I now will now address the issue of why the Riemann zeros - apparently - all lie on the imaginary line (drawn through .5 on the real axis) with each pair of zeros having the complementary form of a + it and a - it respectively.
As we have seen the conventional number line is based on the quantitative notion of number, where component units are viewed in an independent manner.
So once more, illustrating with respect to the natural numbers, when we refer to "3" for example, implicit in interpretation is the view that 3 = 1 + 1 + 1 (where the homogeneous units are viewed as independent of each other).
However a huge unaddressed problem exists with respect to conventional interpretation in that number can equally be used in an alternative fashion (where component units are now strictly interdependent with each other).
Thus we could equally say for example that a number has 3 factors. In this context, as dealt with yesterday, the (composite) number 6 contains 3 proper factors i.e. 1, 2, and 3.
However the very nature of these factors is that they are all defined with reference to their common relationship to 6!
Therefore, we are using 3 (with respect to the three factors) in a different sense!
So when we maintain here that 3 = 1 + 1 + 1 (i.e. as the sum of its component units) these units must now properly be understood as interdependent with each other.
Thus we have illustrated the remarkably simple, yet entirely overlooked fact, that properly understood, in a dynamic interactive context - which is the only appropriate coherent way to view this matter - that number keeps switching from its quantitative aspect (where units are relatively independent) to its corresponding qualitative aspect (where units are interdependent) and vice versa.
Alternatively - to borrow from the language of quantum mechanics - number keeps switching as between its particle aspect (as independent) and wave aspect (as interdependent) and alternately also in reverse manner as between wave and particle aspects.
So if we are to represent the natural numbers as lying on a number line, strictly speaking two distinct lines are required 1) for the quantitative notion of number (where individual units are relatively independent of each other) and 2) for the corresponding qualitative notion of number (where the units share a common interdependence with each other).
This then raises the enormous issue of consistency with respect to both uses.
How, from one perspective, can we be confident that the qualitative (wave) use of number is consistent with the corresponding quantitative (particle) use?
Then, how from the reverse perspective, how can we be confident that the quantitative (particle) use of number is consistent with the corresponding qualitative (wave) use?
And we must keep remembering that in the actual dynamics of number interaction, the quantitative has a likewise (hidden) qualitative and the qualitative a likewise (hidden) quantitative aspect respectively!
Now, when we start from the conventional perspective of the natural number system i.e. as viewed in a quantitative manner, strictly speaking the notion of number, as used with respect to the factors of composites, relates to a different number line (where the unit factors are viewed as interdependent with each other).
So remarkably, we now have the appropriate context for viewing the Riemann zeros i.e. as the (hidden) holistic counterpart of the accepted analytically interpreted natural number system.
And the condition for consistency of the two aspects of the number system (analytic and holistic respectively) is that the Riemann zeros equally lie on a number line.
Now of course this is implied by the Riemann Hypothesis, with the requirement that the imaginary line on which all the zeros are postulated to lie is drawn through .5 (on the real axis).
However there is no way that this can be proven using conventional mathematical axioms, as due to the reduced interpretation implied, it is already assumed that the qualitative use of symbols is absolutely identical with the quantitative!
One might query why the zeros should lie on an imaginary line!
And this is where the holistic interpretation of the imaginary notion is of invaluable assistance.
In holistic terms, to posit is to make conscious and to negate is to make unconscious.
In dynamic terms, unconscious negation already implies that one has already posited an object in a conscious manner.
So just as in physics when an anti-matter particle interacts with its matter equivalent, energy is produced, likewise in psycho spiritual terms, when unconscious negation (of what has been already posited) takes place, intuitive energy is generated.
Now we can accurately refer to this - indirectly expressed through the two roots of 1 - as 2-dimensional experience with both positive conscious and negative unconscious directions.
The imaginary notion in analytic terms then entails obtaining the square root of the negative unit.
In corresponding holistic terms this implies the attempt to express the inherently intuitive understanding (where the identity of complementary opposites is directly apprehended) indirectly in an objective rational manner.
So the key point - as I have been demonstrating in my last few entries - is that the holistic appreciation of the zeros implies a high degree of refined intuitive appreciation (where one simultaneously can literally see from two complementary opposite frameworks). In a much more accessible manner, the appreciation of the paradoxical nature of L and R turns at a crossroads implies similar intuitive insight.
So the requirement that all the zeros lie on an imaginary line really points to the fact that these points represent a circular type of paradoxical understanding (when expressed in an indirect rational manner).
Once again we cannot prove that the zeros lie on an imaginary line! What we can say however is that for consistency to be maintained as between the quantitative and qualitative interpretation of mathematical symbols, we must assume that they all lie on an imaginary line (which is the expression of such consistency).
So each Riemann zero represents a point on the imaginary number line, where the quantitative and qualitative aspects of number interpretation are identical. Expressed more accurately in a dynamic interactive manner, each zero holistically expresses a point where both the quantitative and qualitative aspects of number interpretation approach perfect identity!
So the truth of the Riemann Hypothesis, in this important sense, thereby requires an initial massive act of faith in the subsequent consistency of the whole mathematical enterprise.