Wednesday, October 2, 2013

Where Science and Art Coincide (4)

We have already looked briefly at the nature of 2-dimensional understanding.

Once again from an external physical perspective it contains both analytic (+) and holistic (–) aspects.

Thus analytic appreciation literally implies the positing (+) of phenomena in a conscious rational manner. From a mathematical perspective this relates to the quantitative aspect of understanding. 

Holistic appreciation - by contrast - requires their corresponding negation (–) in an unconscious intuitive fashion, thereby allowing for a new qualitative appreciation of phenomena.
(Indirectly however conscious type symbols are needed to convey such understanding).    

Analytic appreciation essentially relates to the differentiation of discrete phenomena in an independent manner;  holistic appreciation then relates to corresponding integration of such phenomena in a continuous manner (as interdependent). 

Conventional Mathematics - as we have seen - is formally based merely on 1-dimensional understanding.

This implies that the holistic aspect is reduced to the analytic; alternatively it implies that the qualitative aspect is reduced to the quantitative; it also means that the unconscious (intuitive) aspect is reduced to the (conscious) rational; yet again it means that the very notion of integration in experience is confused (through reductionism) with differentiation.

And as our scientific paradigm is firmly rooted in the 1-dimensional interpretation that informs Conventional Mathematics, this therefore implies that it is ill suited for the task of providing us with a properly integrated worldview.

Finally - and perhaps most tellingly - Conventional Mathematics greatly lacks any coherent notion of interdependence.

The very notion of interdependence is qualitative in nature. So once again, in every mathematical context, interdependence is misleadingly interpreted as the quantitative relationship as between independent variables.

Once more this observation is of the very first magnitude in the context of the fundamental relationship of primes to the number system.

As this does indeed require the authentic notion of interdependence, it cannot be properly interpreted in a conventional mathematical manner!

The true holistic mathematical nature of such understanding (which requires the qualitative appreciation of the nature of mathematical symbols) is not yet recognised.

However, it is in fact already widely known and used in variety of contexts.

For example it is implied by all the great mystical traditions with Taoism in particular giving it a fundamental explicit relevance.

It also arises in philosophy, especially with respect to holistic evolutionary understanding e.g. Hegel.

It also very much implied for example in Jungian psychology (which I have always found remarkably compatible with holistic mathematical notions).

It  also arises in modern physics (especially at the quantum level of understanding e.g. wave/particle duality, where unfortunately its true qualitative significance is not yet recognised). It even arises indirectly in Mathematics at a very general level. 

For example Nicholas of Cusa is one Western thinker who sought to apply the "coincidence of opposites" to certain mathematical notions.
One of the earliest Western statements goes back to the Greek philosopher Heraclitus, who had a keen appreciation of its (circular) paradoxical nature. So he understood such paradox as the appropriate logical manner of dealing with the notion of interdependence.

"The way up is the way down;
The way down is the way up"

This in fact is a direct statement of 2-dimensional understanding, So we posit one direction (unambiguously) as "up" and then immediately negate it as "down"; likewise we posit the other direction (unambiguously) as "down" and then immediately negate is as "up".

And I have used this on countless occasions to show what is involved in the recognition of a crossroads. Thus we can clearly decide within arbitrarily fixed frames of reference (where movement is unambiguously in one direction) on what road direction is independently "up" or "down". But when we consider both possible frames of reference simultaneously, then clearly each turn at the crossroads is both "up" and "down" (depending on context). 

So in terms of 2-dimensional understanding, the means by which we unambiguously fix direction as either "up" or "down" separately, corresponds to the 1st dimension (i.e. based on one isolated pole of reference). This requires treating each direction as independent of each other.

However the means by which we understand both each direction as both "up" and "down" simultaneously, corresponds to the 2nd dimension (i.e. based on the dynamic interaction of the two poles of reference).
This requires treating both directions as interdependent with each other.

So we can see clearly here that our knowledge of "independence", in this context, corresponds to the 1st dimension; however our knowledge of "interdependence" corresponds to the 2nd dimension.

Now, I have consistently used 2-dimensional understanding in my interpretation of the relationship of the primes with the natural number system.   

This initially requires using two independent frames (properly of a relative rather than absolute nature). However, just as with Heraclitus' statement, these are seemingly are in direct contradiction with each other.

Thus from the cardinal aspect, each natural number (except 1) represents a unique combination of prime factors.
So 30 for example, can be uniquely expressed as the product of primes (i.e. 2 * 3 * 5).

However from the corresponding ordinal aspect, each prime represents a unique combination of natural number members (again except 1). So for example 5 represents a unique combination (indirectly expressed through the 5 roots of 1) with respect to its 2nd, 3rd, 4th  and 5th members. Again the first member (represented by the 1st root of 1) is not unique, as for every prime number it will be 1!  

So using two opposite poles (i.e. cardinal and ordinal) we are able to give two seemingly consistent explanations (in independent isolation) of the relationship between the primes and the natural numbers.

However when we attempt to look at both explanations simultaneously, paradox arises.

Thus from the cardinal perspective, the natural numbers seemingly depend on the primes as building blocks; however from the ordinal aspect, the primes seemingly depend on the natural numbers as building blocks.

Thus in terms of the 2nd dimension (where the true notion of qualitative interdependence resides), it becomes quite clear that the primes and natural numbers mutually generate each other. In fact both aspects strictly represent phenomenal appearances (as identifiable numbers) with respect to a common nondual reality which is ineffable.

Thus the great mystery of the primes and the natural numbers is that - properly understood - they serve as perfect mirrors of each other (in an identity that is ultimately ineffable).

However we can never hope to understand this from a conventional mathematical perspective, where the focus is solely on the (seemingly) unambiguous cardinal  relationship as between the primes and the natural numbers.

This approach maintains the fiction that the primes have an ultimate phenomenal identity (i.e. as cardinal building blocks) that can thereby be understood in a merely quantitative rational manner.

And this all stems from the 1-dimensional approach adopted!

However when we adopt the 2-dimensional perspective, our view of the relationship as between the primes and the natural numbers (and natural numbers and the primes) changes utterly (ultimately dissolving in total ineffable mystery! 

Looked at from another important perspective, 2-dimensional understanding serves as the appropriate means by which we are able to switch (in the context of 2) as between the ordinal notions of 1st and 2nd.

So in the dynamics of experience (in any relevant context) one object is posited uniquely as 1st; now before the other object can be recognised as 2nd, we must temporarily negate the 1st to posit the other object. Then by referring back to the (relatively) fixed recognition of the initial object as 1st, we can then infer the other object (in this context of relationship) unambiguously as the 2nd.

Thus in the very dynamics of the recognition of 1st and 2nd (in the context of 2), 2-dimensional understanding is implicitly involved, with the 2 roots of 1 providing an indirect quantitative interpretation of the ordinal relationship between the members.

Extending this further, this implies that in the dynamics of the recognition of 1st, 2nd and 3rd (in the context of 3), that 3-dimensional understanding is implicitly involved, with the 3 roots of 1 now providing the indirect quantitative interpretation of the ordinal relationship between the 3 members.

In more general terms, in the dynamics of the recognition of 1st, 2nd 3rd,....nth (in the context of n) where n is prime, n-dimensional understanding is implicitly involved, with the n roots of 1 providing the indirect quantitative interpretation of the ordinal relationship between the n members.

What this implies is truly remarkable!

Explicitly, Conventional Mathematics is formally conducted from a merely 1-dimensional perspective (where - by definition - qualitative is reduced to quantitative interpretation).

However, implicitly, our very ability to consistently understand the ordinal nature of number (as 1st, 2nd, 3rd, 4th, ....) requires 1-dimensional, 2-dimensional, 3-dimensional, 4-dimensional, ...... understanding of a qualitative nature.

Put more simply, our explicit quantitative understanding of number (in a conscious rational manner) implicitly requires the corresponding qualitative understanding of these numbers (in an unconscious intuitive fashion).

Now of course there is no formal recognition - certainly within the mathematical community - that this is the case.

However in a way, this is very easy to explain.

Because the conventional mathematical approach is formally defined in a solely conscious rational manner, this therefore completely blinds practitioners as to the nature of the unconscious intuitive dynamics (underlying all such understanding).

However once we move on to the explicit appreciation of the nature of 2-dimensional understanding, both quantitative and qualitative aspects (implying both conscious and unconscious aspects of understanding) are now formally incorporated in mathematical interpretation.

And as we have seen, the consequences of doing this for Mathematics are truly immense  with every symbol and relationship now potentially understood in a completely new light.   

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