We have seen - when properly interpreted - that the multiplication of numbers always causes a qualitative - as well as quantitative - change in the nature of the relationship.
And appreciation of this is vital in terms of understanding the true (external) relationship as between the primes and natural numbers (and natural numbers and primes).
So again from the conventional mathematical perspective, it is customary to treat the primes as independent quantitative entities (in a cardinal manner).
However once we uniquely combine the primes - as factors of a composite natural number - the very status of the primes change whereby (in the context of this new number) they attain a qualitative interdependent identity.
So again the primes can now seen clearly to possess two complementary aspects (in external terms), of relative independence (in quantitative terms) and relative interdependence (in a qualitative manner). And both of these aspects occur in a dynamic interactive context.
So within such a dynamic context, it is strictly meaningless to insist in absolute fashion on the primes as constituting the "building blocks" of the natural number system.
Rather we now have two complementary perspectives with a strictly relative validity.
Thus again from one relative perspective, the primes do indeed appear to constitute the "building blocks" of the natural numbers (in a quantitative manner).
However from the complementary relative perspective, the ordered nature of the primes is obtained through their relationship with the natural numbers (in a qualitative manner).
Thus if one from one perspective, the (independent) primes appear to "cause" the natural numbers, From the other equally valid alternative perspective, the primes appear to be "caused" by the natural numbers (through the unique interdependent relationships of prime factors).
The big barrier to ready appreciation of this key point - which given the appropriate dynamic perspective is somewhat obvious - is the deeply ingrained notion that numbers (especially the primes and natural numbers) somehow possess an unchanging absolute identity!
However, as I have repeatedly stated, this simply represents the result of a significantly reduced interpretation of number (i.e. where in every context the qualitative aspect - which in truth is equally important - is reduced in a merely quantitative manner).
When one gets used to looking at this issue in a truly dynamic interactive manner, again as I say, it soon becomes pretty obvious that the primes and natural numbers are two sides of the same coin (with no strict identity apart from each other).
We saw, when dealing with the Zeta 1 zeros, that natural number ordinal notions have a truly relative identity.
Indeed we already accept this in many different situations. For instance one can readily appreciate that the achievement of coming in 1st in a one-horse race is somewhat different from that coming in 1st in a race with 40 horses starting. So 1st (in the context of 1) is very different from 1st (in the context of 40). And as we have seen this relativity of ordinal notions is expressed through the Type 2 number system, where an invariant base number of 1 is raised to different numbers, representing dimensions (or powers).
So the two notions of 1st above can be given as 11/1 and 11/40 respectively.
However, what might seem initially surprising is that the prime number cardinal notions likewise have a truly relative identity.
So again for example, let us take the prime number "2" to illustrate.
We can start by giving this a relatively independent quantitative identity as a "building block" that can be used in the generation of (composite) natural numbers.
However, when 2 is then used again with itself or other primes (representing the unique product of prime factors) it assumes a relatively interdependent identity (of a qualitative nature) in that context.
So 2 for example (as a factor of 4) has thereby a relatively distinct identity from 2 (as a factor of 6).
Put another way 2 is uniquely reflected by 4 in the first case, while it is then uniquely reflected by 6 in the second.
In fact the situation here is analogous to the physical world.
For example we could start with hydrogen and oxygen atoms (in a relatively independent state).
However when a unique chemical combination of the hydrogen and oxygen take place (entailing two hydrogen atoms and 1 oxygen) we get a qualitative transformation through the generation of what we commonly recognise as water. Therefore though one could maintain that the two hydrogen atoms in a water molecule are quantitatively the same as two atoms kept in a flask, clearly a qualitative transformation takes place in conjunction with each oxygen atom.
So it is very similar with the prime numbers. Therefore, even though 2 (as an independent prime "building block" in quantitative terms), might appear the same as 2 (now uniquely representing one of the factors of a composite natural number), a qualitative transformation - relating to this unique interdependence of factors - is clearly involved.
And of course this is equally true of every prime that attains a unique qualitative resonance (when used in the factor composition of a natural number).
In fact, in recent years the term "music of the primes" has been frequently used as a metaphor to embrace the wonderful ordered nature of the primes (when viewed in a collective manner).
And this directly suggests their qualitative - rather than strict quantitative - nature.
However from a mathematical perspective, there is no recognition whatsoever of this holistic qualitative pattern, with a solely analytic interpretation conducted within a reduced - merely quantitative - framework.
And this all serves as an essential preliminary in understanding the true role of the Zeta 1 (i.e. non-trivial Riemann) zeros.
For the role of these zeros is essentially - as with the Zeta 2 zeros, though now in relation to the external aspect of the number system - to convert the holistic qualitative nature of the primes (as unique factors of natural numbers) indirectly in a consistent quantitative manner.
No comments:
Post a Comment