There is a further unexpected surprise with respect to the interpretation of the Zeta 1 zeros.
However to appreciate this we must, as always, place our discussion in the dynamic interactive context of development.
I have contrasted before the nature of top-down development (where "lower" dimensions are integrated with respect to the "higher") and bottom-up development (where "higher" dimensions are integrated with respect to the "lower" ).
The former reflects the transcendent aspect of development (with which the Zeta 2 zeros are directly associated); the latter represents the corresponding immanent aspect (with which the Zeta 1 zeros are likewise directly associated).
The transcendent aspect, on the one hand, represents the attempt to achieve the "higher" integration of conscious development (which implies the unfolding of an increasingly important intuitive element).
This ultimately continues until the stage where conscious awareness is so transparent and dynamically interactive, that phenomena no longer appear to arise in experience. This seamless conscious activity can then be integrated with an ever-present pure intuitive awareness (of an unconscious kind).
So this represents the transcendent goal of spiritual development.
In practice severe limitations are likely to affect the attainment of this goal.
Higher level transcendent development is often based on a mistaken hierarchy, where the cognitive activity of reason is ranked as superior to that of affective sense, with pure spirit seen as the apex of the hierarchy.
This in the attempt to control the "lower" senses (in the pursuit of purer contemplative awareness), reason in effect is used to a degree to repress such activity (especially at a primitive unconscious level).
Therefore without this problem being properly addressed, the transcendent goal itself cannot be attained.
So there is a complementary immanent approach to development which inverts this hierarchy of reason as superior to the senses.
Whereas the transcendent aspect is based on the attempted "higher" integration of consciousness, the immanent aspect, in an inverse complementary manner, is based on the corresponding "lower" differentiation of the primitive unconscious.
Put another way, whereas the transcendent aspect represents making the conscious increasingly unconscious, so that all its linear type rigidity is ultimately eroded, the immanent aspect represents the making of the unconscious increasingly conscious in experience, so that all the primitive instincts of the - initially unrecognised - shadow are ultimately fully brought into the conscious light.
Thus once again, the integration of both the (analytic) conscious and the (holistic) unconscious in development, requires this approach from two opposite directions. Thus from the transcendent aspect, the conscious is made fully compatible with respect to the life of the unconscious; from the immanent perspective, in reverse manner, the unconscious is made fully compatible with respect to corresponding conscious activity.
So it is exactly similar with respect to the nature of the number system.
The Zeta 2 zeros reflect this former aspect of integration, where analytic appreciation of the number system (in a quantitative manner) ultimately is made fully compatible with corresponding holistic appreciation (of a qualitative nature). So we move from a linear (1-dimensional) notion of the quantitative nature of number to an increasingly circular (higher dimensional) notion of its qualitative nature.
And this higher dimensional appreciation is directly provided through the holistic interpretation of the Zeta 2 zeros.
The Zeta 1 zeros however reflect the latter aspect of integration where initial confused unconscious interpretation of the number system (in a qualitative manner) ultimately is made fully compatible with corresponding analytic appreciation (of a quantitative nature). So here we move from circular confusion to an increasingly linear (1-dimensional) appreciation i.e. where qualitative appreciation can be fully compatible with linear quantitative interpretation.
So in one case (Zeta 2 zeros) we move from the linear to the mature circular i.e. where quantitative is made fully compatible with the qualitative understanding of number; in the latter case (Zeta 1 zeros) we move from the (confused) circular to mature linear i.e. where in reverse manner the qualitative is made fully compatible with the quantitative nature of number.
This indeed is the very reason why all the Zeta 1 zeros are compelled to lie on a straight line!
These zeros in fact thereby represent the fully differentiated appreciation of the unconscious i.e. holistic nature of the natural number system.
So we have differentiated conscious interpretation in our customary quantitative interpretation of the natural numbers as represented by the real number line.
We then at the other extreme of understanding, we have differentiated unconscious interpretation in this new (unrecognised) appreciation of all the Zeta 1 zeros, as represented by the imaginary number line. And remember in holistic terms, the imaginary notion is used to express indirectly what is unconscious in origin!
Now initially when the primitive impulses of the shadow unconscious are projected into experience they are directly confused with phenomena of a quantitative nature. However, all going well, through a growing holistic qualitative awareness, inappropriate rigid attachment to such quantitative phenomena is gradually eroded. Success in this regard thereby enables one to confront the shadow in an ever-deeper manner with new (formerly unrecognised) unconscious elements continually brought to light.
In this way ultimately all such shadow elements can (in principle) be successfully differentiated.
So - quite literally - all primitive (i.e. prime) elements that have been successfully differentiated in an analytic manner, are thereby now equally successfully integrated in a corresponding holistic fashion.
Remarkably it is exactly similar in dynamic interactive terms with respect to the nature of prime numbers.
Thus the very differentiation of the prime numbers (in a quantitative analytic manner) is ultimately inseparable from their corresponding integration (in qualitative holistic terms).
In other words the Zeta 1 zeros represent the holistic integrative aspect with respect to the prime numbers.
Thus the very ability of the primes to maintain their unique status (as independent building blocks of the natural number system in cardinal terms) is inseparable from the corresponding unique status of the Zeta 1 zeros (which seamlessly maintain the overall interdependent nature of the primes with respect to the natural number system).
And this seamless integration is then demonstrated from a different perspective by the Zeta 2 zeros which now serve as the unique independent build blocks of the ordinal number system.
Thus ultimately the Zeta 1 and the Zeta 2 zeros are themselves totally interdependent.
However what is especially interesting is that the relationship of both sets of zeros is as cognitive to affective (and affective as to cognitive) respectively.
Thus if we initially fix the Zeta 2 zeros with customary cognitive aspect of understanding, then the Zeta 1 correspond with the affective aspect.
Now this will initially appear very strange as up to now, Mathematics - even in ts dynamic interactive operation - has been viewed as a cognitive (rational) type discipline.
So we will address this important issue further in the next blog entry.