Tuesday, February 3, 2015

Intricacies of Addition and Multiplication (6)

We now can begin to address the all important role of primes with respect to the natural number system.
However, once again as we shall see, from the more comprehensive perspective, where both the quantitative and qualitative aspects of number behaviour are explicitly recognised, conventional interpretation is found to be seriously lacking.

In conventional mathematical terms, the primes (and natural numbers) are treated in a Type 1 quantitative manner (that directly conforms to the cardinal interpretation of number).

From this perspective, the primes are viewed as the fundamental "building blocks" of the natural number system.  So every natural number (≠ 1) can be expressed through the unique combination
of  primes (as constituent factors).

In this way the primes appear as the most independent of numbers (from which the composite natural numbers in cardinal terms are derived).

However, as we have seen the primes (and natural numbers) can equally be treated in a Type 2 qualitative manner (that directly conforms to the ordinal interpretation of number).
From this perspective the primes are viewed as the numbers that provide the basis for establishing a unique relationship with every natural number (≠ 1).
In this way the primes appear as the most interdependent of numbers (from which composite natural number interrelationships in ordinal terms are derived).

Therefore from a dynamic interactive perspective, the key feature of the primes is the manner in which both quantitative (cardinal) and qualitative (ordinal) extremes of  behaviour with respect to the number system are embodied in their very nature.

It must be stated at this stage that there are two recognised ways in conventional mathematical terms of deriving the natural numbers.

The first is based on addition as the simple one where every natural number can be expressed through the repeated addition of 1. So 2 for example from this perspective = 1 + 1.

The second is based on multiplication where now every natural number ((≠ 1) can be uniquely expressed as the product of primes. So once again for example 6 = 2 * 3.

However, this approach, as we have seen is reserved solely for the Type 1 quantitative treatment of number (in cardinal terms).
So equally, as we have seen, this must be extended to the Type 2 qualitative treatment of number (in an ordinal manner).

So 2 from this additive perspective 2 (now reflecting the qualitative notion of  2 as "twoness") = 1st + 2nd.

 And 6 from this multiplicative perspective (reflecting the qualitative notion of 6 as "sixness" = 2 (as "twoness") * 3 (as "threeeness").

Now when we recognise the corresponding Type 2 ordinal aspect of the primes, an obvious problem with respect to conventional interpretation immediately arises.

For example from a quantitative (Type 1) perspective, 7 represents a prime (in cardinal terms) and therefore represents an (independent) "building block" of the natural number system.

However from a qualitative (Type 2) perspective, 7 (as "sevenness") already implies 1st, 2nd, 3rd, 4th, 5th, 6th and 7th members). But already two of these members (i.e. 4th and 6th) represent composite number notions.

What this really entails is that the very notion of primes as sole "building blocks" is meaningless from a dynamic perspective.  Rather we must now look at both the primes and natural numbers as being mutually interdependent with each other in a two-way interactive manner.

In any case, the primes are not truly independent - even in a restricted cardinal manner - as they are already dependent for their composition on the Peano based additive system.

However when one equally recognises the Type 2 qualitative aspect of prime behaviour, then it is obvious that they embody both independent and interdependent aspects of number behaviour, that can only be properly appreciated in a dynamic complementary manner.

So from the Type 1 quantitative perspective, the primes appear as the independent building blocks of the natural number system (in cardinal terms).

However equally from the Type 2 qualitative perspective, the natural numbers appear as the interdependent "building blocks" of the primes (in ordinal terms).

So remarkable the ultimate relationship of the primes to the natural numbers (and natural numbers to the primes) is one approaching total synchronicity!

There is another important point which must be made at this juncture.

Once again the conventional treatment of the (Type 1) natural number system (as quantitatively viewed in a cardinal manner) is in terms of a line reflecting the linear (1-dimensional) mode of interpretation adopted.

However when we properly allow for the (Type 2)  treatment of the natural number system (as qualitatively viewed in an ordinal manner) the appropriate means of representation is in in terms of a circle.

In this way both the quantitative and qualitative aspects  of the number system are seen in dynamic terms as linear and circular with respect to each other.

However, understanding must be very refined, for when reference frames switch the linear can attain a qualitative and the circular a  quantitative meaning respectively.

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