We are returning again to interpretation of the - so called - trivial zeros of the Riemann Zeta Function, which occur for the negative even integers of s, i.e. s = - 2 , - 4, - 6, - 8, ....etc.
The first hint as to their nature is given through consideration of the values of the Function for corresponding even integer values of s, i.e. s = 2, 4, 6, 8,... etc.
Now for each of these values, the Function can indeed be given a finite quantitative value (in accordance with accepted linear notions of interpretation that define Type 1 understanding). In other words, in each case the sum of terms for the Function can be seen to converge to a limiting value.
Furthermore since Euler's pioneering contributions, all of these values (with s as an even integer) can be expressed in the form k*(π^s) where k is a rational fraction.
So for example in the best known case where s = 2,
ζ(2) = (π^2)/6.
So in direct terms on the RHS for s > 1, the Riemann Zeta Function (i.e. the Zeta 1 Function) can be given a quantitative interpretation in accordance with Conventional (Type 1) Mathematics (that is defined qualitatively in default 1-dimensional terms). And once again, this always implies that qualitative type considerations are reduced in a quantitative manner.
However when one allows for a uniquely distinctive (Type 2) aspect to Mathematics, this implicitly implies a (hidden) alternative Zeta 2 Function that indirectly can provide a distinctive qualitative interpretation (for Type 1 quantitative results).
So just as π in quantitative terms implies the pure relationship of the circular circumference to its line diameter, in corresponding qualitative (Type 2) terms, π equally implies the pure relationship of circular and linear type meaning.
What this in turn implies - as I have illustrated so many times before with respect to the crossroads analogy - is the ability to give opposite poles in understanding both an independent meaning (as separate) and then a truly interdependent meaning (as complementary and - indeed - ultimately - identical).
And with respect to the simplest example (where s = 2) this implies in Zeta 2 terms, qualitative appreciation of 2 as a dimensional number.
This then quantitatively corresponds with the 2 roots unity which can can be geometrically portrayed as the circle of unit radius, drawn in the complex plane (with a single line diameter that is positive to the right of centre and negative to the left).
Now it has to be understood that when s is positive, understanding is conducted in a rational linear manner. This therefore leads to paradox in terms of the qualitative interpretation of all dimensions (other than 1).
Thus, in attempting to convey the very nature of interdependence in rational terms, one must to a degree keep both poles separate. So therefore the best one can do from a rational perspective is to directly imply that both poles (positive and negative) are indeed ultimately identical (when properly understood as complementary).
Though it is simplest to understand the nature of such circular complementarity (in paradoxical rational terms) for the case where s = 2, the same basic principle applies for all even integer values of s.
What this implies from a quantitative perspective is that roots of 1, for all such even values can be arranged in a complementary manner. So for example if s = 6, this means that 3 roots can be chosen that can then be identically matched in complementary fashion (i.e. by multiplying by - 1) with the 3 remaining roots.
Therefore when we understood the Zeta Function properly in qualitative terms (i.e. through recognition of the alternative Zeta 2 aspect that qualitatively interprets Zeta 1 results) we can then recognise - in a necessarily indirect rational manner - that for all even integer values of s, a perfect complementarity can ultimately be achieved with respect to both independence and interdependence.
In other words, polarised interpretations that arise through initially interpreting in a relatively independent manner (with respect to one isolated frame of reference) are ultimately understood to be fully interdependent with each other when these frames are related.
So again if we take the simplest example (for s = 2) of the two turns at a crossroads, one can initially understand using - relatively - isolated reference frames with respect to one fixed direction of movement, that both left and right turns have an unambiguous meaning.
However when we then relate frames of reference simultaneously (as interdependent) one now appreciate that left and right turns have a purely relative meaning, depending on context.
Thus, what is a left turn travelling up the road will be right when travelling down in the opposite direction; likewise what is right when travelling up the road will be left when approached in the opposite (down) direction.
Now the deeper understanding of such a relationship implies that whereas - in direct terms - independent type distinctions are provided through reason, interdependent appreciation - in direct terms - comes from holistic type intuition.
So though one can indeed indirectly come to a very refined rational appreciation of the nature of interdependence in a circular paradoxical manner, the full realisation of such interdependence can only come directly through (holistic) intuition.
Therefore, to put it another way, though the very nature of Conventional (Type 1) Mathematics - defined as its is by the linear (1-dimensional) approach - is to attempt to understand number with respect to its mere cardinal (quantitative) nature, in truth both cardinal and ordinal aspects always interact (reflecting both the quantitative and qualitative aspects of number respectively).
And though the qualitative (ordinal) aspect can be indirectly conveyed in a refined (circular) rational manner, its direct understanding requires holistic intuitive appreciation!
The shocking indictment therefore that must be made against Conventional (Type 1 Mathematics) is that it cannot possibly deal with both the cardinal and ordinal aspects of number in a consistent manner (as it formally defined solely with respect to the cardinal i.e. quantitative aspect)!
However in the more comprehensive Type 3 approach (which I am attempting to demonstrate here) even when explicitly recognising the quantitative numerical validity of mathematical expressions from the Type 1 perspective (as with the Riemann Function for values of s > 1), one implicitly still recognises the - indirect - need for a corresponding Type 2 interpretation (of a distinctive qualitative nature).
This provides the appropriate perspective to realise that when one then switches to the LHS of the Function, that it is the qualitative aspect that now is given a direct expression (with corresponding numerical values arising, having a merely indirect quantitative interpretation)!
Stating it in an equivalent alternative fashion, though on the RHS of the Function (for s > 1) the interaction of both cardinal and ordinal aspects of number does lend itself explicitly to quantitative type interpretation (and implicitly qualitative), with respect to the LHS of the Function (for s < 0) the position is reversed.
Here the interaction of both cardinal and ordinal aspects of number now lends itself explicitly to qualitative type interpretation of an intuitive kind (with the rational numbers arsising implicitly possessing merely an indirect quantitative meaning).
Appreciation of this very point is truly vital!
For, when properly grasped, this entails that Riemann's famous Functional Equation - when properly interpreted - always establishes a complementary type connection as between quantitative and qualitative (and - in reverse - qualitative and quantitative) type interpretation with respect to RHS and LHS values of the Function!
And this in turn means that we cannot possibly interpret the Function in a comprehensive manner, without recognition of its two vital aspects i.e. the Zeta 1 Function and the Zeta 2 Function respectively!
So again on the RHS (for s > 1) the Zeta 1 Function is explicitly used for interpretation in quantitative terms (with the Zeta 2 Function implicitly used from a qualitative perspective).
Then on the LHS (for s < 0), it is the Zeta 2 that is now explicitly used to interpret numerical values from a qualitative perspective (with the Zeta 1 implicitly used in a quantitative manner).
Now it should be patently obvious that from the conventional perspective, quantitative values of the Riemann Zeta Function for s < 0 make no sense!
For example when s = - 2,
ζ(- 2) = 1 + 4 + 9 + 16 + ....., which clearly diverges from a Type 1 (linear) mathematical perspective. Yet according to the Riemann Functional Equation,
ζ(- 2) = 0.
So much as modern mathematical interpretation attempts to cover over this important issue through technical abstract explanations with respect to analytic continuation, domains of definition, holomorphic functions etc., it cannot properly explain within its own terms such an ambiguous result.
And the true reason for this is that proper interpretation requires recognition of the qualitative - as well as the quantitative - aspect of mathematical interpretation! So quite simply, the result for ζ(- 2) seems counter-intuitive when interpreted in conventional linear terms (where by definition the qualitative nature of such intuition is reduced in a quantitative rational manner).
Indeed the first real indication as to the truly qualitative (ordinal) nature of the Riemann Zeta Function (where s < 0) is provided through appropriate dimensional interpretation of the negative sign for values of s.
I have explained before how dynamic negation (of rational interpretation) is the very means by which holistic intuitive appreciation arises in experience.
So if we wish to truly appreciate the interdependent nature of qualitative (ordinal) type number relationships, then we must switch directly to this intuitive mode (which only indirectly can be given rational expression).
So the important point to grasp is that the trivial zeros of the Riemann Zeta Function directly relate to the qualitative - rather than the quantitative - appreciation of number.
So what the trivial zeros thereby demonstrate is the nature of perfect ordinal interdependence (where number relationships are involved).
Expressed more precisely, the trivial zeros represent a direct qualitative (ordinal) appreciation of the interaction of both the cardinal and ordinal features of number (where perfect integration of both is achieved).
In other words when one appreciates number relationships in such a manner, one does so - not with respect to the initial independence of the numbers involved - but rather from the resulting interdependence arising in the relationship.
So, let us explain again this carefully with respect to the first of the trivial zeros i.e. s = - 2.
Now we already have dealt with the rational counterpart (for s = 2) where the relationship is given its geometric expression (in complementary quantitative terms) with respect to the 2 roots of 1. So we have a circle and a line diameter with the radius portion to the right of the centre positive (and the radius to the left negative). Now of course in dynamic interactive terms, what is positive or negative here is merely of an arbitrary relative nature!
So once again one can - literally - posit a left or right turn at a crossroads in unambiguous linear terms using relatively independent frames of reference i.e. when the direction of approach to the crossroads is fixed in one-way.
So when one moves up the road and approaches the crossroads from this direction, what is left and right (in this linear context) will have an unambiguous meaning!
Then, when having gone through the crossroads one switches direction and heads back down the road, one can again fix left and right turns in an unambiguous manner (from this alternative independent direction).
However when one now simultaneously relates both turns as interdependent, deep paradox arises (from a rational linear perspective). For what is left from one direction is also right from the alternative direction; likewise what is right from one direction is also left from the alternative:
So rational appreciation of interdependence is fully paradoxical (in linear terms). And this is the best one can manage in terms of indirect RHS appreciation of the qualitative nature of the 2-dimensional result.
However when one now switches to the LHS and considers the same dimensional number qualitatively in a negative fashion, this implies direct holistic intuitive realisation of the true nature of such interdependence.
Alternatively it represents the direct appreciation of the pure ordinal nature of number (which implies complete relational interdependence)!
So the startling implication is that the trivial zeros for the Riemann Zeta Function do not - directly - represent a quantitative but rather a qualitative interpretation of number.
And just as the quantitative (cardinal) interpretation of number relates directly to rational appreciation (of a linear kind), the corresponding qualitative interpretation relates directly to holistic intuitive appreciation (which appears as circular when indirectly expressed in rational terms).
The true qualitative nature of results for the non-trivial zeros is even symbolically indicated trough the very use of 0 as the symbol for zero (which directly indicates its circular nature)!
Now this might indeed seem strange (and out of bounds of what normally is considered Mathematics).
But this is precisely my point! For what is normally considered as Mathematics can be seen on close examination to be of an extremely reduced nature and quite frankly not fit for purpose.
And Conventional (Type 1) Mathematics is certainly not suited for proper interpretation of the Riemann Zeta Function.
In fact I now realise that for many years I have been paying Conventional Mathematics a deference which is not in truth warranted. Thus, while recognising the distinctive qualitative aspect for several decades now, I believed that Conventional Mathematics could still preserve a valid place (within its own terms of reference). However I now clearly realise that this position is not in fact tenable!
As it is ultimately impossible to properly understand the quantitative aspect of Mathematics (in abstraction from the qualitative) without confusion upon confusion being heaped on each other, I can now see that whole mathematical enterprise is in urgent need of rebuilding from the ground up (which is exactly what I am now, in my own way, attempting to do).
Of course this does not mean that the very considerable achievements of Conventional Mathematics have all been in vain. Rather it means that they will have to undergo radical reinterpretation, so that they can then be placed in their proper context (as one aspect of a much more comprehensive vision of Mathematics).
Indeed in the very term used for these zeros of the Zeta Function, the customary bias of Conventional Mathematics is clearly demonstrated.
They are referred to as the trivial zeros (implying that - unlike the non-trivial -they have no important role from a quantitative perspective (with respect to clarification of the distribution of primes).
However if one was to obtain a clearer understanding of the true nature of even the first of the trivial zeros (for s = - 2), then one's appreciation of both prime numbers and the Riemann Zeta Function would be changed forever!
Indeed it was precisely this insight, obtained many years ago, that gave me the initial confidence to pursue the Riemann Hypothesis with a view to unlocking its true nature.
From my earlier explorations in Holistic Mathematics - where I was seeking a satisfactory qualitative mathematical manner of mapping out the nature of contemplative development - I had already reached an understanding that the first of these "trivial zeros" related properly to a contemplative state of understanding (rather than rational interpretation).
In this regard I was deeply influenced by the writings of the Spanish mystic St. John of the Cross (which I have recounted in other blog entries). Now St. John is the most profound spiritual writer on the nature of the dynamic negations leading to contemplative development (of a pure intuitive kind). And he literally refers to the nature of such negation as nada i.e. nothing! In other words when one directly obtains a pure realisation of the nature of interdependence, then nothing - by definition - remains in phenomenal terms!
And of course this insight equally applies to the pure nature of numerical interdependence within Mathematics (which is - literally - nothing from an independent quantitative perspective)!
So here we are at the other extreme of the pure ordinal appreciation of number (which is nothing from an independent cardinal perspective)!
Once again the treatment of number in Conventional Mathematics is in terms of its - merely - independent (cardinal) identity, which is misleadingly interpreted in absolute terms.
However if numbers have an absolutely fixed quantitative identity, then there is no way without gross reductionism of giving such numbers a corresponding ordinal (i.e. relational) identity!
So to avoid such confused reductionism, in truth numbers have to be redefined with both independent (cardinal) and interdependent (ordinal) aspects which are now understood in relative terms.
Thus at one extreme we have the quantitative (cardinal) appreciation of numbers (in rational terms) where their - relatively - independent nature is highlighted.
However at the other extreme, we have the qualitative (ordinal) appreciation of numbers (in intuitive terms) where their - relatively - interdependent relational nature is highlighted.
Thus in the context of the Riemann Zeta Function, the trivial zeros represent a state of pure relational number interdependence.
So again if we take the simplest case where s = - 2, this would imply in the context of our crossroads illustration, a pure intuitive realisation of the interdependence of left and right directions (when opposite frames of reference are simultaneously considered).
Though the nature of such interdependence does indeed become more complex for the other negative even integer values of s, in principle the basic position remains the same.
Once again for any even integer dimensional value, we can through the Type 2 Zeta Function, provide a complementary matching set of roots (in quantitative terms) so that they always - literally - cancel out in this manner.
Thus the corresponding negative values for these integers relate to the pure intuitive (ordinal) interpretation of such interdependence (that is directly of a qualitative nature).
So far from being trivial, the true nature of the "trivial zeros" provides a fundamental key for proper comprehension of both the Riemann Zeta Function and its associated Riemann Hypothesis.
For the clear realisation that the trivial zeros relate to a pure qualitative (ordinal) rather than cardinal (quantitative) appreciation of number, immediately suggests the complementary nature of quantitative and qualitative type interpretation that exists with respect to RHS and LHS of the Function.
And the Riemann Hypothesis is then simply seen in this context as the central condition required for preserving the harmonious distribution of the primes (i.e. where quantitative and qualitative interpretations are identical).