Monday, August 5, 2013

Illustrating the Holistic Approach

As I have frequently stated, when understood in appropriate dynamic interactive terms, Mathematics entails two aspects of equal importance i.e. analytic and holistic respectively.

Though it is not quite as clear-cut as this, the analytic - for convenience - can be identified with the quantitative and the holistic with the qualitative aspect of interpretation respectively.

Indeed right away we have a problem with the very use of the word “analytic” which is a specialised more limited interpretation within Conventional Mathematics.

Here analytic relates to the study of infinite series (real and complex), limits, the use calculus notions with respect to such series etc.

However in the wider more universal scientific use of the term, analytic - in any context - applies to the breaking down of a whole into its component parts.

And because of the very nature of present scientific method this implies a reduced (i.e. quantitative) notion of a whole which is seen merely as the sum of its constituent parts.

So this is the sense in which I use the word analytic in a mathematical context (which would include all analysis in the narrow more specialised sense in which the terms is used).

Indeed the analytic aspect of understanding concurs perfectly with what I characterise as the 1-dimensional approach.

So once again what this precisely means is that where the interpretation of any relationship in concerned that only one polar frame of reference is used.

Therefore all of current Mathematics - at least what is formally accepted as valid Mathematics - is one dimensional (in this qualitative interpretative sense).

Thus in relation to the first key polarity set i.e. external and internal, mathematical symbols and relationship are given a mere external (i.e. objective) identity in absolute terms that is not influenced through internal (subjective) interpretation.

Therefore though in dynamic interactive terms external and internal polarities of experience are positive (+) and negative () with respect to each other. However the very essence of absolute interpretation is to ignore this distinction in effect treating meaning in a merely positive (+) fashion.

So this is a perfect example of what is meant by the 1-dimensional approach where rational interpretation is linear (and unambiguous) in just one positive direction!

This 1-dimensional approach equally characterises treatment in conventional terms with respect to the second key polarity set i.e. quantitative and qualitative.

Here symbols and relationships are given a mere quantitative identity, again in absolute terms that is not influenced through qualitative interaction.

There is huge confusion in present Mathematics with respect to this fundamental point.

For example the natural numbers 1, 2, 3, 4,  etc are defined as independent with respect to their mere cardinal identity in quantitative terms.

However the very ordering of numbers (whereby we meaningfully can place numbers in relation to each other) requires a distinctive ordinal identity (of a qualitative nature). In other words in ordinal terms a number only has meaning in relation to other members of a number group.

So the meaning of 3rd for example only has meaning in the context of a number group (which can arbitrarily vary in size). Therefore, 3rd in the context of 3 numbers clearly carries a very distinct meaning from 3rd in the context of 300!

Thus ordinal identity therefore relates to the notion of an interdependent - rather than independent - number identity.

So the huge reductionist assumption that is made in Conventional numbers is that we can order the cardinal numbers in a merely quantitative manner.

However once we depict the cardinal numbers e.g. as successive points on a number line, we are thereby assuming a dimensional context that properly relates to a qualitative ordinal identity!

Thus the very essence of 1-dimensional interpretation is that it directly confuses the quantitative notion of individual numbers (as independent units) with the qualitative dimensional notion of the overall general order or collective interdependence as between various numbers.

And let’s be utterly frank here. Our cherished notions of the number system (that have developed now over several millennia) are thereby based on a fundamental confusion (i.e. where qualitative is reduced to quantitative interpretation).

So the qualitative nature of number is not just something vague, as I have often read, such as number personalities or even number archetypes, but as referring directly to the fundamental issue of the means by which we are enabled to achieve an overall order with respect to the number system.

Therefore without properly recognising a qualitative dimension we cannot strictly derive meaningful quantitative notions of number.

Put another way the reduced (i.e. merely quantitative) notions of number we have inherited are defined in a merely 1-dimensional analytic manner.   

Now if pressed sufficiently professional mathematicians may eventually concede that the mental constructs we use to interpret mathematical reality are strictly of a subjective rather than objective nature; with much greatly difficulty they may even concede that ordinal notions strictly refer to qualitative rather than quantitative meaning.

However they will then go on to happily assume a direct absolute correspondence as between (i) objective mathematical reality and subjective mental interpretation and (ii) quantitative objects of an independent nature and a overall qualitative dimensional context (that implies interdependence between these objects).

So in both cases a basic reduction in meaning is involved. This does not entail that no useful benefit can be achieved through such reductionism. Clearly the development of conventional Mathematics proves otherwise. However it does entail that mathematical edifice has been built on a limited and - ultimately - faulty foundation.

Therefore the essence of the analytic approach is to clearly attempt to separate on the one hand

(i) objective truth from subjective mental interpretation and

(ii) quantitative (independent) objects from a qualitative (relational) context.

Then having attempted this clear separation a direct correspondence is assumed as between both in absolute terms.

So once again such analysis entails the implicit belief that mental constructs directly correspond with the mathematical reality (thereby interpreted) and also that general relationships between objects correspond with the (assumed) independent identities of these objects.

Now if one thinks clearly about it such assumptions are untenable and even farcical.

Surely it offends common sense to maintain for example that numbers are independent (in an absolute sense) when clearly numbers can be placed in relationship with other numbers!

The essence of the holistic approach by contrast is that it is inherently dynamic and interactive in nature.

Now the remarkable fact that immediately arises is that all numbers (and indeed all mathematical symbols) can be given a holistic - as well as analytic - identity.

Thus associated with each number for example can be defined a unique mathematical reality in holistic terms. Therefore for example, when the lens of interpretation keeps changing the corresponding mathematical reality to which it corresponds likewise keeps changing (in a relative manner).

So all numbers in holistic terms (except 1) are associated with unique holistic interpretations of mathematical reality (defined in a merely relative manner).

The significance of the number 1 in this context is that it represents the special limiting case where mathematical reality is defined in an absolute manner.

And this is the number that defines all conventional mathematical interpretation. So once again Conventional Mathematics can be precisely characterised as 1-dimensional (in qualitative terms).

So with all other dimensional numbers (other than 1) we have a unique dynamic interaction as between opposite polarities ,constituting in turn unique holistic interpretations.

Now the significance of this for interpretation of the Riemann Zeta Function is immense.

As we know , the one value for which the Function remains undefined in analytic (quantitative) terms is where s (the dimensional value) = 1.

However we now have a corresponding qualitative interpretation where the Function likewise remains undefined in holistic (qualitative) terms for s = 1.

What this means quite remarkably is that the Riemann Zeta Function remains uniquely undefined when we seek to interpret it in a mere analytic (i.e. conventional mathematical) manner.

This therefore implies that the Riemann Zeta Function - when appropriately interpreted in a dynamic interactive fashion  - provides an ingenious means of reconciling analytic and holistic interpretation of the number system i.e. its cardinal and ordinal aspects.

And clearly again this cannot be done in an absolute analytic manner!

Now to keep it simple, I will illustrate briefly the holistic approach with reference to  the (qualitative) dimensional notion of 2 (which in a very special manner is the most important).

Now the essence of 2 in this sense is that it is defined by a 1st and 2nd dimension.

I have explained before how indirectly we give quantitative expression to such ordinal notions through obtaining the corresponding two roots of 1 i.e. (of 11 and 12).

So writing these roots (in reverse order) we obtain + 1 and 1.

Now the 1st dimension relates to analytic interpretation. So we interpret both numbers in an independent quantitative manner.

However the 2nd (new) dimension relates directly to qualitative interpretation (of an interdependent nature).

So + 1 and 1 now represent the dynamic manner in which we switch as between the opposite poles of experience (with 1 representing each pole).

For example with relation to the first polarity set ,we posit the external pole (i.e. by making it conscious) to get objective knowledge of mathematical relationships.

However in order to switch to the internal pole of mental interpretation (of these objects) we must negate the external pole (thereby rendering it unconscious).

If in reverse, we start by positing the internal pole, we then switch to the external (by negating the internal pole).

So there is a dynamic interdependence as between both poles with - what is labelled as - positive or negative depending purely on context.

And as all mathematical experience necessarily entails both external objects and internal interpretation, then the dynamic interpretation of this experience requires 2-dimensional - rather than 1-dimensional – interpretation.

So in this context the very number 2 takes on a remarkable new holistic meaning.

Thus once again in analytic (cardinal) terms 2 = 1 + 1 (with no qualitative distinction as between units).

However in holistic (ordinal) terms 2 = 1st + 2nd which indirectly is represented as (+) 1 1 = 0. Thus the qualitative interdependent nature of the relationship is exemplified by the absence of a quantitative result!

In actual experience the analytic (cardinal) understanding of numbers relates directly to rational interpretation (of a linear kind).

The holistic (ordinal) understanding relates however directly to intuitive appreciation (which indirectly is expressed in a circular i.e. paradoxical rational manner).

Therefore in actual experience (including of course mathematical) there is always a sense in which opposite poles remain separate. However there equally is an important sense in which they overlap as interdependent.   

Whereas rational understanding relates directly to their separation, intuitive appreciation arises from realisation of their mutual interdependence.

So the analytic aspect of understanding is directly of a linear rational nature (though indirectly requiring, especially in creative work, a degree of intuition).  

However the holistic aspect of understanding is directly of an intuitive nature (though indirectly requiring rational expression in a circular paradoxical manner).

Once again we can see the highly reduced nature of Conventional Mathematics, which is formally defined merely in terms of linear rational notions. So intuition - which is directly associated with qualitative type understanding - is formally reduced to (linear) reason. And this again is why qualitative meaning is reduced to quantitative!  

From this reduced perspective, mathematical symbols and relationships are given but one unambiguous interpretation.  

However when we move - at a minimum - to 2-dimensional interpretation, all mathematical symbols and relationships are given two unique interpretations (in both analytic and holistic terms).

I have illustrated, in this entry, the nature of both interpretations - analytic and holistic - for the number 2.

However in principle every mathematical symbol and relationship can equally be defined for both, with comprehensive mathematical understanding arising from the interaction of twin aspects.

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