Yesterday, we looked at the distinction as between the analytic and holistic approaches to number interpretation.
Once again with the analytic, an absolute type interpretation results. At a deeper level this represents a linear (i.e. 1-dimensional) approach, implying in any context single unambiguous polar reference frames. So the external is clearly separated from the internal pole; likewise the quantitative is clearly separated from the qualitative aspect.
Therefore the customary analytic interpretation of number is with respect to its independent (external) objective identity in a merely quantitative sense!
However with the holistic, a relative type interpretation by contrast results, which always entails the dynamic interaction of opposite polarities.
So number from this perspective, necessarily entails a dynamic interaction as between internal (mental) and external (objective) aspects; equally it entails a dynamic interaction as between quantitative (independent) and qualitative (interdependent i.e. relational) aspects.
In extremes, the holistic aspect in itself represents the complementary opposite of the analytic aspect.
Whereas the analytic extreme leads to the absolute interpretation of number as unchanging phenomena of form, the holistic extreme leads to the purely relative interpretation of numbers as energy states (that ultimately are ineffable).
Insofar as the natural numbers are concerned, the analytic interpretation corresponds directly with standard cardinal notions.
However a major (unrecognised) problem relates to the corresponding conventional treatment of ordinal numbers, which properly requires holistic - rather than analytic - appreciation.
When one reflects carefully on it, the ordinal notion of number implies qualitative as well as quantitative considerations.
Say we are trying to rank the exam results of pupils in a class of 20!
Now this requires the quantitative notion of 20 and the recognition of each pupil as independent.
So standard cardinal notions here apply with 2 = 1 + 1, and 3 = 1 + 1 + 1 and so on.
However the attempt to rank each student as 1st, 2nd, 3rd, etc strictly implies the qualitative notion of interdependence, whereby we can place the student positions in relationship to each other.
Therefore in the standard analytic interpretation of number, ordinal notions are therefore necessarily reduced in a merely quantitative type manner.
Indeed, in this context it is interesting how the qualitative notion is never even mentioned in connection with the ordinal treatment of numbers, where the seemingly safer more neutral term of "rankings" is employed.
However, once again, rankings necessarily entail the qualitative notion of the relationship between numbers.
And if we view natural numbers as absolutely independent (in a cardinal manner), well then this begs the significant question of how these numbers can be related with each other!
So clearly there is an enormous question, regarding the very nature of ordinal numbers, which is completely overlooked in conventional mathematical terms.
Now if we confine ourselves for the moment to a finite group of numbers, we can begin to appreciate how ordinal rankings are merely of a relative nature (that keep changing depending on context).
So if for example we envisage a class of 2, obtaining 2nd place might not appear an achievement.
as in this context it represents last place!
However as the size of the class increases, the relative significance of 2nd changes. So 2nd in the context of 20, has very different connotations from 2nd in the context of 2. Again 2nd in the context of 200 would appear even more impressive!
So the meaning of 2nd - as indeed the meaning of every ordinal number - continually changes as the corresponding cardinal magnitude of the finite group to which it relates, itself is increased.
It is only when we view the cardinal number set as infinite, that the ordinal nature of number appears unambiguous.
So therefore if we consider 1st, 2nd, 3rd, 4th etc as ordinal members of an infinite class, then their meaning appears as invariant, whereby they can be represented in identical terms with the corresponding cardinal point of 1, 2, 3, 4 etc. on the number line.
In this (1-dimensional) linear manner - which again underlines all standard analytic interpretation -
ordinal notions can seemingly be successfully reduced to their cardinal counterparts in similar fashion.
However there is n an important unappreciated paradox about what is involved here.
When for example, we refer to the cardinal numbers 1, 2, 3 and 4 for example, these - by definition - are given a limited finite identity.
However, when we refer to the corresponding ordinal numbers 1st, 2nd, 3rd and 4th, these, by contrast necessarily pertain - with respect to their unambiguous identity - to number relationships which entail an infinite class (in cardinal terms).
So ultimately, the seeming correspondence as between cardinal and ordinal numbers in conventional mathematical terms, is based on the fundamental reduction of infinite to finite notions. And this strictly, is the same problem that underlines the attempted reduction of all qualitative notions in merely quantitative terms.
Thus the remarkable conclusion that we have reached - which has profound implications for conventional mathematical understanding - is that the ordinal notion of number is inherently associated with the Type 3 mathematical worldview (where both quantitative and qualitative notions are related).
Ordinal notions thereby represent both the quantitative notion of the independence (of each individual natural number) with the qualitative notion of the interdependence (with respect to the relationship between these numbers).
Thus in Type 3 terms, we at last realise clearly that the very notions of independence and interdependence (with respect to number) are themselves necessarily of a relative nature.
What this means is that both cardinal and ordinal notions mutually imply each other in a relative -rather than absolute - manner.
Using Jungian notions, thus there is a hidden shadow side to the interpretation of the qualitative aspect of number (in a quantitative fashion).
Also, there is also a hidden shadow side to the quantitative aspect of number (in a qualitative manner).