Tuesday, May 8, 2012

Through the Looking Glass of Number

Let us sum up for a moment the key issues with relation to the number system!

The basic reason why the relationship as between the primes and the natural numbers remains so unclear is because the very nature of the number system is fundamentally misrepresented in conventional mathematical interpretation.

In keeping with the linear rational approach adopted, numbers are treated in independent terms as absolute quantities that are fixed. This accords with the cardinal nature of number. However properly speaking numbers possess also a relational capacity of interdependence with other numbers which strictly is qualitative in nature. This in turn accords with the ordinal nature of number.

In conventional Mathematical terms the qualitative ordinal aspect is misleadingly reduced in quantitative terms leading to much confusion that simply cannot be properly addressed from this perspective.

So the basic starting point for a coherent interpretation is the recognition that the number system is properly of a dynamic interactive nature, with complementary aspects that are quantitative and qualitative with respect to each other.

Thus from one perspective, numbers possess a relatively independent status in cardinal terms as number quantities; from the equally important opposite perspective numbers equally possess a relatively interdependent status in ordinal qualitative terms (through their relationship to other numbers).

Whereas the independent aspect can be represented directly in linear terms, the interdependent aspect is revealed indirectly in a circular manner.

However it is vital to understand that this latter circular aspect corresponds to a distinctive circular logical approach to understanding!

I have already demonstrated that the ordinal notion of a number is highly ambiguous and dependent on an overall holistic context. Thus the meaning of 2 in ordinal terms (i.e. 2nd) will change depending on the size of the number grouping to which it relates!. So the 2nd of two members is relatively distinct from the 2nd of – say – 20 members!

And likewise this is true for all ordinal numbers (with an unlimited amount of possible interpretations).

Now one might still wonder in what sense the meaning of cardinal numbers is relative!

In what sense for example is 2 merely relative?

Well, to appreciate this, we must remember the dynamic relationship of whole and part (and part and whole) in experience. Thus any particular number such as 2 only has meaning in the context of the number concept which is potentially unlimited in scope.

Thus in this dynamic interactive context, the number “2” and indeed any specific number has no precise meaning as it is necessarily defined in terms of a concept (that is potentially unlimited).

In other words the very nature of number cannot de defined in an absolute finite manner and so must always remain to a degree indeterminate!

So strictly speaking all cardinal number quantities are merely independent in a relative quantitative sense; equally all ordinal numbers are merely interdependent in a relative qualitative sense.

Thus our actual experience of number is inherently dynamic entailing the continual interaction of both cardinal (quantitative) and ordinal (qualitative) aspects.

And crucially both of these aspects correspond to distinctive forms of mathematical understanding.

So the former (Type 1) aspect – once again – corresponds with rational interpretation of a linear kind.

The latter (Type 2) aspect corresponds directly with intuitive appreciation (that indirectly can be expressed in a rational circular manner). And with respect to each number as a dimension (other than 1) a distinctive configuration with respect to circular understanding can be defined (which always necessarily includes linear type understanding in a refined manner).

When the number system is appropriately understood in this dynamic interactive manner (with both quantitative and qualitative aspects that are relative) the true nature – though not the mystery – of the prime numbers resolves itself revealing a looking glass reality of mirror number reflections.

So again from the – relatively – quantitative (Type 1) perspective, the primes are seen as the essential building blocks of the cardinal number system with all natural numbers in cardinal terms representing a unique combination of prime components.

However again now from the – relatively - qualitative (Type 2) perspective, each prime is defined in ordinal terms through a unique combination of natural number components.

Then in Type 3 understanding, where both aspects are harmonised in complementary fashion, the prime numbers and natural numbers now increasingly are seen as perfect mirrors of each other (in an ultimate identity that is identical). And the Riemann Hypothesis simply points to this ultimate identity (of the quantitative and qualitative aspects of the number system).

However there is no way that this two-way mirror relationship as between the primes and natural numbers (and natural numbers and the primes) can be understood in – mere – conventional (Type 1) terms.

Thus the true significance of the Riemann Hypothesis is that its proper appreciation will eventually entail a profound revolution in the very manner the number system - and indeed all Mathematics - is understood.

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