Tuesday, January 27, 2015

Intricacies of Addition and Multiplication (2)

We saw yesterday that all numbers - indeed all mathematical notions - possess both quantitative (analytic) and qualitative (holistic) aspects, which dynamically interact with each other in complementary fashion.

Therefore from a comprehensive perspective, every number is thereby defined in terms of both a base and dimensional value (where with respect to the expression ab, a is the base and b the dimensional value respectively.

From this perspective the natural numbers can be defined with respect to both Type 1 and Type 2 aspects respectively.

With the Type 1 aspect, where the base aspect varies (as quantity), the (default) dimensional number is 1; with the Type 2 aspect, by contrast, where the dimensional aspect (as quality) varies, the default base number is 1.

The cardinal notion of number is then directly associated from this perspective with the Type 1 aspect; the ordinal notion is then directly associated with the Type 2.

Crucially therefore both the cardinal notion (implying independence) with respect to number quantities and the ordinal notion (implying, by contrast, interdependence with respect to the qualitative relationship between numbers) represent two distinct types of number appreciation, which are dynamically relative in nature with respect to each other.

However it is equally possible to switch reference frames. Then with respect to the natural numbers of the Type 1 aspect, the base value now varies and is defined in qualitative (holistic) terms with respect to a (default) dimensional value of 1; then, in relative terms, the dimensional aspect varies, and is now defined in quantitative (analytic) terms with respect to a default base value of 1.

In this way, both base and dimensional values take on quantitative and qualitative meanings that dynamically interact with each other in a complementary fashion.

And if you think that this seems somewhat far-fetched, let me remind you that this is what continually takes place with respect to our experience of number!

So from one perspective we are able to recognise objects in natural number fashion with respect to their cardinal (quantitative) characteristics. So if I point to a group of 3 people on the street, such recognition entails quantitative recognition of number. If however I now specifically identify - say in terms of a criterion such as height - 1st, 2nd and 3rd members of that group - this now entails corresponding ordinal recognition of a qualitative nature.

And likewise it is similar with respect to numbers, now serving as dimensions. So the recognition of 3 dimensions (now serving as more general categories within which specif number objects can be defined) implies cardinal recognition of a quantitative kind. However the distinction as between the 1st, 2nd and 3rd dimensions (according to some criterion) implies ordinal recognition of a qualitative kind.

So with respect to actual experience we effortlessly switch as between both quantitative (analytic) and qualitative (holistic) recognition with respect to natural numbers serving as both objects and dimensions respectively.

However when it comes to conventional mathematical understanding, we are offered but a reduced - and ultimately highly distorted - interpretation of the nature of number.

So rather than two aspects - quantitative and qualitative - being explicitly recognised that are of equal importance, the qualitative is effectively reduced to the quantitative  aspect in a very limited manner.

And then rather that the more accurate relative appreciation of number, as the consequent dynamic interaction as between opposite polarities, the inaccurate notion of number as absolutely existing is an abstract manner has now unfortunately become firmly embedded in consciousness. Indeed it will take the greatest revolution yet witnessed in our intellectual history to enable the required conversion in viewpoint to a truly dynamic perspective.

For what I am really addressing here is the urgent need now now for a completely new kind of mathematical appreciation that can in no way can be incorporated as some kind of extension of existing understanding.

It may help to provide further context for this dynamic interpretation of number to now look at the issue more closely again in terms of the psychological dynamics that underlie number experience.

All phenomenal experience - including of course mathematical - is conditioned by fundamental sets of opposite polarities.

The first set relates to external and internal poles. So what we identify as objectively existing strictly has no meaning independent of the internal mental constructs used to interpret such experience.
Thus in truth, experience represents a continual dynamic interaction with respect to objects as external and corresponding mental constructs, which relatively of an internal nature.

Thus all objective understanding necessarily reflects a certain (arbitrary) interpretation with respect to such experience.

So for example the belief that numbers represent absolute objects enjoying an abstract existence in a universal mathematical Heaven, ultimately reflects a distorted interpretation (i.e. that numbers can have an objective existence independent of interpretation).
Therefore once we recognise the necessary interaction of (external) objects with (internal) interpretation, mathematical reality must then be understood in a dynamic relative fashion.

The second set of key polarities relates to the interaction of whole and part, which equally manifests itself in terms of quantitative and qualitative aspects that are general and specific with respect to each other.

It is this second set that is directly relevant to the dynamic interpretation of number here outlined.
In the experiential interaction of polarities, we always necessarily experience these poles - to a degree that can greatly vary  - as both independent from us and yet interdependent with us in experience.

For example to recognise and object as external, we thereby need to experience it as - relatively - independent from us. However it cannot be experienced as completely independent as this would exclude any mental interaction (thereby eliminating the possibility of experience).

Therefore, with respect to our conscious recognition, we experience the object as independent;  however with respect to our unconscious recognition (of which we may not be explicitly aware) both external and internal aspects are recognised as interdependent with us.

Then with respect to mathematical understanding, the independent aspect (of conscious recognition) is directly identified in analytic fashion with rational understanding (strictly linear rational understanding).

However the corresponding interdependent aspect (of unconscious recognition) is directly identified in holistic manner with intuitive appreciation (which indirectly can be given a paradoxical circular rational expression) .

Therefore the recognition that number has dual aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other, equally implies the incorporation of (conscious) reason with unconscious (intuition).

Therefore Mathematics can no longer be identified as merely a conscious rational pursuit. Rather the much greater requirement now exists to successfully reconcile both (conscious) reason with (unconscious) intuition with respect to all mathematical understanding .

Though intuition may indeed be informally recognised as important for creative mathematical discovery, in formal terms it is reduced to reason. Thus the failure to recognise both quantitative (analytic) and qualitative (holistic) aspects of number as clearly distinct, directly parallels the corresponding failure to explicitly recognise (conscious) reason and (unconscious) intuition as distinct with respect to understanding.

The relationship between number as representing base and dimensional objects respectively, corresponds directly in psychological terms with the corresponding relationship as between perceptions and concepts.

In the dynamics of understanding, when a number perception takes place in quantitative, terms, a complementary conceptual recognition of that number takes place in a qualitative manner.

So when the natural number perception is of a cardinal nature, the corresponding conceptual recognition is - relatively - ordinal.

However. equally when the number perception takes place in a qualitative manner, the complementary conceptual recognition is qualitative.

Therefore when the natural number perception is now ordinal in nature, the corresponding conceptual recognition is - relatively - cardinal.

In this manner, through the dynamics of experience we keep switching as between both cardinal and ordinal recognition of natural numbers in base and dimensional number terms.
In other words we can recognise natural numbers as applying to both specific objects and general dimensions in both a cardinal and ordinal manner!

However when a distorted interpretation is imposed on this number process, whereby only the quantitative aspects is recognised (thereby explicitly conforming to linear rational interpretation), it sets severe limits with respect to the nature of interaction than can take place. Remember smooth interaction requires recognition of complementary quantitative and qualitative aspects!

So in conventional mathematical terms, the interaction between opposite polarities (quantitative and qualitative) becomes so rigid that the qualitative aspect is no longer even recognised.
The mistaken belief in a merely quantitative interpretation (corresponding to linear rational interpretation) then prevails.

This has become so ingrained - representing the deep unrecognised shadow of Conventional Mathematics - that I would thereby expect enormous resistance with respect to the appropriate dynamic interpretation of number (that properly concurs with experience) .

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