Saturday, May 27, 2017

Number and Development (12)

We are accustomed through conventional mathematical training to view the primes in quantitative terms as the independent "building blocks" of the natural number system.

However what is not all realised is that very nature of the primes changes when they exist - not individually - but rather as factor components of the unique product combinations that generate the composite numbers.

It is in this manner therefore that the qualitative nature of the primes arises i.e. through their interdependence with other prime factors.

So for example both 2 and 3 (as separate individual primes) can in a valid - though strictly relative - sense be viewed as independent "building blocks" in quantitative terms.

However when 2 and 3 are then combined through multiplication to uniquely generate the composite number 6, i.e. 2 * 3 = 6, both 2 and 3 now acquire a relative interdependent meaning in this context, which is thereby of a qualitative (holistic) nature.

And of course there is ultimately no limit to all the relative contexts in which each of the primes can be used (with respect to unique factor combinations with other primes). So 2 for example must necessarily exist as a factor with respect to every even composite number!

Just as we saw earlier that there is an inherent paradox in terms of the definition of each individual prime (with complementary quantitative and qualitative aspects), equally this is true with respect to the collective relationship of primes with respect to the natural number system.

Again from the conventional quantitative perspective, we are automatically trained to see the relationship between the primes and natural numbers unambiguously in a one-way manner.

So clearly from this perspective, the natural numbers appear to depend for their existence on the primes.
However implicit in this view is an unexpected problem which is rarely recognised, as our very understanding of the primes already requires the natural numbers for their proper comprehension.

In other words, the very ability to spatially separate in a meaningful fashion the primes from each other already implies a notion of order that applies to the natural numbers.

So the positioning of each prime already depends on the composite ordering of prime factors.

And if we cannot meaningfully assign a position to each prime, then equally we cannot meaningfully provide it with a definite numerical identity!

Thus again we have two complementary perspectives.

From the quantitative perspective, the natural numbers appear to depend on the independent primes as "building blocks".

However, from the qualitative perspective, the positioning of each prime appears to depend on the  interdependence of prime factors that uniquely generate the natural numbers.

Therefore to properly appreciate this paradox, we must once again move to a dynamic interactive appreciation of number behaviour, entailing both quantitative and qualitative aspects as equal partners.

This then leads inevitably to the realisation that - just as with micro  - the macro behaviour of the number system entails the synchronistic behaviour of both quantitative (analytic) and qualitative (holistic) aspects, which is ultimately ineffable.

And of course as micro behaviour (associated with the Zeta 2 function) and macro behaviour (associated with the Zeta 1 function) are themselves dynamically complementary, ultimately neither has a meaning independent of the other    

Thus properly understood, in dynamic interactive terms,  both the primes and natural numbers (and natural numbers and primes) mutually co-determine each other. both with respect to micro and macro aspects, in a synchronistic manner.

So again the key fallacy with respect to  conventional understanding is the attempt to view the number system in an absolute - merely quantitative - manner where the relationship as between primes and natural numbers is misleadingly viewed as one-way and unambiguous. 

We have already seen how the solutions to the Zeta 2 function provide an indirect quantitative means of expressing the qualitative nature of each individual natural number member of a prime number group (at the micro level).

Likewise - again in true complementary fashion - the solutions to the Zeta 1 (Riemann) function, provide an indirect quantitative means of expressing the qualitative nature of the collective interdependence of prime factors with respect to the natural number system (at a macro level).

So this is the key revelation that can now be made with respect to the Riemann zeros.

Remember how Hilbert, when once queried as to most important problem in Mathematics replied,
" the problem of the zeros of the zeta function, not only in Mathematics but absolutely the most important!"

And the true reason why these zeta zeros are indeed so important is that indirectly they express the hidden qualitative nature of the natural number system.

And in even more precise terms we can say that  the Zeta 2 zeros indirectly express the qualitative (holistic) nature of the ordinal natural number system, whereas the Zeta 1 (Riemann) zeros express the corresponding qualitative nature of the cardinal natural number system, And ultimately, both ordinal and cardinal aspects are completely interdependent  with each other.

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