Tuesday, May 1, 2012

Nature of Number System (2)

It is customary to view primes as the basic building blocks of the number system; however this merely reflects the quantitative bias that is inherent in the conventional mathematical approach to number.

So from this approach numbers are viewed essentially as quantities (without qualitative distinction applying to individual members). And the natural numbers are then defined in terms of the product of unique combinations of prime numbers.

However the very notion of a prime number strictly has no meaning in the absence of its individual members.

So for example when I identify 3 as a prime number this implies a grouping with a 1st 2nd and 3rd member.

This ranking of members in an ordinal fashion relates directly to the qualitative rather than the quantitative notion of number.
Also it quickly becomes apparent that no absolute distinction can apply to such ordinal rankings which have merely a relative meaning depending on the number grouping in question.

So the 2nd of 3 objects in a group implies a different meaning than the 2nd of 5. Thus the very notion of a specific number (in this ordinal sense) keeps varying depending on context.

And if the meaning of what is ordinal (in terms of qualitative distinction) is merely relative, then this implies that the cardinal notion (in terms of quantitative distinction) itself is likewise relative.

Thus the very precondition for relating the cardinal and ordinal features of number i.e. the quantitative and qualitative is that both aspects are understood in relative terms.

Now the standard quantitative notion of a prime number is necessarily based on a reduced interpretation of its qualitative nature.

For as soon as we even admit to the very notion of individual members of the prime group, we must necessarily use the natural numbers in an ordinal sense.
So once again to illustrate this point 5 as a collective prime number quantity implies 1st, 2nd, 3rd, 4th and 5th individual members (i.e. the natural numbers from 1 – 5 in an ordinal sense).

So from the quantitative perspective we start out attempting to explain how the natural numbers are derived from the prime numbers (in quantitative terms).

However when we allow for the individual qualitative distinction that is necessarily implied through the ordinal rankings of the members of the prime number group, then we realise that the natural numbers are already uniquely contained within this prime number grouping.

Therefore when we properly allow for both the cardinal and ordinal nature of number (i.e. quantitative and qualitative aspects) we are presented with two complementary perspectives with respect to the relationship as between the primes and the natural numbers.

From the standard quantitative perspective the natural numbers are collectively understood in cardinal terms through unique combinations of prime number components.

However from the (unrecognised) qualitative perspective the prime numbers are collectively understood in ordinal terms through unique combinations of natural number components.

In other words when quantitative and qualitative are properly recognised in a balanced interactive manner, the prime numbers and natural numbers can be seen in complementary terms as perfect mirrors of each other. And the pure experience of this identity is ultimately of an ineffable nature (where no distinction as between quantitative and qualitative remains).

The Riemann Hypothesis is simply a statement pointing to the nature of this identity!

When seen in this light it is not only futile trying to prove the Riemann Hypothesis in a merely quantitative manner; it is futile even trying to understand its true nature in this manner.

The price to be paid in properly incorporating the qualitative aspect is that our conception of the very nature of Mathematics is in need of radical revision. However such a revision will then open up marvellous new vistas of understanding that are presently unimaginable.

No comments:

Post a Comment