We are accustomed to thing of Mathematics - especially in its treatment of number - in an absolute unambiguous manner (where the meaning of symbols remains fixed).

However in truth an unlimited number of relative type interpretations can potentially apply, with the standard conventional approach representing just one special limiting case.

Put another way the standard interpretation is of a 1-dimensional nature, whereby qualitative type considerations with respect to mathematical symbols are reduced in a merely quantitative manner (within a rigidly fixed framework).

However associated with every mathematical symbol is a unique qualitative manner of interpretation.

Thus Analytic i.e. Conventional Mathematics relates to the quantitative interpretation of mathematical symbols whereas Holistic Mathematics relates to their qualitative inetrpretation. And the integration of both approaches - in what I refer to - as Radial Mathematics, requires a dynamic interactive approach, whereby both quantitative (analytic) and qualitative (holistic) interpretations are inherently understood in a complementary manner.

So the huge unacknowledged problem is that just one special limited interpretation (where the qualitative meaning of symbols is reduced in a merely quantitative manner) has become synonymous with "all Mathematics", thereby completely blinding us to the infinite array of riches that would readily unfold through recognition of its hidden qualitative aspect as an equal partner.

So in psychological terms, the vitally important unconscious aspect of understanding has been completely blotted out in the misleading attempt to formally portray Mathematics as merely a rational (conscious) discipline.

However so far we have been looking at merely the "micro" aspect of number behaviour.

In other words we have started from the notion of an individual prime as a quantitative "building block" of each natural number to discover its hidden qualitative meaning as a number group whose individual ordinal members are uniquely defined in a natural number fashion.

So the inherent paradox of the nature of each prime is thereby clearly revealed. Thus from the quantitative perspective, the primes appear to unambiguously generate the natural numbers (in a cardinal manner); however from the (hidden) qualitative perspective each individual prime already is defined by an unbroken sequence of natural numbers (in an ordinal manner).

Thus from a dynamic interactive perspective - entailing both cardinal and ordinal aspects - it becomes clear that both the primes and natural numbers are ultimately co-determined in a synchronistic manner (which is ultimately ineffable).

However this realisation can have no meaning within the conventional mathematical perspective that misleadingly insists on viewing the nature of number in a merely quantitative manner!

However as well as the "micro" aspect of each individual prime, we likewise have the "macro" aspect of the collective behaviour of the primes with respect to overall natural number system.

And of course both individual "micro" and collective "macro" aspects of prime (and natural number) behaviour are themselves complementary in a dynamic interactive manner.

So strictly speaking we neither start with each individual prime (as somehow pre-defined) or the entire natural number system (as likewise somehow pre-defined).

Rather they both arise through mysterious dynamic interaction patterns that then subsequently enable their separate identities to be abstracted in a fixed manner.

In now looking at the "macro" behaviour of the number system, we make direct contact with Riemann's zeta function and the famed (non-trivial) zeros.

However, it was intense investigation over many decades relating to the less investigated ordinal nature of number (briefly outlined in the 10 previous blog entries) that eventually provided me with the appropriate framework to understand the true nature of these Riemann zeros.

However the starting point goes back to a classroom revelation when attending primary school in Ireland, regarding an unacknowledged problem regarding the nature of multiplication.

As we know in quantitative terms all natural numbers can be uniquely expressed as the product of primes.

So if we take the number "6" to illustrate this is uniquely expressed as 2 * 3.

Now the number "6" is represented as a point on the number line. In other words 6 is represented in a 1-dimensional manner as 6

^{1}.

However if we represent 2 and 3 here in geometrical fashion - say as two sides of a rectangular tables (measured) in metres - then the area represented by 2 * 3 is given in square (i.e. 2-dimensional) terms. In other words, though each side relates to a measurement in 1-dimensional, the resulting area relates to a corresponding measurement in 2-dimensional units.

However with respect to the standard treatment of multiplication, this qualitative transformation in the nature of units is simply edited completely out of consideration.

Thus the result of 2 * 3 (and by extension every factor combination of primes) is given in a merely reduced (i.e. 1-dimensional) quantitative manner.

However though this early insight still remained fully valid, it was only at a later date that I realised an additional - perhaps even more fundamental - problem with the conventional nature of multiplication.

When we speak of the area of a table (as for example in my illustration) we are still operating at the analytic level of understanding dimensional numbers.

However as we have seen, all such numbers have a corresponding holistic interpretation and this too is intimately involved with the very nature of multiplication.

If for these purposes we have two rows with 3 coins in each row, each row would represent in analytic terms a dimension (e.g.. length) and each column (of 2 coins) another dimension (e.g. height).

Now without specific reference to rows or columns, we could attempt to treat each coin in an independent manner and obtain the result by adding up each of the (independent) units.

However the key point about multiplication that in order to use the operator of 2 we must recognise the mutual interdependence of the two rows (with 3 coins).

Thus crucially whereas with addition, we can proceed by recognising the independence of each individual item, for multiplication we must also recognise the mutual interdependence of rows and columns.

So when we recognise the two rows as interdependent we use 2 as operator to obtain 2 * 3.

Equally when we recognise the three columns as interdependent we use 3 as operator to obtain 3 * 2.

So conventional multiplication attempts to represent mutiplication - very misleadingly - as a short-hand form of addition.

So with respect to addition when we add the two rows we get 3 + 3 = 6. And is each row is defined in a 1-dimensional manner the result through addition is likewise 1-dimensional in nature.

Thus 3

^{1}+ 3

^{1}= 6

^{1}.

Conventional multiplication then attempts to "speed up" this process. Because 3 is repeated 2 times with respect to addition, we now multiply 3 two times i.e. 2 * 3 to apparently get the same result.

Now this "speeding up"is of course not so evident when the operator is 2. However imagine if 3 was to be added to 3 one hundred times, then the expression of this through multiplication as 100 * 3 would indeed be much more efficient.

However the crucial unrecognised problem is that the very switch from addition to multiplication requires the recognition of the rows and columns (representing the two dimensions) not only as each containing independent items but also that these dimensions themselves as mutually interdependent (and thereby freely interchangeable with each other).

And this latter recognition requires the qualitative (holistic) interpretation of the nature of a dimension.

So I was already well primed - to excuse the pun - to see that there was a distinct qualitative aspect to the nature of multiplication, which was completely unrecognised in conventional mathematical terms. And I was already confident that - when appropriately understood - the Riemann (Zeta 1) zeros directly related to this hidden qualitative aspect of the cardinal number system.

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