There is a very important consequence arising from yesterday’s entry!
As I have stated, considerable reductionism exists in conventional Mathematics with respect to ordinal notions (which basically are identified on cardinal lines).
So the natural numbers in cardinal terms are readily identified with the corresponding members from an ordinal perspective;
So 1, 2, 3, 4 etc. are readily ranked as 1st, 2nd, 3rd, 4th and so on.
However the true original relationship relates to that between the primes and natural numbers (and natural numbers and primes) which is quantitative as to qualitative (and qualitative as to quantitative etc.)
Now once again in the expression a^b, a can be defined as the base number and b as the dimensional number (which are quantitative as to qualitative in relation to each other).
However when we take each in isolation, they appear solely as quantitative numbers.
So therefore with respect to the base number, the relationship as between cardinal primes (and ordinal natural numbers) ranks 2 (as 1st), 3 (as 2nd), 5 (as 3rd), 7 (as 4th) etc.
Likewise with respect to dimensional numbers (as powers or exponents) the relationship as between cardinal primes (and ordinal natural numbers) ranks again 2 (as 1st), 3 (as 2nd), 5 (as 3rd), 7 (as 4th) etc.
However in dynamic relationship the each other what seems unambiguous (as within isolated frames) becomes deeply paradoxical.
The upshot of this is that the consistency of cardinal and ordinal relationships does not derive from either the primes or natural numbers in isolation, but rather from their simultaneous relationship (strictly in a phenomenal context, the dynamic interaction between both approaching simultaneity).
This highlights the significance of the non-trivial zeros as the basic requirement for any consistent ordering of numbers (in either cardinal or ordinal terms).
Again as the Riemann Hypothesis relates to a particular statement regarding the nature of the non-trivial zeros, it is futile attempting to either prove or disprove such a statement in conventional mathematical terms.
This again is due to the fact that the basic condition enabling such axioms to be used in a consistent manner itself depends on the prior nature of the non-trivial zeros.
So we cannot meaningfully approach the validity of a statement from axioms which already depend on this statement!
As I stated yesterday the importance of the non-trivial zeros can hardly be overstated as they once again serve as the basic requirement for ensuring consistency with respect to both the quantitative and qualitative interpretation of mathematical symbols in all phenomenal circumstances (with both aspects ultimately identical in an ineffable manner).
However we now have another surprising result!
The non-trivial zeros derive from the conventional (Type 1) quantitative aspect of approach to Mathematics.
So far we have been at pains to indicate that they cannot however be properly interpreted without reference to a complementary Type 2 aspect!
However when we follow this Type 2 aspect to its own logical conclusion, it leads to the generation of an alternative set of “non-trivial zeros”.
As we have seen, both Type 1 and Type 2 systems appear as the direct inverse of each other. So the natural number as base quantity in Type 1 represents the dimensional number in Type 2; likewise the dimensional number in Type 1 represents the base quantity in Type 2.
So again for example, 5^1 in Type 1, is represented as 1^5 in the Type 2 system!
Now the Riemann Zeta Function (which I refer to as Zeta 1) is defined with respect to the Type 1 system involving a sum of natural number terms defined with respect to a fixed complex number s (as dimension).
Then we set the infinite series of terms for ζ(s) = 0 to derive the solutions for s (with both trivial and non-trivial solutions).
So for convenience this equation ζ(s) = 0
can be written,
1^(– s) + 2(– s) + 3(– s) + 4^(– s) + …… = 0.
However we can derive a fascinating alternative Type 2 formulation - which I refer to as Zeta 2 - where the base natural number quantities now appear as dimensional numbers and the dimensional number s as base quantities.
So we start with 1 = s^n i.e. 1 – s^n = 0 in order to find all solutions for s!
As one of these i.e. 1 = s, is not unique we divide 1 – s^n by 1 – s to get
1 + s + s^2 + s^ 3 + …. + s^(n – 1) = 0.
Multiplying by s (giving s = 0 as a solution) we get,
s^1 + s^2 + s^ 3 + …. + s^n = 0;
This is Zeta 2.
The natural numbers as base quantities in Zeta 1 are now number dimensions in Zeta 2.
Also the number dimensions in Zeta 1 are now base quantities in Zeta 2 (with – s replaced by s).
Also whereas Zeta 1 is an infinite series, Zeta 2 is finite though the number of terms n can be extended indefinitely.
So the solutions for s (which represent the n unique roots of 1) represent a second set of zeros (i.e. Zeta 2 zeros).
Interestingly however, whereas the solutions for s (in Zeta 1) represent complex numbers with an imaginary part that is transcendental, in the case of Zeta 2, the solutions for s represent complex numbers that are algebraic irrational! (Indeed it would seem that this may well be an important clue explaining the differing nature of ζ(s) for (positive) even and odd integer values of s respectively!
Properly understood, the non-trivial zeros (for Zeta 1) cannot be understood in isolation from the corresponding Zeta 2 solutions (and vice versa).
One fruitful way of understanding their respective role is as follows!
Whereas the Type 1 show us how an incredibly dynamic set of numbers is already inherent in the number system (that conventionally appears static and absolute), the Type 2 show us how understanding of mathematical symbols, that starts in (1-dimensional) static terms, can be progressively raised to higher dimensional interpretation (which likewise becomes incredibly dynamic in nature).
So the understanding of the number system itself as dynamic and interactive in nature, properly requires a method of understanding that is also dynamic and interactive relating both quantitative and qualitative aspects.
So the ability to properly see the non-trivial zeros as already immanent or inherent in the conventional number system, requires the corresponding ability to fully transcend conventional linear interpretation (of this system). And the Zeta 2 zeros relate directly to the precise qualitative nature of all these higher dimensional interpretations (which are potentially unlimited) which then serve as the dynamic means for appropriate quantitative interpretation of the nature of the (recognised) Zeta 1 zeros!
One further interesting point!
The designation of the Zeta zeros in Type 1 for s = – 2, – 4, – 6, … etc. as trivial represents an unfortunate misnomer due to lack of appreciation of the role of the Zeta 2 Function.
In fact the true explanation of the quantitative nature of the “trivial” zeros in Type 1 springs directly from the qualitative appreciation of the “non-trivial” zeros in Type 2.
Likewise appreciation of the qualitative nature of the “trivial” zero in Type 2 (where s = 1) springs directly from appreciation of the quantitative nature of the “non-trivial” zeros in Type 1!