Friday, May 23, 2014

More on Randomness and Order (1)

They key point to grasp with respect to understanding of the number system is that randomness and order are complementary notions, which can only be properly appreciated in a dynamic interactive manner.

Indeed randomness in this context implies non-order (i.e. lack or order), whereas from the opposite perspective, order implies non-randomness (i.e. lack of randomness).

Crucially therefore both notions imply each other and cannot be interpreted - except in reduced fashion - in the standard absolute manner that characterises the very paradigm of Conventional Mathematics.

Indeed in the deeper sense, both randomness and order imply the key notions of independence and interdependence respectively. And as I have repeatedly stated, this requires - in terms of a consistent mathematical approach - both analytic (Type 1) and holistic (Type 2) aspects of interpretation in equal balance.
As once again Conventional Mathematics is formally defined solely in terms of the analytic (Type 1) aspect, it is severely limited therefore in its capacity to unravel the true nature of these issues.

In effect, in its attempt to confine interpretation within the Type 1 aspect, it thereby adopts a distorted perspective (which robs these very notions of randomness and order of their true dynamic nature).

It can help to demonstrate the relative nature of randomness and order by initially looking again at the tossing of an unbiased coin.

Now with respect to each individual toss, the random nature of the outcome is in evidence.
Thus a high level of unpredictability attaches to the outcome (which can either H or T).

However when we now look at the other polar extreme of the collective nature of a combined series of tosses, the ordered nature of the outcome now comes sharply into evidence.

Therefore as the number of tosses increases, without finite limit, the  proportion of Heads and Tails recorded will approximate ever closer to 1/2.

Now, of course, in terms of finite events, we do not get a result of 1/2 in absolute terms. Rather the result is strictly relative and approximate approaching ever closer to 1/2 (which however is not exact in a finite manner).

So the collective notion of order here is strictly of a relative approximate nature.

However, when we view the matter appropriately, this implies that the notion of randomness (attaching to each individual toss) is also necessarily of a relative approximate nature.

Now one might attempt  maintain that the probability of a H or T is exactly 1/2 for each separate trial.
However, strictly in any practical experimental context, it would be impossible to ensure conditions guaranteeing this absolute lack of bias. So some degree of bias, arising from the nature of the coin, weather conditions, manner of tossing etc. will inevitably arise (regardless of how small).

More importantly, the very notion of independence (randomness) with respect to each individual trial implies the corresponding notion of interdependence (order) with respect to the collective group of trials. So once again, though we cannot successfully predict the outcome of any individual trial, we can expect to predict to a high degree of accuracy the overall result attaching to a combined group of trials!

Thus we can sum up so far, by once more stating that both the random nature of each individual trial (in tossing a coin) and the ordered nature of the collective outcome (of a combined group of tosses) are strictly of a relative approximate nature. Randomness and order are thus complementary notions that can only be properly understood in a dynamic interactive manner.

Now the relationship of the cardinal primes to the natural numbers can be understood in a similar manner.

When we look at the individual nature of the primes, they are strongly characterised by their random nature.
This in turn corresponds with the view of the primes as the independent building blocks of the natural number system.
However quite clearly the primes are not random in any absolute sense. There are certain restrictions on this randomness. Apart from 2, no even number can be prime; then in the denary system, apart from 5, no other number ending in 5 can be prime!

Then we look at the collective nature of the primes, they are now in complementary fashion strongly charcterised by their ordered nature, exhibiting a stunning overall degree of regularity. However once again this order is of a relative rather than absolute nature. Thus, like the frequency of Heads and Tails in the tossing of coins, we can equally predict the frequency of primes (and by extension the composite natural numbers) to a high - though never absolutely total - degree of accuracy.

Even when we allow for the refined modifications to such predictions using the Riemann zeros, the estimation of the number of primes (to a certain number) strictly remains of a relative - rather than absolute - nature. Now in theory one might attempt to maintain that given infinite adjustments (through using the total set of zeros) that our predictions would then be absolute. However by their very nature, we can only have access to a finite number of such zeros. Thus such total adjustment, by its very nature is not possible.

So once again the notions of randomness and order in the cardinal number system are relative and approximate based on complementary poles, which can be only properly appreciated in a dynamic interactive manner.

One of the great problems with the way that we view the natural number system in cardinal terms is that customarily it is viewed as being composed of individual numbers that are independent of each other.

Thus though it may well be accepted that the primes serve as the building blocks of the (composite) natural numbers, once they have been derived from the primes, they are then misleadingly treated as independent units in a similar manner.

So 1, 2, 3, 4,.... are all treated as independent number entities in all subsequent calculations. This then leads to the gravely mistaken impression that the number system itself exists in abstraction as some absolute system frozen in space and time.

This distorted tendency is then greatly accentuated through the very paradigm that defines conventional mathematical thinking that is linear (1-dimensional) in nature (based on single isolated poles of reference).

Thus the very paradigm on which Conventional Mathematics is based is utterly unsuited to come to grips with the true dynamic interactive nature of the number system based on complementary poles of reference.

Thus internal and external constitute one key set of dynamic polarities. We cannot form for example the notion of a mathematical object such as a number in an external manner, without a corresponding mental perception of the number that is - relatively - of an internal nature.
So we cannot have objective truth in the absence of subjective interpretation. The very illusion of absolute truth in Mathematics itself reflects the mistaken belief that objective truth can exist in the absence of such subjective interpretation! So this mistaken belief itself reflects an important - though ultimately untenable - form of interpretation!

Individual (part) and collective (whole) aspects then constitute another key set of dynamic polarities. So we cannot form a notion of an individual number perception in the absence of the collective conceptual notion of number. From the opposite perspective, we cannot form the general concept of number in the absence of particular number perceptions. Thus, the number with respect to its individual and collective aspects, is dynamic and interactive (based on complementary opposite poles).
However, Conventional Mathematics is once again based on the reduced notion that these two aspects can be abstracted from each other in a static absolute manner.

Though the natural number system - from the cardinal perspective - intimately depends on the primes, the natural numbers are then treated as exhibiting even more randomness than the primes.

Thus in a lottery where 100 people are given tickets numbered 1 - 100 respectively, if we draw 25 tickets (allowing replacement) each ticket has an equal chance of being chosen on each draw (though as we have seen this must strictly be interpreted in a relative, rather than absolute manner). So the tickets are randomly chosen in this manner.

However, the composite natural numbers depend on the unique order exhibited by the primes. So it seems somewhat paradoxical that we should then consider all such numbers in a merely random fashion.
This simply therefore reflects a reduced - and unsatisfactory - linear way of attempting to view the number system.

Quite clearly the randomness of the system cannot be understood in the absence of its corresponding order; likewise the order of the system cannot be understood in the absence of its corresponding randomness.

This notion of true interdependence with respect to the number system requires holistic appreciation of a circular kind (rather than analytic understanding in a linear manner).

Then in true appreciation of the - relatively - independent and interdependent aspects of the number system both analytic (Type 1) and holistic (Type 2) understanding are required.    

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