In the previous blog entry, I commented on the dynamic complementary nature of both randomness and order with respect to the number system.
So from one perspective (i.e. Type 1), the individual nature of cardinal primes reflects the extreme of random behaviour; however the collective nature of primes (with respect to the number system) reflects the opposite extreme of ordered behaviour.
Then when we look at prime behaviour from the corresponding (Type 2) ordinal perspective, these relationships are completely reversed.
For example, if we take the prime number 3, it thereby is composed individually of its 1st, 2nd and 3rd members respectively.
So - by definition - these individual members are thereby ordered in a natural number interdependent manner.
Thus the notion of 2nd therefore arises through its relationship with 1st and 3rd; likewise 3rd arises through its relationship with 1st and 2nd and finally, 1st arises through its relationship with 2nd and 3rd members respectively.
So the individual notions of 1st, 2nd and 3rd (as the 3 ordinal natural number members of 3 are thereby highly ordered in a relative manner.
However, once again this ordering cannot be absolute, for a certain arbitrary fixing of identity is always required in locating the 1st member of a group.
So again 3 as a prime group entails 3 individual members. Now, in principle any of these 3 could be identified as the 1st member So once again, in any relative context with respect to identifying 2nd and 3rd members respectively, the initial position with respect to the 1st member must be fixed!
Now this is similar to the cardinal approach, in the sense that 1 is itself excluded from the relationship of the primes with the natural numbers. So all other natural numbers (except 1) can be uniquely expressed as the combination of primes! Therefore though the number 1 is not specifically included, it underlies in a fundamental manner the nature of all other natural numbers.
So returning to the ordinal approach, the individual nature of the natural number members of a prime group are of a highly ordered nature.
Then in reverse, the combined collection of these prime groups lies at the opposite extreme of exhibiting the extreme of random behaviour.
What this means in effect is that in this context, the order in which we take the primes is of little consequence with the same basic conclusion applying to the ordinal behaviour of its natural number members.
In other words if one picked a prime at random from the collection of all primes then the behaviour of each of its individual natural number members lies at the extreme of a totally ordered identity (in relative terms).
However we are now left with a considerable dilemma.
Once again from the (Type 1) cardinal perspective, the individual primes represent a random extreme (with respect to the overall number system) whereas the collective behaviour of the primes represents the opposite ordered extreme (in relative terms); then from the (Type 2) ordinal perspective, the individual natural numbered members, comprising each prime, represent an ordered extreme, whereas the collection of all primes lie at the other extreme of random behaviour (in relative terms).
So both sets of results are fully paradoxical in terms of each other.
In other words, what is random with respect to the primes from the cardinal perspective (Type 1) is ordered from the alternative ordinal (Type 2) perspective; and what is random from the ordinal perspective (Type 2) is ordered from the alternative cardinal perspective (Type 1).
And it is vital to recognise that our actual appreciation of number is necessarily of a dynamic interactive nature combining both cardinal and ordinal type appreciation.
So, for example, the explicit recognition of 3 as a cardinal prime implicitly entails corresponding ordinal recognition of its constituent individual members in a natural number fashion; equally the explicit recognition of the 1st, 2nd, and 3rd members of 3, implicitly entails cardinal recognition of the number 3.
Now, in direct terms, cardinal and ordinal recognition are quantitative as to qualitative - entailing notions of independence and interdependence - respectively.
The huge limitation of the accepted conventional mathematical approach is that it attempts to abstract the quantitative from the qualitative aspect of appreciation. Thus it reduces an inherently dynamic interactive relationship as between two complementary aspects of number, in a merely reduced absolute type manner.
Thus the standard approach to viewing the relationship of the primes to the natural numbers is to treat both with respect to their mere cardinal identities.
This leads therefore to a substantial misinterpretation of the true relationship between both aspects.
In other words the ultimately mysterious relationship as between the primes and the natural numbers is the same mysterious relationship as between their cardinal and ordinal identities.
Indeed ultimately this points to the fundamental connection as between both the quantitative notions of independence and qualitative notions of interdependence that underlie the behaviour of the number system (and indeed everything in phenomenal creation).