We looked yesterday at the holistic explanation for ζ (– 2), the first of the trivial zeros.
Indeed the holistic rationale used here is equally important with respect to appreciation of the non-trivial zeros.
We have seen for example that in holistic terms ζ (– 2) = 0 represents a psycho-spiritual energy state (i.e. as pure intuition).
And as in holistic terms, psychological and physical aspects are complementary, this of course also entails that ζ (– 2) = 0 has a corresponding meaning in terms of the appreciation of a physical energy state!
So what happens in this holistic generation of energy is that two analytic quantitative values (having unambiguous meanings within independent (1-dimensional) frames of reference are brought together and now holistically understood - in terms of each other - as directly complementary. So like matter and anti-matter particles in physics they negate or cancel each other out resulting in pure energy.
So the holistic ability to recognise - in any context - the complementarity of relationships is directly intuitive in nature though this can subsequently be given an indirect (circular) rational explanation.
Therefore in dynamic interactive terms, the direct qualitative - as opposed to quantitative - appreciation of a mathematical relationship is holistic in nature. However this qualitative appreciation arises from the complementary pairing of quantitative type relationships (that have an unambiguous analytic meaning within fixed independent frames of reference).
So the true mathematical appreciation of the holistic qualitative aspect of the relationship requires a more refined quantitative type interpretation in analytic terms (where one recognises that it takes place within frames of reference that strictly are always of an arbitrary nature).
So once again using our crossroads example we can unambiguously define a turn as left or right (assigning it a value of + 1 or – 1 respectively if the crossroads is approached from just one direction). However when we recognise that the crossroads can be simultaneously approached from two opposite directions, then left and right (i.e. + 1 and – 1) now have a paradoxical meaning (where + 1 = – 1 and – 1 = + 1).
Thus it is in this moment of simultaneous recognition (from two complementary reference frames) that the holistic qualitative nature of the relationship is directly appreciated in intuitive fashion as 0 (i.e. nothing in dualistic phenomenal terms).
The huge significance of all this is that the fundamental nature of the relationship as between the primes and the natural numbers (and natural numbers and primes) is necessarily conditioned by polar opposites (such as external/internal and whole/part) that are complementary with each other in dynamic interactive terms.
So conventional analytic understanding always takes place within isolated independent frames of reference that are necessarily of an arbitrary contingent nature!
In fact at this point it might be of value to proceed a little further by now explaining the holistic mathematical significance of the second of the trivial zeros i.e. ζ (– 4).
Again ζ (– 4) = 0.
What this means is that appropriate holistic appreciation now entails two sets of complementary relationships (that indirectly can be represented by the four roots of 1).
In other words at the 4-dimensional level of understanding one understands that all experiential relationships are conditioned by both real and imaginary polarities that are positive and negative with respect to each other.
At this level one would recognise that all mathematical relationships have both an objective and subjective aspect (i.e. as mental interpretation).
Therefore - though this is frequently overlooked - we can have no objective meaning (in mathematical terms) without corresponding mental interpretation. Thus the view - as so strongly elucidated for example by Hardy - that primes are objective in an absolute manner - is itself the product of a particular mental interpretation (that is of a crucially limited nature).
Likewise, as we have seen, we cannot - though again completely overlooked in conventional terms - strictly have quantitative mathematical meaning (in the absence of a corresponding qualitative aspect).
Indeed this entails the fundamental relationship of whole and parts, in dynamic interactive terms, is always quantitative and qualitative with respect to each other. Thus the key reductionism that defines conventional mathematical treatment of number is the attempt to define the whole (in any context) merely in terms of its quantitative parts!
However at the 4-dimensional level of appreciation, one now sees mathematical objects and corresponding mental interpretation in complementary terms as positive and negative with respect to each other.
Then one further understands the whole and part likewise in even more refined holistic mathematical terms as real and imaginary with respect to each other.
Thus from a qualitative perspective whereas the conventional analytic approach represents the "real" aspect of mathematical understanding, the unrecognised holistic aspect represents its corresponding "imaginary" aspect.
Therefore though both real and imaginary aspect (complex numbers) are now recognised in quantitative terms, there is as yet no corresponding recognition of an imaginary aspect (in a qualitative manner).
So the holistic aspect of mathematical understanding on which I am currently elaborating, represents this unrecognised imaginary aspect (in qualitative terms).
And future Mathematics will be required to move to a new complex rather than the current real rational approach (in qualitative terms) that characterises the conventional treatment of all relationships.
Therefore it requires a more refined intuitive state to properly recognise the truly complementary nature of both the real and imaginary aspects of mathematical relationships (in positive and negative terms).
However in principle it is very similar to 2-dimensional understanding (where now opposite reference frames cancel out in both real and imaginary terms).
And of course negative 4-dimensional understanding then entails the direct intuitive recognition of the coincidence of opposite poles (free of secondary rational interpretation).
Now meaning can likewise be given to all the other negative even dimensions (representing trivial zeros) where complementarity can be established as between one set of reference frames and another set that are directly opposite in all cases.
All these "higher" even dimensions entail various configurations entailing real and imaginary values (as indirectly expressed through the corresponding roots of the number).
So therefore in holistic terms, these likewise represent varying configurations, in dynamic interactive terms, with respect to both objective recognition (and corresponding mental interpretation) and holistic (qualitative) and analytic (quantitative) recognition.
Therefore with respect to the prime and natural numbers the understanding corresponding to ζ (– 4) = 0, relates to the direct intuitive recognition that ultimately these numbers have no objective existence independent of mental interpretation. In other words their ultimate nature is ineffable!
Likewise it entails that both the prime and natural numbers ultimately can have no quantitative meaning independent of their corresponding qualitative aspect. This implies that a remarkable synchronistic interaction dynamically characterises their relationship, which ultimately again is ineffable in nature.