In yesterday's blog entry I explained how the very making of the simplest ordinal distinctions at a rational conscious level, implicitly requires unconscious recognition of an intuitive kind.
Equally what this entails, is that a dynamic interactive process entailing both the internal and external aspects of experience (and quantitative and qualitative) is necessary in the making of such ordinal distinctions.
However though our actual experience of number is thereby dynamically interactive in this sense (entailing conscious and unconscious), from a conventional mathematical perspective, a reduced - and thereby distorted - interpretation of the process is given in a merely conscious rational fashion!
When one appreciates this clearly, then nothing but a complete overhaul in the manner in which we interpret number (and by extension all mathematical relationships) is thereby required.
And from the perspective of this required new interpretation, the very nature of the zeta zeros (Zeta 1 and Zeta 2) becomes very apparent.
We are already moving in the direction of such clarification.
You may recall that the qualitative nature of the meaning of 2nd (in the context of 2) arose as the solution to the simplest version of the Zeta 2 (finite) equation
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime) = 0, where t = 2.
i.e. 1 + s1 = 0 so that s1 = – 1.
Now – 1 has indeed an indirect quantitative meaning that geometrically is represented as a point on the circle of unit radius.
However its true 2-dimensional significance arises from the fact that this now represents a dynamic interaction as between (complementary) quantitative and qualitative aspects. And once again the qualitative aspect entails the dynamic negation of conscious recognition (according to one isolated pole of reference). And this is the very means through intuitive (unconscious) recognition is generated.
Now once again it is vital to appreciate that this cannot be interpreted within the conventional mathematical framework (which formally is 1-dimensional in nature).
So + 1 as the one root of 1 here has a merely quantitative meaning (in conscious rational terms.
Thus the quantitative number as solution for the (one) root of 1 is identical with the dimensional number (which is also 1).
Thus the number 1 as dimension is utterly unique in this sense as it is the only number where the qualitative notion of dimension is identical with the quantitative meaning of its root.
And once again this is the precise reason why Conventional Mathematics is defined merely in reduced quantitative (i.e. 1-dimensional) terms.
However with every other dimensional number ≠ 1 its corresponding (non-trivial) roots are distinct from the dimensional number. The crucial importance of this is that ordinal interpretation now entails a dynamic interaction as between quantitative and qualitative aspects.
So what is explicitly interpreted in quantitative terms (in an ordinal relationship) equally entails qualitative type considerations (of an unconscious) nature .
Now these dynamics - though necessary for recognition - remain merely implicit at the customary level of conventional mathematical experience.
So ordinal recognition of numbers is thereby misleadingly interpreted in a reduced quantitative manner.
Thus what I am attempting to do here is to make such implicit dynamics now explicit in a more integrated coherent interpretation of number.
Once again the simplest case of ordinal recognition relates to the notion of 2 as dimension (i.e. where t = 2 in the Zeta 2 equation).
Now the next case of ordinal recognition i.e. where 2nd and 3rd are given meaning (in the context of 3) arises from the (finite) Zeta 2 equation where t = 3.
Now on the face of it this seems much more difficult to interpret as we need to give the two non-trivial roots (of the 3 roots of 1) both a quantitative and qualitative interpretation.
However as I shall show later, fortunately for our purposes, all the other prime numbered (non-trivial) roots can be explained as more refined variations on the simplest case of 2.
Now you might already be wondering about the (non-trivial) roots of equations where t is not a prime number.
For example what about the important case where t = 4?
Well all I will say at this stage is that such cases entail interaction with the Zeta 1 equation so that ultimately both Zeta 1 and Zeta 2 are themselves seen as totally interdependent.
However, having said this it is initially valid to consider Zeta 1 and Zeta 2 as relatively independent of each other.
Thus the important conclusion that we can reach is that the Zeta 2 zeros in fact represent the holistic (unconscious) basis of our ordinal recognition of numbers (at the conscious rational level).
Now we are accustomed once again to interpreting this process (of ordinal recognition) in a merely quantitative manner.
However in truth - as we have seen - experiential recognition entails a dynamic interaction of both quantitative and qualitative aspects.
So once again the Zeta 2 zeros provide the indispensable (unconscious) holistic qualitative basis of ordinal number recognition that is interpreted in an analytic quantitative manner at a rational (conscious) level.
Thus properly understood the Zeta 2 zeros thereby are as important as the ordinal numbers. In fact they represent an intrinsic component of the recognition of such numbers!
Once again I draw your attention to the key fact that the one value for which the Riemann Zeta Function is undefined is where s (the dimensional value of the Zeta 1 Function) = 1.
Now the significance of this can only be properly explained in a dynamic interactive manner.
In other words, for all values of s ( ≠ 1) a dynamic interaction is entailed as between the quantitative (analytic) and qualitative (holistic) aspects of the number system.
When properly understood the Riemann Zeta Function provides an ingenious way of mapping quantitative to qualitative (and qualitative to quantitative) interpretations respectively.
The Riemann Hypothesis then expresses the condition for the mutual identity of both meanings (which is vitally necessary to ensure consistency with respect to all subsequent numerical operations).
Once again the Riemann Zeta Function remains undefined (where s = 1).
Putting it bluntly, this simply means that not alone can the Riemann Hypothesis neither be proved (nor disproved) in conventional mathematical terms, but much more importantly it cannot be successfully interpreted in this manner.
Thus the real message of the Riemann Hypothesis is that we are now in need of a radical reformulation of the very nature of Mathematics.