We will finally, in this blog entry, further probe the precise holistic significance of the Riemann Hypothesis, where all non-trivial zeros of the zeta function are postulated to lie on the imaginary line through 1/2.
The significance of 1/2 is fascinating from a holistic context. What it in fact entails is that the notion of mathematical truth in objective terms is dynamically inseparable from the corresponding notion of this same truth as (subjective) interpretation!
Though from the conventional mathematical perspective, we are accustomed to interpreting number objects in an absolute objective fashion, this makes no sense in experiential terms. And remember all mathematical understanding is necessarily experiential in nature!
So for example when one becomes objectively aware of a number - say "2" - this necessarily corresponds with the mental perception of "2". So strictly speaking in experiential terms, "2" represents a dynamic interaction governed by two reference poles that are - relatively - external and internal with respect to each other.
So, if we refer to the external (objective) pole in quantitative terms, then the corresponding internal (subjective) pole as perception is - relatively - qualitative in nature.
So once again the understanding of "2", in our lived experience, properly entails both quantitative and qualitative aspects in dynamic interaction with each other.
Now the very nature of conventional mathematical interpretation is the attempt to freeze this interaction in an absolute manner (through recognition of just one polar reference frame).
This then leads to the mistaken notion that mathematical objects enjoy an absolute existence in an objective space (which can be - literally - abstracted from experience).
So "2" is now defined absolutely in a merely static objective manner.
If pressed, a mathematician may reluctantly agree that one cannot form knowledge of a number object, without a corresponding mental construct!
However any consideration of the necessary dynamic interaction as between object and construct will then be quickly dismissed as irrelevant in a mathematical context.
Thus in effect the assumption is made that the mental constructs absolutely correspond with their objective counterparts.
In other words the qualitative aspect of appreciation is thereby reduced in a mere quantitative manner!
Though it clearly makes no sense to insist that these two polarities of understanding (external and internal) can be absolutely separated in an independent manner from each other, this is the unwarranted key assumption that currently underlies all accepted mathematical interpretation.
For even when this assumption (of the abstract nature of mathematical objects) is insisted on in formal terms, the actual experience of understanding implicitly requires that some dynamic relationship (however small) be maintained as between external and internal poles.
So properly understood, the standard analytic approach to Mathematics really represents just one limiting extreme, where one attempts to approach as close as possible to an absolute type appreciation of mathematical objects (which however always remains at the level of relative approximation)!
So again even though one may interpret "2" as representing a number with an absolute identity, this can never be strictly true, for the very nature of mathematical communication represents a form of social consensus that can only be approximate.
Indeed it should be very clear that in this discussion, I am directly challenging the very nature of this consensus. This of course does not mean that satisfactory communication still cannot take place (within well-defined contexts). However once again, mathematical truth remains always of a relative - rather than absolute - nature.
So the analytic extreme requires the attempt to keep that opposite poles that underlie all phenomenal experience (such as external/internal and whole/part) as independent as possible.
Thus once again, in any context, just one pole of reference is used in an unambiguous manner, with relationships now literally conforming to analysis through linear (i.e. 1-dimensional) reason .
However, the holistic extreme represents the complete opposite;
Here, rather than concentrating on the relative independence of opposite poles (such as external and internal) one rather focuses on their complementary identity (through interdependence). Then in extremes - which conforms directly to highly refined intuitive understanding - one approaches a pure energy state (largely free of any phenomenal notion of number).
Attaining this nondual state - which again can only be approximated in relative manner in experience - requires achieving the golden mean as between the two poles (external and internal) so that both are harmonised as close as possible with each other.
As we have seen, 1-dimensional entails sole recognition of just the pole (i.e. the external pole of "objective" quantitative recognition).
Now in a wonderful holistic manner, 1/2 represents the equal splitting of this dimension as between both the external (quantitative) and internal (qualitative) poles.
So the true holistic significance of the fact that the non-trivial zeros are postulated to all lie on the line through 1/2, is that this condition is necessary to ensure the identity of both the quantitative and qualitative aspects of number recognition.
And what is vital to appreciate is that this requirement can only be properly appreciated in a holistic intuitive manner (where both quantitative and qualitative aspects of recognition are directly harmonised in experience).
As I explained in an earlier blog entry, we can then indirectly translate this intuitive recognition in a circular logical fashion, through paradox.
So once again, the zeros represent points on an imaginary number line, where both the primes and natural numbers are identical, where randomness and order with respect to the number system are identical, where the cardinal and ordinal interpretation of number is identical and finally where addition is identical with multiplication. And all of this results from being simultaneously able to "see" number relationships with respect to two complementary poles of reference!
However I must stress once again that conventional mathematical ability (of an analytic nature) is of no avail in this regard. Indeed as it is based in extreme fashion on just one independent pole of recognition it is only likely to hinder such ability.
So again, Holistic Mathematics represents an utterly distinctive form of mathematical recognition, whose very development initially requires substantial initial withdrawal from the unfortunate "indoctrination" imposed through conventional mathematical procedures.
I know very well from experience how deeply ingrained is the conventional mathematical "ideology" in our culture.
But for the fact that I formed a strong conviction at the age of 10, regarding a fundamental unaddressed problem with respect to multiplication, I doubt if my "conversion" to Holistic Mathematics could have readily taken place.
However though this has led to many wonderful discoveries, that have greatly enriched my life, it has always remained very much a solo voyage. In fact, remarkably few are yet prepared to seriously question the key construction faults that deeply underlie the mathematical edifice.
I will address now briefly the holistic reason why the zeros lie on an "imaginary" line.
I have already mentioned that one can indirectly express the holistic intuitive appreciation of the nature of the zeros indirectly in a circular logical fashion (through rational paradox).
Now in holistic terms, the "imaginary" line is the appropriate means to express notions of an inherently circular nature in an acceptable linear manner.
When I was addressing the numerical values of the zeros I showed that they were closely related to the divisors (natural factors) of each number.
However this required converting from a circular scale (represented by the circumference of a circle) to a corresponding linear scale (represented by the radius).
In similar holistic fashion, zeros that lie on an "imaginary" line represent a means through rational discourse of expressing the fact that these points conform to a different logical system (i.e. that is holistic - rather than analytic - in nature).
In normal discourse "imaginary" - as opposed to "real" - is generally used to indicate what is of unconscious origin. It is very similar in holistic mathematical terms.
So the postulate that all the zeros lie on an "imaginary" line implies that the holistic unconscious basis of the (cardinal) number system is fully consistent with its corresponding analytic interpretation. Again this is an assumption that is already implicit in the use of the standard axioms of Conventional Mathematics! (In other words, the truth of the Riemann Hypothesis is assumed in the these axioms, which means of course that it supersedes them and cannot be thereby proved or disproved through their use). Thus once again, the Riemann Hypothesis points to the key requirement that both quantitative and qualitative aspects of interpretation can be fully reconciled with respect to the number system.
Finally I will briefly address the fact that the non-trivial zeros can be given negative as well as positive values.
This once again points to the pure holistic nature of these zeros.
In holistic terms, the fact that each positive zero has a matching negative value, implies that in their very understanding the momentary positing of a zero (as consciously independent) must be immediately negated (in an unconscious manner). It is only then that its pure holistic nature (as a psychological energy state) can be continually maintained. And of course this has a complementary interpretation in terms of a physical energy state (within the atomic structure of matter).
So once again we are here at the holistic extreme to number (which is based on the intuitive recognition of the pure interdependence of number) which complements - in relative fashion - the analytic extreme (of its pure independence).