It is important to keep restating my central purpose on these blogs.
Once again I am demonstrating how mathematical relationships are inherently of a dynamic interactive nature entailing complementary opposite polarities.
In actual experience the key polarities then relate firstly to external (objective) and internal (subjective). We cannot for example have knowledge of a mathematical entity such as a number in an (objective) external manner without the corresponding mental perception of this number which - relatively - is of a (subjective) internal nature. So we cannot therefore in Mathematics have "objective" truth in the absence of corresponding "mental" interpretation (both of which dynamically interact in experience).
Therefore the standard mathematical approach of attempting to treat mathematical objects in an an absolute abstract manner is ultimately untenable. Of course I will readily admit that, very much in the manner of Newtonian Physics, such an assumption can prove extremely valuable in approximating truth in a partial manner. In other words the notion of objective validity in an absolute type manner represents one very useful - though ultimately limiting - interpretation of mathematical truth.
Secondly, all experience (including of course mathematical) necessarily involves the dynamic interaction of the two fundamental poles of whole and part. This can be expressed alternatively as the interaction of quantitative and qualitative, individual and collective and perhaps most crucially of notions of independence and interdependence respectively.
Now, as it stands, the accepted mathematical enterprise is but of a very reduced nature where, in every context, qualitative notions of interdependence are reduced in an independent merely quantitative manner.
Admittedly huge progress has been made, though again necessarily in a limited manner, through its reduced quantitative assumptions.
However, it is doomed ultimately to failure with respect to understanding the key relationship as between the primes and the natural numbers.
So it will take an enormous paradigm shift, such as never has occurred before within Mathematics to start dealing with this fundamental issue in an appropriate manner.
In fact as I have been repeatedly stating in these blog entries it requires an inherently dynamic interactive approach based on complementary poles of reference to properly understand the two-way relationship as between the primes and natural numbers.
This therefore requires both an analytic (quantitative) and holistic (qualitative) method of interpreting all mathematical variables. And it has to be said that given the increasingly abstract nature of mathematical developments that the holistic manner of understanding - which in truth is equally important with the analytic - has been all but eliminated in formal terms from Mathematics.
The current obsession to prove the Riemann Hypothesis reflects very much the highly reduced quantitative bias of Mathematics. Here an attempt is made to treat both primes and natural numbers as independent entities in a merely quantitative cardinal manner.
However both the primes and natural numbers equally have an ordinal - as well as cardinal - identity. And whereas the cardinal identity directly focuses on such numbers as independent entities, the ordinal aspect relates by contrast directly to the qualitative notion of number as interdependent (where meaning is necessarily derived from their relationship with other numbers).
So properly understand the relationship of the primes with the natural numbers entails the two-way dynamic interaction within the number system of both quantitative notions of individual numbers as independent and qualitative notions of the collective interdependence of all numbers.
I have been illustrating in the past few blog entries this two-way relationship from yet another perspective.
Staring with the Type 1 perspective which focuses directly on the quantitative nature of the number system, I have been at pains to show its hidden ordinal aspect.
So we start with the well known set of cardinal primes (which I refer to as Order 1 Primes).
However these implicitly contain a natural number basis (in ordinal terms) i.e. so that they can be ranked in order as 1st, 2nd, 3rd, 4th and so on.
By then switching to the prime rankings within this natural number ordinal ordinal set, we are then able to identify a new subset of primes in a cardinal manner (which I identify as Order 2 Primes).
And we can continue on interactively in this manner switching as between primes and natural numbers in cardinal and ordinal manner to identify an unlimited number of subsets of Higher Order Primes.
And I demonstrated how the Prime Number Theorem would then apply to each of these subsets.
So we have the strange paradox here that what we consider as cardinal primes implicitly can also be viewed as ordinal natural numbers. Thus the relationship as between the - ultimately unlimited - subsets of cardinal primes throughout the number system, can equally be viewed as the relationship between corresponding subsets of ordinal natural numbers!
We then viewed from the Type 2 perspective the much less recognised relationship as between the ratio of natural to prime factors per number (up to n).
Now in Type 1 terms the Prime Number Theorem relates to a quantitative notion of frequency i.e. the number of cardinal primes up to n, on the natural number scale.
However in Type 2 terms the Prime Number Theorem relates to a qualitative notion of frequency, in that we are comparing prime with natural number factors (which reflects an ordinal notion).
This is a crucially important point. However because of the great neglect of ordinal type notions - which are mistakenly assumed to be derived from cardinal - within Conventional Mathematics, the appropriate mathematical language does not even exist to discuss this issue coherently.
We have in fact two distinct notions of the ordinal. The first is within a prime group of numbers. So 3 is a cardinal prime; however the group of 3 necessarily contains 1st, 2nd and 3rd members in natural number ordinal terms. And the notion of addition connects these 3 members so that the group of 3 (in cardinal terms) = 1st + 2nd + 3rd members in an ordinal manner. So again, we see that the explicit notion of a cardinal prime, implies entails natural number notions (in an ordinal manner).
However the 2nd - largely unrecognised - notion of ordinal relates to the unique combinations of prime numbers, through which all composite natural numbers are derived.
So when I express 6 (as a cardinal natural number) as the product of 2 * 3, I am strictly using the prime numbers 2 and 3 in an ordinal fashion. In other words a unique interdependence is established as between these two primes resulting in the composite number 6, which we then view as an independent cardinal number. And the notion of multiplication ordinally connects, in this case, the relationship between the primes.
So 6 in cardinal natural number terms = 2 * 3 (from an ordinal prime perspective).
Thus, when we look appropriately at the matter, a double paradox in revealed with respect to the number system.
From the Type 1 perspective, each (explicit) cardinal prime implicitly entails natural numbers in an ordinal fashion.
Then, from the complementary Type 2 perspective, each (explicit) cardinal natural number, implicitly entails primes in an ordinal fashion.
When one appreciates this two-way interaction appropriately, it then becomes obvious that the primes and natural numbers are ultimately totally interdependent with each other (in both cardinal and ordinal fashion).
Indeed it is only the phenomenal tendency to attempt to view relationships within isolated reference frames - especially pronounced within Mathematics - that creates the illusion of a causal relationship between them.
So Conventional Mathematics is still firmly stuck in this totally one-sided notion of the primes as the (cardinal) building blocks of the (cardinal) natural number system.
However when we properly include both Type 1 (cardinal) and Type 2 (ordinal) perspectives, which are quantitative and qualitative with respect to each other, the number system is understood in a dynamic interactive manner, with prime and natural number notions, and cardinal and ordinal notions, ultimately fully interdependent with each other in an a priori ineffable manner.
Of course, as I have been saying from the very beginning of these blog entries, this means that the attempt to prove or disprove the Riemann Hypothesis is strictly futile.
Quite simply, The Riemann Hypothesis by its very nature transcends the limits of the conventional mathematical approach. It points in fact to the ultimate condition necessary for the the consistent reconciliation of both cardinal (quantitative) and ordinal (qualitative) notions within the number system. And clearly this cannot be achieved within a paradigm that does not (formally) recognise a distinct role for the qualitative. So the a priori consistency (as between quantitative and qualitative notions) to which the Riemann Hypothesis applies, is already necessarily assumed in the very use of the conventional mathematical axioms!
However there is a much bigger issue to be faced here here than the role of the Riemann Hypothesis important as it admittedly is! This is that the very paradigm on which Conventional Mathematics is built is crucially flawed .
Rather than representing all valid mathematical inquiry, it represents just a small - though admittedly very important - special case.
We are now in need - not alone of properly understanding the number system - but indeed all mathematical relationships - of an unparalleled paradigm shift to a truly dynamic interactive manner of appreciating mathematical relationships, that entails both quantitative (analytic) and qualitative (holistic) aspects of appreciation in equal balance.
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