As I have repeatedly stated, proper comprehension of the Riemann Hypothesis has the most far reaching consequences possible for the true nature of Mathematics.
In fact, to put it bluntly, what we know as Mathematics is built on a massive lie!
In other words, though there are two equally important aspects to all mathematical understanding that are quantitative and qualitative with respect to each other, Conventional Mathematics is built on the reductionist illusion that only one of these i.e. the quantitative is relevant.
Indeed the modern development of Mathematics can be likened to a gross form of propaganda where at every turn reference to the qualitative has been expunged so as to leave conventional wisdom unchallenged.
Imagine presenting the history of a country comprising two proud races of equal importance in terms of the contribution of just one! Worse still, imagine that great pains have been taken to avoid ever making reference to the existence of this second race. We would perhaps see such propaganda as indeed very distorted!
The same charge can be made against Conventional Mathematics. Unfortunately we have now been told the same propaganda for so long that we accept it utterly without question as the total truth.
I am writing this blog entry to proclaim "The Emperor Has No Clothes".
Once again the notion of the quantitative is built on the fallacy of numbers possessing an objective independent existence (in absolute terms).
This notion is enshrined in its cardinal use. So if we refer to a group of objects - say 3 cars - we are referring to them in a quantitative manner (i.e. as independently existing).
However if I now refer to an object in ordinal terms - say the 3rd object - this has no meaning in itself but must be given a wider general context with respect to the related group of objects.
So the ranking of this object depends on a more general context which refers to a qualitative - as opposed to quantitative - distinction.
Now amazingly you will never find it mentioned in a mathematical textbook that with ordinal rankings, we have now shifted to the qualitative notion of number. This of course would immediately raise serious questions regarding the accepted - merely quantitative - notion!
So to avoid this conflict abstract mathematical terminology has been skilfully developed so as to preserve the "quantitative illusion".
Last night I looked up the Oxford English Dictionary to quickly find "rank" given as one of the definitions of "qualitative". However again you will not see this mentioned in mathematical textbooks!
Therefore the first thing to clearly grasp - which is perhaps the most important of all - is that the cardinal and ordinal aspects refer to the quantitative and qualitative aspects of number respectively.
Both of these aspects continually interact in experience. We cannot apply the cardinal aspect to number (within an implied ordinal aspect). We cannot in turn apply the ordinal aspect to number (without an implied cardinal aspect).
Thus the notion of number is properly of a dynamic interactive nature with aspects that are - relatively - independent and interdependent with respect to each other.
The next key issue to grasp is that the ordinal i.e. qualitative nature of number (when properly recognised) is based on an entirely distinct logical system to that of quantitative appreciation.
Conventional Mathematics is defined by its merely 1-dimensional nature (in qualitative terms). What this means is that the interpretation of relationships in any relevant context is based on just one (independent) polar reference frame. So typically for example, mathematical objects are viewed from an external reference frame (thus avoiding interaction with a corresponding internal aspect); likewise mathematical objects - as we have seen - are viewed from a merely quantitative framework (thereby avoiding qualitative interaction)
However once we accept the inherent relative nature of mathematical relationships i.e. where opposite polarities dynamically interact in two-way fashion in experience, we move to higher dimensional interpretation.
So just as 2, 3, 4, 5,.. have a quantitative meaning in mathematical terms (where in qualitative terms interpretation remains 1-dimensional), likewise 2, 3, 4, 5 have a qualitative meaning (where interpretation now takes place in accordance with such dimensions).
Each higher dimension from a qualitative perspective represents a distinctive manner of configuring the opposite polarities of experience that is inversely related to its corresponding roots of 1).
Thus to give meaning to ordinal number distinctions (in quantitative terms) we must use these higher dimensions.
For example to give meaning to 1st and 2nd (in the context of a group of 2) requires 2-dimensional interpretation (inversely related to the 2 roots of 1).
More generally to give meaning to 1st, 2nd, 3rd ....nth (in the context of a group of n) requires n-dimensional interpretation (inversely related to n roots of 1).
The importance of the primes in this context is that each prime number is associated - apart from the common root of 1 - with a unique set of non-trivial roots.
In this sense associated with each prime is a unique natural number arrangement in ordinal terms.
These correspond with - what I refer to as - the Zeta 2 (non-trivial) zeros.
It is vital to grasp that the proper nature of these zeros is of a dynamic holistic nature (where both the quantitative aspect of independence and the qualitative aspect of interdependence are perfectly reconciled).
For example the simplest prime number 2 is associated with 1st and 2nd members in ordinal terms.
Now each of these members can be given an indirect independent quantitative expression (on the circle of unit radius) as + 1 and – 1 respectively. Then the qualitative interdependence of both members is indicated through their sum = 0.
Thus in this way a total harmony is established as between quantitative and qualitative aspects (for this prime number 2).
Therefore, once again the holistic significance of the Zeta 2 zeros resides in the fact that for each prime, a unique circle of relative independence and interdependence exists (with respect to its natural number members). So quantitative and qualitative aspects are perfectly harmonised in this manner for ordinal number members with respect to each prime.
Now whereas the Zeta 2 operate on the micro scale - as it were - with respect to the internal composition of each prime (in terms of natural number members in ordinal terms) the Zeta 1 zeros operate in reverse on the macro scale with respect to the external composition of the natural numbers, (in terms of prime constituents in cardinal terms).
In other words the Zeta 1 (non-trivial) zeros translate, as it were, the cardinal nature of the relationship of the primes to the natural numbers i.e. where each natural number can be expressed uniquely in terms of prime factors, indirectly in an ordinal qualitative manner through a corresponding unique set of numbers.
Once again the proper nature of these zeros is dynamic and holistic. In other words through each zero (as independent) indirectly represents a point of pure qualitative interdependence (as a numerical energy state) the combined set of all these zeros represents the locally independent quantitative nature of the primes (in opposition to the common shared relationship of primes and natural numbers).
In this way a perfect harmony is preserved as between both the quantitative (independent) and qualitative (interdependent) aspects of the overall number system (through the relationship of the primes to the natural numbers).
Ultimately of course the Zeta 1 and Zeta 2 zeros are identical in an ineffable manner.
Therefore in Zeta 2 terms, the unique holistic nature of (ordinal) natural numbers to primes internally, where quantitative and qualitative aspects are fully reconciled for each number, is inseparable in Zeta 1 terms from the unique holistic nature of (cardinal) primes to the natural numbers, where quantitative and qualitative aspects are reconciled externally for the number system as a whole.
However, as the zeta zeros (Zeta 1 and Zeta 2) are intrinsically of a dynamic holistic nature, indicating both internally and externally within the number system how quantitative (cardinal) and qualitative (ordinal) aspects are ultimately fully reconciled, it is pointless trying to understand their role in a merely quantitative manner.
Indeed put simply the Zeta 1 and Zeta 2 zeros point (from two complementary perspectives) directly to the unrecognised qualitative aspect of the number system.
Once again just as in quantitative terms, the Riemann Zeta Function remains uniquely undefined in where s (as dimensional number) = 1, likewise, the Riemann Zeta Function remains uniquely undefined in qualitative terms where s (as dimensional number) = 1.
I cannot stress how important this is! What it means is that the Riemann Zeta Function (and Riemann Hypothesis) remain uniquely undefined, when we attempt to understand number in a merely absolute quantitative manner!
All other values for s (≠ 1), refer to dynamic relative interpretations (where number has both quantitative and qualitative aspects)!
The implications could not be more fundamental. Our present understanding of number (and by extension all mathematical notions) is based on a reductionist sham i.e. that quantitative meaning can be given independent of a general context that is necessarily qualitative. So in truth both dynamically interact in all meaning!
Not only Mathematics, but indeed all the Sciences are now deeply contaminated with the same fundamental falsehood.
We need to start facing up to this critical issue immediately. A successful future for our civilisation will ultimately depend on it!