confusion of quantitative with qualitative meaning. And this is inevitably the case as the very basis of such Mathematics operates on the reduction – in any context – of qualitative to quantitative type interpretation.

If we look at our crossroads analogy again we can perhaps see the problem more clearly.

Within isolated reference frames we can give a general proof a valid quantitative interpretation; then again in a particular situation we can give its application in particular cases a valid quantitative type interpretation. However in mutual relation to each other, both the general and particular are as quantitative and qualitative (and qualitative and quantitative). So the situation in Type 1 Mathematics is like the person who cannot see that the turns at a (horizontal) crossroads are necessarily left and right and right and left with respect to each other. And the problem again relates to isolated frames of reference (which defines the 1-dimensional approach).

Thus we can conclude from a Type 3 perspective that all mathematical proof is of a relative nature (and subject to uncertainty). And this uncertainty, which is akin to the particle and wave nature of sub-atomic particles, arises out of the complementary quantitative and qualitative type nature of mathematical understanding.

Indeed the deeper root of the Uncertainty Principle within Quantum Mechanics arises from this more fundamental uncertainty at the heart of Mathematics!

Mathematical proof when correctly viewed represents but a special form of social consensus among the mathematical community who agree to share the same basic assumptions. However the very fact that I am strongly questioning here the nature of such assumptions bears testament to the fact that it is of a merely relative (rather than absolute) nature.

However if one accepts the limited assumptions of Type 1 Mathematics (which is indeed valid for dimensional 1 interpretation) then most propositions such as the Pythagorean Theorem do indeed have a valid proof that appear absolute from that perspective.

However one key implication of the new Zeta Function is that Type 1 Mathematics is uniquely unsuited to clarification of the very nature - not alone proof - of the Riemann Hypothesis. Of course it is still highly valuable as a means of clarification of the quantitative features with respect to the Zeta Function. However as the central issue properly relates to a prior relationship as between quantitative and qualitative type meaning it is clearly not appropriate for this task.

So all mathematical proof within a Type 3 mathematical perspective is of a merely relative nature (and subject to uncertainty).

Within the narrow confines of Type 1 Mathematics, most propositions are indeed capable of proof. However the Riemann Hypothesis which requires a Type 3 perspective for its adequate interpretation, clearly does not belong to this category!

Saturday, February 25, 2012

Riemann Zeta Function (4)

(I have reworded the headings on the three blogs – that referred to the non-trivial zeros - to reflect more accurately the scope of enquiry contained therein).

We mentioned how in deriving finite values for the Riemann Zeta Function for s > 1, that a basic reductionist process is involved whereby the infinite notion is in effect treated as a linear extension of the finite.

We also commented on the fact that the value of ζ(2) = (pi^2)/6.

Now in a reduced quantitative manner, this certainly appears to be the case. However when we look more closely, we begin to realise that there is necessarily also a qualitative aspect to this result with the value of pi in truth representing a dynamic interaction as between both quantitative and qualitative aspects (with an inherent uncertainty attached to its value).

ζ(2) represents the sum of terms of 1 + 1/4 + 1/9 + 1/16 + …

If we take the actual sum of any finite number of terms in this series, we clearly get a rational answer and by taking a sufficiently large number of terms we can indeed approximate very closely the “true” value for (pi^2)/6.

However pi does not represent a rational - but rather an irrational (transcendental) - number.

So clearly a qualitative change in the nature of the number is involved in moving from a finite to an “infinite” number of terms.

In truth the infinite aspect here does not relate to quantitative measurement but rather to a potential quality that is inherent in the number quantity.

So the very nature of pi - and by extension (pi^2)/6 - is that it necessarily combines both quantitative and qualitative aspects in its very nature. Indeed this is strictly true of all numbers (when appropriately interpreted).

We can never precisely ascertain the quantitative value of pi (in actual terms). We can indeed approximate this value to any required degree of accuracy by treating it in a rational manner. However it is not in fact a rational number and this is because of a qualitative aspect (which is inherent in its transcendental status).

So though Conventional Mathematics attempts to treat pi solely as a number quantity, in truth it represents a relationship between finite (quantitative) and potential (infinite) aspects with an inevitable uncertainty thereby attached to its actual nature.

This is indirectly revealed in quantitative terms by the fact that its decimal sequence continues indefinitely (with no fixed pattern).

We can indeed see more precisely into the nature of pi by looking initially at its definition from a quantitative perspective.

So in this context pi represents the relationship between the circular circumference and its line diameter.

Now if we place a point at the centre to spearate both axes, we would create two poles with measurement to the right positive and to the left negative respectively.

However the very essence of the linear approach is to treat both poles as positive. So both directions of the radius – from the centre of the line diameter - are thereby treated in uni-polar fashion as positive, with pi then representing the ratio of the length of the circumference to its line diameter.

However both line and circle have equally a qualitative meaning where they relate to linear and circular understanding respectively.

It would be useful here to once again employ our crossroads analogy with the centre of the line representing the intersection of the two roads.

Now Conventional Mathematics – in the same manner as the treatment of the diameter in quantitative terms - gives the same designation in both directions.

So on this occasion if I travel up a road through the centre of a horizontal crossroads, I can unambiguously identify a right turn; then when later coming down the same road, I can again unambiguously identify a right turn. So using uni-polar reference frames, both turns at the crossroads are designated as right!

And again this is precisely how Conventional Mathematics necessarily operates (recognising in formal terms solely a quantitative aspect). So for example in relation to the primes when studying their individual nature, it adopts a quantitative interpretation; then when it switches direction with respect to general frequency of the primes, again it adopts a quantitative interpretation.

However when we view both turns as interdependent, they are necessarily right and left and left and right with respect to each other respectively.

Likewise when we view the two directions of mathematical enquiry as interdependent, they must necessarily be quantitative and qualitative (and qualitative and quantitative) with respect to each other.

The devastating implications of this – when properly appreciated – is that by definition, the very notion of interdependence cannot be properly interpreted within the accepted mathematical framework!

Therefore it is strictly futile even attempting to understand, for example, the two-way relationship of the prime numbers to the non-trivial zeros within the conventional linear framework of Mathematics.

And of course this implies that the very attempt to approach the Riemann Hypothesis from the standpoint of Conventional (Type 1) Mathematics is equally futile.

So the circular notion from a qualitative (Type 2) mathematical perspective relates to interdependent polar reference frames, which in their simplest form requires two poles!

So whereas the lines drawn from the centre represent linear interpretation (based on uni-polar reference frames (as independent), the circumference represents circular appreciation (based on bi-polar reference frames) as interdependent.

Now clearly both approaches are needed in Mathematics. Before we can relate two aspects together as interdependent, we must recognise them separately (as independent).

However once again by its very nature, Conventional Mathematics is inherently unsuited to the treatment of interdependence.

In fact this can be illustrated in a manner which is utterly devastating with respect to the nature of Conventional Mathematics.

The only dimensional value for which the Riemann Zeta Function is undefined is s = 1. And this is the very dimension that defines the conventional mathematical approach. So for every other value of s the Function is indeed defined.

So Mathematics can in truth be defined from an unlimited number of dimensions (each of which represents a unique manner of interpreting its symbols).

Now all these alternative dimensional interpretations entail a distinctive relationship as between quantitative and qualitative aspects of interpretation.

And this in turn is precisely why the Zeta Function can be defined for these values!

This clearly establishes – when correctly interpreted – that the Riemann Zeta Function is really about establishing a precise complementary relationship as between quantitative and qualitative type aspects, with the Riemann Hypothesis lying right at its centre as the condition necessary for establishing the ultimate identity of both aspects.

So the one interpretative system that is uniquely unsuited to proper comprehension of the Riemann Zeta Function (and its associated Riemann Hypothesis) is the 1-dimensional approach (that defines Conventional Mathematics).

So we can validly say without a hint of hyperbole, that just as in quantitative terms the Riemann Zeta Function remains (uniquely) undefined where s = 1, likewise in qualitative terms it remains (uniquely) undefined where s = 1.

And as we have seen, this once again means that the conventional mathematical approach – defined by its 1-dimensional approach - is by its very nature unsuited to the unlocking of the rich secrets of the Riemann Zeta Function (and of course the Riemann Hypothesis).

At a more fundamental level, this points directly to the unsatisfactory manner in which finite and infinite notions are treated. Though these notions represent two distinct poles (with respect to understanding), they are necessarily reduced in terms of each other from a conventional mathematical perspective.

So from one direction the infinite is treated as a linear extension of the finite as with its treatment of convergent series on the Riemann Zeta Function (where s > 1).

Alternatively the finite is treated as a linear extension of the infinite (which has great relevance for treatment of corresponding Zeta values where s < 0).
So therefore when we attempt to add a finite to an infinite number in conventional (reduced) terms the infinite number remains unchanged!

This in fact represents a fundamental mistreatment of the very notions of finite and infinite!

Properly understood the finite aspect always has an actual (specific) meaning, whereas the infinite has a potential (holistic) meaning.

And in dynamic interactive terms, the potential aspect always remains inherent within actual numbers.

We demonstrated this earlier with respect to pi.

Pi has indeed a quantitative aspect (which in actual terms is subject to uncertainty and therefore not fully determinate); however equally inherent within it is a qualitative holistic aspect, where in Jungian terms it serves as a number archetype.

So the true transcendental nature of pi is meaningless in the absence of both quantitative and qualitative aspects.

Indeed just as pi is so important from a quantitative perspective, equally - when appropriately viewed - it should likewise be seen as an important key number archetype pointing to the fundamental relationship as between linear and circular notions (i.e. independence and interdependence).

Finally it should be apparent that when we define the Zeta Function with respect to s = 2, we do indeed obtain a finite quantitative result that in actual terms is - relatively - determinate. However the expression (pi^2)/6 also necessarily contains a qualitative aspect as the relationship of circular to linear understanding.

And because we are dealing with 2-dimensional understanding, only two poles are involved.

So inherent in the very result for ζ(2) is a significant clue as to the qualitative nature of interpretation that is appropriate in this case. In other words the proper interpretation of the result for ζ(2) i.e. (pi^2)/6, entails both a linear (quantitative) and a circular (qualitative) relationship. And in this case it entails just two (real) poles of understanding (which is the precise meaning of 2-dimensional).

We will return to further exploration of these issues in future blogs.

Thursday, February 23, 2012

Deep Prime Connections

I mentioned in a previous blog that whereas the quantitative aspect of primes relates to their individual aspect (as distinct prime numbers) their qualitative aspect relates, by contrast, directly to the natural numbers i.e. in the manner in which each natural number can be expressed in terms of a unique combination of prime factors. (And of course the key observation I was making in this context was that the multiplication of distinct primes necessarily entails dimensional change of a qualitative nature!)

So the quantitative aspect relates to the individual independence of the primes (with respect to the natural numbers), whereas the qualitative aspect relates to their capacity of relatedness through which they can uniquely generate the natural numbers as prime factors.

Now the simplest manner in which the quantitative and qualitative aspect of number manifests itself is through cardinal and ordinal interpretation respectively.

So if we write down the natural numbers as representing the ordinal ranking of the primes, then

1, 2, 3, 4, 5,.... thereby represent the 1st, 2nd, 3rd, 4th, 5th,... primes respectively which correspond in cardinal terms to 2, 3, 5, 7, 11,....

So in this important respect, the ordinal ranking of the natural numbers (as qualitative) corresponds directly with the cardinal ranking of the primes (as quantitative).

We could carry this process further by now seeking in turn to obtain an ordinal ranking of the primes.

So this ordinal ranking of 2, 3, 5, 7, 11,... would thereby correspond with the 2nd, 3rd, 5th, 7th, 11th ... primes, corresponding to 3, 5, 11, 17, 31, ...etc. And we could continue this process indefinitely in a progressive thinning out of the primes by once again labelling our latest set of primes 1, 2, 3, 4, 5,... and then once again obtaining the new prime set corresponding to ordinal prime rankings with respect to these numbers, then relabelling them in natural number fashion before proceeding to look at again at the numbers corresponding to the new ordinal prime rankings!

However it is enough for our purposes here to confine ourselves to the initial ordinal ranking of primes.

As is well known the sum of the reciprocals of the natural numbers i.e. the harmonic series

1 + 1/2 + 1/3 + 1/4 +.....approximates log n + γ (where γ = Euler-Mascheroni constant = .5772..).

And log n, as we know approximates the average gap or spread between cardinal primes (which continually improves as n becomes progressively larger).

And as the contribution of the Euler-Masheroni constant becomes increasingly less important for very large n, then the sum of the reciprocals of the natural numbers approximates well the average gap as between the prime numbers!

What is equally fascinating is that the sum of reciprocals of the prime numbers (cardinal)

i.e. 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + .... is approximated by log log n + B1 (where B1 = Mertens Constant i.e. .261497...)

So just as the sum of the reciprocals of the natural numbers approximates the average gap as between cardinal primes (as independent numbers) for very large n, in like manner the corresponding sum of the reciprocals of the prime numbers should approximate the relational capacity with respect to the ordinal primes for n (i.e. as the number of unique prime factors for n)

This would suggest that the number of prime factors of very large - and I mean very large - natural numbers, would thereby approximate well with log log n.

Now, Hardy and Ramanujan proposed this some time ago (though I am not aware of the precise details of what they had in mind).

However the very point of this exercise is to show how the same conclusion can be reached through holistic recognition of the qualitative aspect of the primes!

Indeed it would be tempting to conjecture that perhaps a more precise measurement could be given in relation to the number of prime factors of a very large number, by using the original approximation for the sum of the reciprocals of primes,

i.e. log log n + .261497...

Wednesday, February 22, 2012

Riemann Zeta Function (3)

We concluded yesterday that the numerical values of the Riemann Zeta Function for every finite value of s (with the exception of 1) on the real axis, can be given - relatively - both a quantitative (specific) and qualitative (holistic) interpretation. This means that when we change reference frames, numerical values now have a quantitative (holistic) and qualitative (specific) interpretation.

So depending on context, numerical values can be given both specific and holistic interpretations. Likewise, again depending on context, corresponding psychological interpretation of such values can be given both specific and holistic interpretations. And though this may not indeed be clear yet without further ample illustration, it is absolutely key to appreciating the true nature of the Riemann Zeta function (and indeed its associated Riemann Hypothesis).

Also we mentioned that the Riemann Functional Equation establishes an important complementary relationship as between values for ζ(s) on the RHS > .5 and corresponding values for ζ(1 - s) on the LHS < .5.

What this means in effect is that when we define a specific quantitative value for ζ(s) with respect to a linear frame of reference on the RHS (where s > .5) the corresponding value with which it is related for ζ(1 - s) on the LHS (where s < .5) is necessarily defined with respect to an alternative circular frame of reference.

So for example the value for ζ(2) on the RHS = (pi^2)/6 is defined in terms of standard linear interpretation.

The corresponding value through the Riemann Functional Equation on the LHS is
ζ(- 1) = - 1/12. Though this indeed has a specific numerical value, the result has no meaning in terms of the standard linear interpretation of a series. So what has happened is that the frame of reference has shifted here to a distinctive circular mode of interpretation!

Therefore though the quantitative results with respect to both series can be given specific values they correspond to differing modes of interpretation. So in the language that I customarily now use, (pi^2)/6 corresponds to a Type 1, with - 1/12 in relative terms, corresponding to a distinctive Type 2 interpretation.

So once again we can see the insuperable difficulties that Conventional Mathematics faces in appreciating the true nature of the Riemann Hypothesis. Because it is defined in merely Type 1 terms, it has no way of properly interpreting all the numerical values on the LHS of the Function (for s < .5) and thereby has no means of interpreting what the whole Function is really about!

When I spent some time reading about the Riemann Hypothesis I was amazed at how little attention was actually given to philosophical interpretation of the strange values that are thrown up for the Function through analytic continuation.

This lack of proper understanding is carefully hidden behind a forbidding wall of technical jargon regarding the nature of continuation. So we hear a lot about differing domains of definition, holomorphic and meromorphic functions, uniquely defined values etc. all of which of course at the Type 1 level is perfectly valid.

However, the reality remains that mathematicians are still unable to provide a satisfactory explanation with respect to half the values defined by the Riemann Zeta Function! And the reason is that is requires the (unrecognised) Type 2 aspect of Mathematics for such interpretation!

And finally we saw in yesterday's blog that both quantitative and qualitative interpretations - which are separated for other values - ultimately coincide where s = .5!
This means that for all the points that lie on the imaginary line drawn through .5, quantitative and qualitative likewise coincide.

And as all the non-trivial zeros (of the Zeta Function) necessarily relate to points on this imaginary line if the Riemann Hypothesis is true, this then implies a coincidence of both quantitative and qualitative aspects with respect to the primes.

In other words the Riemann Hypothesis is the condition necessary to ensure that we can perfectly reconcile the individual nature of each prime (that is finite) with the overall collective nature of the primes (in infinite terms). And this ultimately points to an ineffable state!

Now in looking more carefully at the Riemann Zeta Function (and of course the non-trivial zeros) we will first demonstrate in more detail the manner in which quantitative and qualitative interpretations operate.

In doing this we will initially consider values of the Function on the RHS of the real axis for s > 1.

We will then look at its complementary partner values with respect to the Function where s < 0.

And finally we look at the all-important critical strip as between 0 and 1 (where ultimately in the complex plane all the non-trivial zeros lie).

Our first task seems relatively easy and uncontroversial as Euler had already mapped out the Function on the real axis (for s > 1) long before Riemann.

So the first thing we can determine is that the value of the Function (for all values > 1) results in convergent series with finite values.

However even here some subtlety with respect to interpretation of numerical results for the Function is required.

Now, it is important to keep reminding ourselves that finite and infinite constitute distinctive concepts, which properly relate to quantitative and qualitative understanding respectively.

So the finite strictly relates to actual (specific) numbers, whereas the infinite properly relates to the (holistic) potential that is inherent in all numbers. However, as I have frequently stated, the very essence of the linear approach is that it reduces the infinite in actual terms, effectively treating it as a linear extension of the finite!

And this is amply demonstrated by values for the Zeta Function (where s > 1).

So for example to illustrate with the series (where s = 2) we generate a sequence of actual (specific) terms 1 + 1/4 + 1/9 + 1/16 + ...

It quickly becomes apparent that the size of these terms diminishes very rapidly with their sum converging towards a limiting value. So, after some point (which we can arbitrarily choose) the addition of further terms makes little difference to the overall sum, and we then conclude that the infinite set of terms thereby converges to the same limit.

I have already made the observation that intuition - insofar as it is informally used in Type 1 Mathematics - does so in a supporting role (so that formally it can be ignored).

And that is precisely what happens here.
Now if we want to express in a subtler manner the nature of convergence with respect to infinite type series, we rationally interpret that the series approaches some limiting value, and then intuit that the infinite series of terms will attain the same value.

From a conventional Type 1 perspective, the final step in moving from the finite to the infinite conclusion is missed, with the whole process misleadingly interpreted in rational terms!

You might consider that such subtlety in interpretation is unnecessary and constitutes merely pin pricking.

However you would be wrong! for it is the very identification of what is involved here, that will enable us to make the decisive leap in interpreting corresponding values for the Zeta Function for s < 0.

For example when s = - 1, we generate the sum of series of the natural numbers i.e. 1 + 2 + 3 + 4 + .....

Now from the standard linear approach, it is quite obvious that this series will diverge in value so that we cannot give its sum a finite value.

Yet, in the context of the Riemann Zeta Function, it acquires the rational finite result of - 1/12.

And as the Riemann Functional Equation establishes an important relationship as between such values on the LHS (with no meaningful interpretation in linear terms) and corresponding values on the RHS (with a finite linear interpretation) then we cannot even properly understand these values on the RHS) without properly understanding corresponding values on the LHS.

And once again, because of the absence in formal terms of a holistic qualitative (Type 2) aspect to Conventional Mathematics, it thereby cannot provide such interpretation.

So, the deeper significance of what I am attempting through this exploration of the Riemann Zeta Function is the key fact that Conventional Mathematics is sadly lacking as a comprehensive means of overall interpretation.

And its limitations are especially exposed in tackling the Riemann Hypothesis.

Thus, properly understood, the two aspects of Mathematics (quantitative and qualitative) dovetail together in perfect harmony with respect to appreciation of the Riemann Hypothesis. And this synchronised use of Type 1 and Type 2 Mathematics, which I am attempting to demonstrate here, represents but the most preliminary introduction to Type 3 Mathematics (which is what Mathematics should truly be about).

So to sum up this contribution. The finite and infinite notions correspond to two distinctive notions, which are quantitative and qualitative with respect to each other.

Conventional Mathematics - by definition - applies a strictly linear (1-dimensional) approach, where effectively the infinite notion is reduced in a finite manner (or alternatively where the qualitative aspect is reduced to the quantitative).

And then, we get numerical results for the Zeta series, which seemingly correspond with common sense (that is likewise based on linear interpretation).

So Conventional Mathematics is based on 1-dimensional interpretation (where qualitative is reduced to quantitative). However I have already made the important observation that an unlimited set of alternative dimensional interpretations of mathematical reality exist corresponding to all other numbers ≠ 1.

And here quantitative and qualitative - while being necessarily related - maintain a distinctive identity.

This likewise entails that in all such cases, finite and infinite notions of number likewise maintain a relatively distinct identity.

And where this is the case, the actual finite nature of series does not necessarly concur directly with their infinite values.

And when we come to investigate values of the Zeta Function (for s < 1) this is especially relevant.

So again (for s = - 1) the actual nature of the terms in the series 1, 2, 3, 4, etc. would strongly suggest that its sum diverges to infinity (from a linear perspective).

However this is not the case with the infinite sum = - 1/12!

Thus, we will need to investigate this numerical value in the alternative manner suggested (where linear notions of interpretation thereby have to be abandoned).

Tuesday, February 21, 2012

Riemann Zeta Function (2)

Though I initially started to use the analogy of crossroads as a means of highlighting the "missing" qualitative aspect of Mathematics, I have since come to realise that in fact it serves a more direct purpose in its ability to properly identify key features with respect to the Riemann Hypothesis.

The very depiction of the Riemann Hypothesis is in the form of a crossroads.

So in horizontal terms we have the real axis (corresponding in turn to the real dimensional values (s) of the Riemann Zeta Function. Then we have a vertical line drawn through the real axis where s = .5 which represents the imaginary part of the complex number values for s that generate the non-trivial zeros.

And of course if the Riemann Hypothesis is true, the imaginary part of all non-trivial zeros will lie on this vertical line through .5!

Now, as we have seen at a crossroads we are faced with opposite polarities that can shift location (depending on convention). Thus from one valid perspective, travelling "up" a road when I unambiguously label one turn as left, the other turn at the crossroads (in relation to this turn) is necessarily right; then travelling "down" the road again from above the crossroads, when I unambiguously label one turn as left the other turn is then - relatively right.

However when combining both terms of reference "up" and "down" (and "down" and "up") we create a purely circular logical interpretation, where what is left is also right and what is right is also left. And such circular logic is the very essence of the interpretation of interdependence (which cannot be approached with isolated reference frames).

And of course this poses a central problem for conventional mathematical interpretation (based on independent frames in accordance using linear rational understanding).

In particular it poses intractable problems with respect to interpretation of prime numbers (whose central feature relates to such interdependence).

Now putting it more mathematically i.e. in holistic (Type 2 terms), left and right represent a specific example of positive and negative polarities (which in dynamic interactive terms change location depending on context).

And in the present context positive and negative apply to the dynamic interaction of both quantitative and qualitative type meaning.

Now a full understanding requires that we combine both 1-dimensional (using independent frames of reference) with 2-dimensional interpretation (where reference frames are interdependent). This requires in turn that we employ linear logic (in the case of the fixed frames) and circular logic (in the terms of the simultaneous combination of these frames as interdependent).

So we will cut right to the chase now (before elaborating later in considerable more detail).

Riemann came up with a fascinating transformation formula (Functional Equation) that associates the values for ζ(s) to the RHS of .5 on the real axis, with corresponding values on the LHS of .5 for ζ(1 - s) with the one exception where s = 1.

Now remember .5 in this context represent the intersection of our crossroads with - literally - right hand and left hand designations.

The crucial fact to grasp is that right hand and left hand correspond here - relatively - with both quantitative and qualitative interpretation respectively.

In other words if we fix values on the real axis to the right of s with quantitative interpretation of values of the Zeta Function, then values to the left of s thereby must - in this context - be given a qualitative interpretation.

Likewise if we alternatively fix values to the left of .5 with quantitative interpretation of the value of the Zeta Function, then values to the right must - in this alternative context - be given a qualitative interpretation.

So the Zeta Function therefore can be given both quantitative and qualitative interpretations on both the LHS and RHS of .5 respectively.

Now the midpoint (.5) of this horizontal crossroads represents the pure intersection of both quantitative and qualitative type interpretations (as simultaneously identical).

And this again is what the Riemann Hypothesis is all about i.e. in establishing the key condition for the identity of both quantitative and qualitative aspects (which are inherent in the primes).

So the additional requirement that the value of the Zeta Function be then = 0, requires that in the case of all the imaginary parts of s for which this is true, that the real part = .5. In other words the imaginary part values of s, for which both ζ(s) and ζ(1 - s) = 0, must necessarily lie on the imaginary axis through .5.

Now, this in no way constitutes a proof, as the Hypothesis is prior to any proof and already inherent in any axioms that might be used for its establishment. Rather it is a demonstration of the nature of what already necessarily is the case!

As I say, I will tease out in more detail the exact nature of this relationship as between quantitative and qualitative in future blog entries!

I will just leave here with one revealing observation.

The only point on the real axis where ζ(s) is not defined occurs for s = 1.

1 in this context directly refers to a dimensional value and as I have stated so many times before, Conventional (Type 1) Mathematics is defined - literally - by its 1-dimensional approach, where the qualitative aspect is thereby reduced to the quantitative.

Foe all other dimensional values, both quantitative and qualitative preserve distinctive meanings (that are not directly confused with each other).

So once we accept that the Riemann Zeta Function properly relates to the complementary matching of both quantitative and qualitative aspects of interpretation, then this cannot apply in the case of 1 (as by definition it can be given no matching qualitative interpretation!)

Putting it more starkly, it highlights how unsuited conventional (Type 1) interpretation in fact is as a means of appreciating the nature of the Riemann Hypothesis.

So once again I will return to the nature of this complementary quantitative/qualitative matching in future blog entries with a view to much greater clarification of what is involved.

Monday, February 20, 2012

Riemann Zeta Function (1)

When asked once what was the most important problem in Mathematics - as claimed in Constance Reid's book - the great mathematician Hilbert replied!

"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important"

Now Hilbert would have been referring here to the non-trivial zeros.

Funny, it was the trivial zeros that proved more crucial in my own case in realising the true nature of the Riemann Hypothesis, which is ultimately concerned with the condition for the identity of quantitative and qualitative meaning.

It seems that Hilbert might even have had some intuition that this was the case which would make his quote especially apt.

For as all created phenomena represent the relationship of both quantitative and qualitative aspects with their roots in the primes, what could be more important than discovery of the ultimate relationship between both aspects?

However in formal terms Hilbert considered Mathematics very emphatically as being based solely on the masculine principle (of linear logical interpretation).

Indeed he held the belief that all problems in Mathematics could in principle be solved within its axiomatic system.

In this regard he was quickly proven to be mistaken, when Godel showed that there would always be important problems which would remain undecidable (which could neither be proved nor disproved). This was subsequently demonstrated in 1963 with respect to the Continuum Hypothesis (which was the very first problem on Hilbert's famous list of 23 unsolved problems!)

Incidentally Godel apparently held the belief that the Riemann Hypothesis would also likely fall into this category of undecidable propositions!

As for Riemann, it became subsequently clear that he did not think of Mathematics as a specialised abstract pursuit but rather combined his mathematical interests with problems from Physics.

So it would be very easy to believe if Riemann had been aware of subsequent developments with respect to Quantum Mechanics, that he would have quickly appreciated the relevance of the non-trivial zeros for physical quantum systems. Indeed it is likely that he would have been the first to make such a suggestion!

So we will return over the next few blogs to the key issue of what in fact the non-trivial zeros of the Riemann Zeta Function actually mean, where we will finally conclude that they are every bit as significant as Hilbert postulated (though for reasons that would strongly conflict with his definition of Mathematics).

So we start with the prime numbers 2, 3, 5, 7,.. which are the most independent of all numbers (with no factors other than each prime and 1). In this regard the individual primes represent the extreme example of the masculine principle serving as the building blocks for the natural number system.

However there is an equally fascinating aspect to prime numbers in that they are likewise the most interdependent of all numbers.

What this precisely relates to is the Fundamental Theorem of Arithmetic, where each natural number can be uniquely expressed as the product of one or more prime numbers.

Now, because of the merely quantitative bias of Conventional Mathematics, I believe that the true significance of this latter aspect of the primes has been continually overlooked.

As I have stated many times before, when we multiply numbers together a qualitative (dimensional) - as well as quantitative - transformation is involved.

In Conventional Mathematics the qualitative aspect is then ignored with the result expressed in a merely reduced quantitative manner.

So what is entirely missed therefore with respect to multiplication, is that inherently it entails a qualitative process (which - by definition - cannot be encapsulated in conventional mathematical terms).

Therefore it is no wonder that Brian Conrey and Alain Connes have been quoted as frankly admitting that a central issue exists with relation to the link between addition and multiplication that is not understood!

Well the simple answer is that whereas addition can indeed be considered within a merely quantitative perspective, multiplication necessarily entails both quantitative and qualitative aspects of transformation.

In my journey towards appreciation of the true nature of the Riemann Hypothesis, I had already formed a strong realisation of this fact from childhood.
I knew there was something wrong with the conventional interpretation of multiplication (and that both quantitative and qualitative aspects were involved). However obviously at the time I had not sufficient intellectual capacity to develop the implications of that insight further!

So when I later returned to the issue my first instinct was to develop an alternative mathematical approach (where the missing qualitative aspect could be properly explored).

So the wonderful thing about prime numbers is that while representing the extreme example of independence in isolation, yet when combined, they represent the extreme collective example of interdependence (and can generate every natural number in a totally unique manner). And this unique relational capacity represents the embodiment of the (unrecognised) feminine principle.

Therefore prime numbers combine both extreme autonomous (quantitative) and relational (qualitative) capacities in their very identity.

The overall pattern of the individual primes is now well recognised, where they become progressively less densely populated as we travel up the natural number scale.

However it is the reverse with respect to the non-trivial zeros - or more precisely the imaginary part of the dimensional numbers that give rise to such zeros - which become more densely populated as we progressively travel up (or down) the imaginary number scale.

The clue to what these zeros actually represent comes from their relational capacity.

Now for very low numbers on the scale say up to 30, relatively few primes are required to generate the natural numbers involved; likewise few options exist in terms of possible unique combinations of such primes.

However as we progress to larger natural numbers, the number of primes required for their generation steadily increases (and likewise the possible combinations entailing such primes).

So the non-trivial zeros can be appropriately understood as measuring this relational capacity of the primes (in their unique collective configurations with the natural number system).

As I have stated before, this relational capacity, by comparison with the individual nature of the primes, is properly of a qualitative (rather than quantitative nature).

Two key indications of this are already apparent with respect to their quantitative representation!

Some weeks ago I marvelled at how easy it was to move from the prediction of the (average) spread as between individual primes to the corresponding (average) spread as between non-trivial zeros. The key distinction is the manner in which 2pi plays a role in the formula for the latter!

Now 2pi precisely measures the circumference of the unit circle (on which the qualitative aspect of the number system is based).

Therefore from a qualitative perspective, this clearly implies that we move from a linear interpretation with respect to the spread as between individual primes to a circular interpretation with respect to the spread as between non-trivial zeros.

And as circular interpretation is the indirect rational means of portraying intuitive appreciation of a holistic kind, this clearly establishes that we require two distinctive forms of understanding for the interpretation of the individual primes on the one hand and - relatively - the non-trivial zeros on the other.

Put another way the non-trivial zeros relate to the qualitative (holistic) nature of the primes (which can be given an indirect quantitative measurement).

I have shown in other blog entries how the imaginary (from a qualitative perspective) then serves in turn as the indirect means of expressing circular type notions in a rational linear manner.

An lo and behold once again the non-trivial zeros line up obligingly as quantitative points on an imaginary scale of measurement.

We must however keep returning to the crossroads analogy to remind ourselves of what is happening.

Once again, when using isolated poles of reference, when I travel "up" a road I can label a turn at a crossroads unambiguously left. Then when I later travel down the same road, I can unambiguously label the other turn also as "left". So both designations as "left" are valid within isolated poles of reference.

However when we consider both turns as interdependent (in two-way relationship to each other), they are necessarily left and right (and right and left) with respect to each other.

Now, it is exactly analogous in mathematical terms. When we approach the primes from an individual perspective, they do indeed display quantitative aspects. Then when we approach the primes from the opposite general perspective (with respect to their overall frequency) again they display quantitative aspects. However when we consider both individual and general characteristics (in a two-way interdependence), they are necessarily quantitative and qualitative with respect to each other.

So when we look at either the individual or collective nature of the primes - or indeed equally the individual or collective nature of the non-trivial zeros - within isolated frames of reference, they will indeed all display quantitative characteristics. However when we consider these in two-way interdependence with each other, in each case, they are necessarily quantitative and qualitative (and qualitative and quantitative) with respect to each other.

So once again the primes comprise two extreme aspects with respect to their very identity; with respect to their individual nature, we have an extreme autonomous capacity (as independent); then with respect to their collective identity we have
an opposite extreme relational capacity (as interdependent).

However properly understood in relation to each other, these capacities are as quantitative to qualitative (and qualitative to quantitative) respectively.

However because of the mere quantitative bias of Conventional Mathematics based on linear logical notions, it has no way of properly interpreting the equally important qualitative aspect. So it necessarily reduces the qualitative aspect (in the holistic communication capacity of the primes) to mere quantitative interpretation.

And in doing this it misses out completely on the true nature of the primes!

I referred to Alain Connes' metaphor of the missing female heroine (relating to his quest to solve the Riemann Hypothesis) in an earlier blog entry.

Thus, though the heroine in truth is, when viewed properly, centre stage in ardent embrace of our hero, from a conventional mathematical perspective, she appears entirely absent.

Sunday, February 19, 2012

Prime Numbers (Local and Non-Local Identity)

It is my intention in these blogs - for anyone who cares to read - to combine elements of the teacher, preacher, prophet, iconoclast, radical, antagonist - indeed whatever it takes - to challenge existing mathematical perceptions and in some measure awaken readers to a wonderfully enlarged mathematical universe that still remains totally undiscovered.

Though the discussion on this blog - as is the intention - necessarily centres around the Riemann Hypothesis, this really is being used to serve a much greater purpose relating to the true (unrecognised) nature of Mathematics.

The insights that I offer here do not come from idle speculation, but rather arise from a deep struggle in my late teens with the nature of Mathematics leading to a significant personal transformation, the implications of which I have continued to explore for some 45 years.

We are accustomed to viewing electromagnetic radiation in physical terms as a spectrum comprising many different bands; likewise this is true from a psycho spiritual perspective.

And just as natural light comprises one narrow band with respect to the physical spectrum, equally the natural intuitions that inform the vast bulk of intellectual discourse in Western society likewise come from one narrow band (around the middle of this psychological spectrum).

And Mathematics - as conventionally understood - represents par excellence, an extreme specialisation with respect to the thinking associated with this narrow band.

So as I have stated so many time before, in a very precise qualitative manner, Conventional Mathematics is based on a 1-dimensional logical approach (where isolated uni-polar reference frames of enquiry are adopted).

However on the full spectrum, representing the latent possibilities for psycho-spiritual development, many further bands of ever more refined forms of intuitive energy exist.

In the past such bands were always universally associated with spiritual contemplative development, which tended to be strongly wedded to the belief systems of the various mystical traditions. Though many detailed accounts have been left by courageous voyagers who successively traversed these realms, invariably they are couched in a religious manner (employing the symbols of their respective traditions).

So, throughout history the deep implications of the contemplative vision for Mathematics and the Sciences have therefore been largely ignored.

Thus, my own particular concern with respect to contemplative type development has been significantly related to its implications for scientific type understanding and especially for mathematical interpretation.

So from the perspective I now customarily adopt, I can see clearly that what we conventionally term Mathematics represents in fact just one narrow band on the overall mathematical spectrum (where qualitative type meaning is directly reduced in quantitative terms).

However potentially an unlimited number (i.e. dimensions) for other mathematical systems exist, that represent - precisely defined - dynamic configurations with respect to both quantitative and qualitative meaning.

Putting it another way, Conventional Mathematics is formally based on exclusive recognition of the conscious aspect of recognition (which gives it an absolute type appearance); however in all other mathematical systems both conscious and unconscious are involved in a dynamic relative manner.

Stating it yet another way, Conventional Mathematics necessarily entails the confusion of potential (infinite) with actual (finite) notions of meaning, whereas in all other systems a careful distinction between both is maintained.

Or finally, to emphatically make the point here, no formal recognition is given to the role of intuition in Conventional Mathematics (where again its use is reduced to rational type interpretation). However in all other systems, intuition is recognised as operating in a vitally distinct holistic manner, and cannot be encapsulated in a linear (i.e. 1-dimensional) rational fashion.

It might be instructive in this regard to explain how my holistic appreciation of the nature of prime numbers emerged.

When I was initially engaged in in attempting to map out the psychological spectrum, not surprisingly I spent some considerable time dealing with earliest infant development and the processes by which phenomenal understanding emerges.

I then began to appreciate that there were close connections as between the meaning of the words "primitive" in a psychological context and the corresponding holistic notion of prime numbers.

In earliest development, phenomena enjoy but a fleeting momentary existence, and the reasons for this are very interesting to probe.

Because neither conscious nor unconscious aspects of personality have yet been properly differentiated, holistic notions (relating to the unconscious) are directly confused with specific phenomena. In other words the neonate continually confuses the general dimensions (of space and time) in experience with specific objects. Therefore because - quite literally - phenomenal objects cannot yet be placed in a proper dimensional context - they enjoy but an immediate transient existence.

I hasten to add that this insight is also very relevant to the nature of sub-atomic particles, which likewise enjoy but a fleeting momentary existence. This clearly implies that at this level of material organisation, the natural dimensions of space and time have not yet properly formed. Of course we do not properly recognise this fact when we insist on imposing perceptions of dimensions appropriate to the macro level on sub-atomic reality!

So we can accurately characterise - in both physical and psychological holistic terms - earliest reality as existing in a - mathematical - prime framework, where both the individual (object) and collective (dimensional) identity of the primes is confused.

Putting it more precisely from a physical perspective, the prime numbers contain two aspects relating to an infinite potential ground (as their dimensional attributes) and to finite actual entities (where they can be identified as individual numbers).

And in corresponding psychological terms with respect to their manner of interpretation, the prime numbers again contain two corresponding aspects relating to an unconscious (with respect to their holistic) and a conscious (with respect to their specific) nature respectively.

Incidentally when I read some years ago that a quantum physical basis had been discovered for the Riemann non-trivial zeros, I was not at all surprised; in fact due to the holistic nature of my approach, I had already formed the conclusion several years earlier that this was necessarily the case!

So with reference to the whole spectrum of development, we can see clearly that there are necessarily two aspects to the primes (with matching physical and psychological correspondents).

And this means in turn that from a dynamic interactive perspective - which is the proper context of experience - prime numbers have no strict reality apart from the interpretations that are brought to bear on them!

Now mathematicians such as Hardy would have recoiled in horror at such a revelation.

From their perspective, the identity of numbers was as absolute as one could get (eternally existing - as it were - in some sort of immutable Universe).

And of course the reason they could think this, relates to the specialised nature of rational development that only unfolds at a later stage of human development.

So the supposed absolute nature of numbers, simply reflects the absolute nature of the rational paradigm from which they are interpreted!

Or to put in an equivalent manner, reality inevitably reflects the manner we look at it; so when we formally exclude the role of the unconscious altogether, numbers do indeed appear as absolute fixed entities!

However once we begin to incorporate the unconscious appropriately with conscious interpretation, this comforting worldview irretrievably breaks down.

Thus, stating it in more nuanced language (incorporating both conscious and unconscious aspects) there is an inherent potential in prime numbers which only then later in human history can become actualised in a finite manner. And as such actualisation remains a continual on-going process, the prime numbers can be accurately seen - without hyperbole - as embracing the entire course of created evolution!

So for example, since Riemann's path breaking discoveries in 1859, it is known that underlying the actual identity of specific primes (that had long been observed) is a - hitherto unknown - harmonic structure.

Therefore though this potential was always inherent in the nature of the primes, we had to wait till 1859 for its very first revelation in actual terms.

Furthermore we can safely say that the full implications - which are truly mind boggling - of this revelation, have not yet evolved.

If we revert to Quantum Mechanics, we can perhaps appreciate that though the foundations were laid in the 1920's, the continued reverberations with respect to the implications of this new theory lasted for a considerable period afterwards (and have in no way yet been fully realised).

For example, one fascinating issue related to the apparent non-local effects of sub-atomic particles, so that a particle such as a light photon seemingly could communicate with another photon (even at a great distance).

Einstein in his (unconscious) desire to maintain a strict deterministic approach railed against such new findings. However in the 80's, though controversy necessarily remains, it was experimentally demonstrated that non-local effects with respect to particles do in fact occur.

Now properly understood, this is also true in an even more profound manner with respect to the nature of prime numbers.

Though prime numbers in their (actual) individual identities, seem the most "local" of all numbers; yet with respect to their (infinite) potential identity (in relation to all other primes) they are equally the most "non-local" of numbers. So they combine, in their very nature, extremes with respect to two opposing modes of identity!

Thus again, at the level of (actual) individual primes they are the most independent of all numbers. However equally at their collective (potential) level they possess the ability to communicate with all other primes, thereby preserving a perfect synchronicity with respect to overall behaviour.

However the deeper implication of this finding requires the surrender of an exclusive rational approach to Mathematics!

Quite simply the present paradigm is geared to deal merely with local effects; by its very nature it cannot deal with non-local effects, which correspond to a distinctive holistic mode of behaviour (and matching interpretation).

Put another way, we are now truly at the crossroads in Mathematics with respect to the prime numbers. The unconscious now urgently needs to be explicitly incorporated (with conscious understanding) in the recognition of a distinctive qualitative aspect to their behaviour. And of course by extension this recognition necessarily applies to all Mathematics!

Then, once again, the Riemann Hypothesis points to the condition for the mutual identity of both the quantitative (local) and qualitative (non-local) aspects of the primes.

And eventually, it will be realised that this new feature of the primes had always been potentially inherent in their very nature (which however could only be actualised when appropriate human evolution with respect to corresponding unconscious understanding had taken place)!