Tuesday, May 6, 2014

Randomness and Order Among the Primes

It is often stated that rather like the tosses of an unbiased coin that the prime numbers are distributed in an fully random manner among the natural system.
However this clearly is not the case.

The random tossing of a coin implies independent events so that outcome on any toss is uninfluenced by what went before. So the probability of H or T on each toss therefore = .5.

However though it is indeed customary to refer to the primes in independent terms as  "the building blocks" of the natural number system, such independence is of a merely relative nature that needs careful qualification.

There is indeed from one perspective, a valid (i.e. cardinal Type 1) sense in which the primes can be viewed as independent. However what is not properly recognised is that when viewed from the equally valid (i.e. ordinal Type 2) perspective, each prime number can be viewed as a unique interdependent group of related members.

Therefore from the dynamic interactive perspective - which is the appropriate way of viewing the matter - the number system is characterised necessarily by the complementary notions of independence and interdependence respectively.

In other words the number system is defined by a delicate dynamic balance as between notions of randomness and order.

We can perhaps illustrate better the limitation of the notion of randomness as applied to primes with the following example.

Imagine on an estate 100 houses numbered from 1 - 100 with the owners entered into a raffle  where 25 prizes are on offer.

So from the truly random perspective, where each house is equally likely to be chosen (and where all houses are included in each draw) there is a chance of 1 in 4 (or probability of 1/4) of an owner winning a prize.

Therefore, if for example no. 23 is the first number drawn and I am the owner of house no. 24, I still have an equal chance with every one else of being successful in subsequent draws.

However imagine a scenario where the 25 winners are chosen on the basis of the 25 prime numbers from 1 to 100.

Clearly therefore in this scenario, if I  live in an even numbered house (other than 2), I have no chance of being successful in the draw.

So prime numbers are not truly random in this sense as - again apart from 2 - all even numbers are excluded from consideration. Also in a base 10 system, by definition, a number ending in 5 (apart from 5) cannot be prime.

So the randomness of prime numbers is clearly of a qualified relative nature.

The most correct way of stating the issue is that the individual nature of the primes is as random as is possible while maintaining the equally important requirement of their overall collective relationship with the natural numbers being as ordered as is possible.

And both of these characteristics - order and randomness - relate to the two equally important aspects of independence and interdependence respectively.

Unfortunately, the very paradigm that characterises conventional mathematical understanding is fundamentally unsuited to grasping the dynamic relative nature of the number system.

Because of an inherent absolute nature, it is thereby biased towards an over-emphasis of the independent aspect of numbers. So notions of order and relatedness can thereby only be dealt in conventional mathematical terms in a reduced manner.

Thus from a proper dynamic perspective, independence (randomness) and interdependence (order) are complementary opposite notions with a merely relative validity.

So again, one could validly maintain in dynamic relative terms, that from the cardinal (Type 1) perspective, the individual primes are distributed as randomly as possible as is consistent with their overall collective order with the natural numbers; equally we could say that the primes are as collectively ordered as is possible that is consistent with their unique individual identities.
Indeed this points once again to the central mystery of the number system, that relates to the marvelous manner in which both the quantitative aspect of independence and qualitative aspect of interdependence, though uniquely distinct, can be yet consistently reconciled with each other.

From another equivalent dynamic perspective, the random nature of the individual  primes is replicated (as their shadow identity) by the ordered collective nature of the non-trivial (Zeta 1) zeros.

Then from the opposite complementary perspective, the ordered nature of the collective overall  set of primes is replicated (again as their shadow identity) by the individual nature of each non-trivial (Zeta 1) zero.

However we equally - and of course in a balanced approach - should also look at the relationship between the primes and natural numbers from the alternative ordinal (Type 2) perspective.
Here, each prime number is viewed as a collective group of related ordinally ranked  members. So the prime no. 3 is thereby composed of the set of 1st, 2nd and 3rd members.

Here the random nature of each prime (representing an internal grouping of members) is replicated (as their shadow identity) by the ordered collective nature of the Zeta 2 zeros.

Thus for example the sum of the 3 roots of 1 (representing the Zeta 2 zeros for the prime number 3) = 0

i.e. the sum of 1,  .5  + .866i and  –  .5   .866i  = 0.

Now strictly speaking one of these roots (i.e. 1) is trivial in the sense that it it is not unique and in fact inseparable from the cardinal notion of 1.

So this in fact illustrates my very point that notions of independence and interdependence are of a merely relative nature.

We can only deal with interdependent notions with reference to a fixed notion (that is independent).

Likewise we can only deal with independent notions against the background assumption of overall order with respect to the number system (that assumes interdependence).

However from the Type 2 perspective, the Zeta 2 zeros do also possess an individual ordinal identity, while once again the collection of primes (now representing groups of members) possesses an ordered collective identity.

(I illustrated this Type 2 collective identity of the primes recently in "Alternative Approach to Frequency of Primes").

When one understands this complementary two-way relationship with respect to the random (independent) and ordered (interdependent)  nature of the primes with respect to the number system, and then appreciates this in both Type 1 (cardinal) and Type 2 (ordinal) terms, then it becomes quickly apparent that the ultimate relationship of the primes to the natural numbers is one of pure interdependence (in an absolute ineffable manner).

The illusion of some definite causal relationship as between both (i.e. primes and natural numbers) stems from attempting to view such a relationship in a limited partial context (where dynamic complementarity does not operate).

We can indeed from a partial context trace the relationship between the primes and the natural numbers in both Type 1 (cardinal) and Type 2 (ordinal) terms.

However though left (E) and right (W) turns at a crossroads may indeed appear unambiguous when approached from merely one direction (either North or South), when both North and South are recognised as complementary directions, then left and right turns are rendered paradoxical (with a merely arbitrary identity in any limited defined context).

In this important sense, the relationship between the primes and the natural numbers is exactly similar.