Tuesday, December 22, 2015

Wholes and Parts (7)

The holistic nature of number, which we have just encountered, is directly relevant for the coherent interpretation of its ordinal aspect.

Once again, in conventional mathematical interpretation, the ordinal nature of number - though inherently of a qualitative nature - is directly reduced in an absolute quantitative manner.

So for example if we take the simple case of "2" to illustrate, in cardinal terms this can be absolutely expressed as the sum of its quantitative part units.

Thus 1 + 1 = 2.

However if one was to express this in the conventional ordinal manner,  we would say that
with respect to the two units,

1st + 2nd = 2.

Thus implicit in this interpretation is the identification of 1st and 2nd with 1 (unit) in each case.

Therefore to spell it more fully this implies that 1st (unit) = 1 and 2nd (unit) = 1 respectively.

So we thereby have a direct reduction of ordinal meaning in a quantitative cardinal manner!

However, once we move to the relative notion of number - where both quantitative (analytic) and qualitative (holistic) aspects necessarily interact in a dynamic interactive manner - we require a new interpretation of the true ordinal nature of number.

In fact, when one reflects a little on the matter, the merely relative nature of ordinal identity should become quickly apparent!

For example, if one was to state that a horse came in 1st (in a one-horse race) this would not be considered as a worthwhile achievement. However if one was then to say that in another race, the horse came in 1st (with 40 horses participating) this would indeed appear much more impressive.

Thus the relative meaning of an ordinal ranking depends on the cardinal size of the group (to which it is related).

Thus we have an unlimited range of possible relative interpretations of 1st, 2nd, 3rd, etc. depending on the cardinal size of the group involved.

And once again, to define these various holistic meanings, we simply obtain the n roots of 1 (where n is the cardinal size of the group in questions) and then interpret these (roots) in the appropriate holistic manner.

Now, as we know, one of these roots will always = 1. What this means in effect is that the notion of interdependence must necessarily always start from the corresponding notion of independence (as in like manner the notion of independence always implies the corresponding starting notion of interdependence).

This means that one of the solutions (for the n roots of 1) will always reduce down to 1.
And in fact it is this default solution that is the only root considered in the conventional treatment of ordinal rankings (where holistic qualitative meaning is in turn reduced in an analytic quantitative manner).

I will illustrate this now a little further with respect to the simple case of the 2 roots of 1.

These therefore provide - when appropriately interpreted in holistic manner - the true qualitative meaning of 1st and 2nd (in the context of 2 members).

Now these would be represented as 11/2  and  12/2, which gives – 1 and + 1 respectively.

Therefore 2nd in the context of 2 is + 1.

So the default interpretation of ordinal rankings - where they are effectively reduced in a quantitative manner - implies that we always define the nth ordinal ranking (in the context of a cardinal group of n).

However clearly we can define the nth ordinal ranking likewise in terms of any group > n!

For example instead of considering 2nd (in the context of 2), we could consider 2nd (in the context of 3, 4, 5,.....) where these all now acquire a true holistic identity. So an unlimited number of holistic interpretations are available for 2nd (and indeed by extension for any ordinal number).

Thus 2nd (in the context of 3) would be represented as 12/3 , which correct to 3 decimal places is
  – .5 + .866i. Then this numerical measurement would be given the appropriate holistic interpretation (as representing a certain unique configuration with respect to the two fundamental polar pairings (i.e. internal/external and whole/part respectively).   

The key to appreciating the holistic interpretation represents an inescapable paradox (from the standard 1-dimensional dualistic perspective).

Now in relative terms 1st (in the context of 1) creates no paradox and reduces to standard linear interpretation.

However 1st and 2nd (in the context of 2) creates this inescapable paradox, as either unit (of the cardinal group) can potentially be 1st or 2nd.

In fact this type of potential recognition where an ordinal position can holistically range over the entire group of cardinal unit members), necessarily informs our common sense recognition (at an implicit unconscious level).

For example, say one is ranking cars as to size (with largest ranked 1st) and we have two models - a Fiat Panda and standard Mercedes - the Mercedes (in this context) will be ranked 1st and the Fiat 2nd. However let's say we switch to ranking by age (with newest ranked 1st) and that the Fiat is registered in 2015 and the Mercedes in 2010. Then the Fiat (in this new context) will be ranked 1st and the Mercedes 2nd.

So before any ranking takes place, implicitly we must be able to accept the notion that 1st and 2nd have a merely arbitrary meaning (depending on context). In other words what can be 1st in one context can be 2nd in another and vice versa. And this is paradoxical in terms of standard linear logic (which is unambiguous in a dualistic fashion).

Of course in any actual context (framed by just one polar reference frame), linear logic will apply.
However implicitly, all possible rankings must apply, before actual rankings (in any given context) can be explicitly made.

Thus once again the deeper significance of all this is that the unconscious level of understanding directly underlines true holistic - as opposed to analytic - interpretation.

Sooner or later, Mathematics as a discipline will have to face the deeply uncomfortable fact that a truly coherent meaning (with respect to any of its concepts) implies the proper integration of holistic (unconscious) with analytic (conscious) interpretation. 
If readers can at least clearly grasp the supreme importance of this one fundamental fact (at present completely denied by the Mathematics profession) this blog will have not been in vain.     

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