Wednesday, November 30, 2011

Cardinal and Ordinal Numbers

I have frequently referred to the fact that all numbers have both quantitative and qualitative aspects which are dynamically interdependent in experience.

However Conventional (Type 1) Mathematics, in giving formal expression to the merely quantitative aspect, thereby distorts appreciation of the true nature of number.

One manifestation of the relationship of quantitative to qualitative aspects pertains to the relationship as between the cardinal and ordinal use of number.

The very nature of the conventional linear (1-dimensional) approach is that it reduces qualitative to quantitative type interpretation.

This is exemplified by the manner in which we look at natural numbers.

In cardinal terms we can speak of the natural numbers as 1, 2, 3, 4,... as marked along the standard number scale.

However equally in ordinal terms we can speak of the 1st, 2nd, 3rd, 4th .... numbers as marked along the same scale.

So here there is a direct correspondence as between each successive number (as used in both a cardinal and ordinal sense).
Thus 1 corresponds with the 1st number, 2 with with the 2nd number, 3 with the 3rd number, 4 with the 4th number etc.

Indeed, without implicit recognition of the ordinal use of number, it would be strictly impossible to define the cardinal numbers.

However there is a subtle difference with respect to both types. The cardinal number 4 for example suggests a multiple of part units (that are literally 1).

So in concrete terms 4 could be used to denominate a collection of any four units (e.g. 4 stones).

However 4 in ordinal terms i.e. as the 4th unit inherently conveys the opposite relationship of part to whole. So the 4th unit specifically refers to just one unit which is seen as part of - what in cardinal terms - is a whole that comprises 4 units. So implicitly to give a ranking of 4 (as the 4th) necessarily implies recognition of 3 other units that are 1st, 2nd and 3rd. In this sense it represents the inverse (i.e. reciprocal) of the cardinal notion (i.e. 1/4).

Therefore when seen in this light, the relationship as between the cardinal and ordinal notion of number in experience represents the dynamic interaction as between whole and part (and part and whole).

Now, when written more fully 4 is represented as 4^1.
In other words the default quantitative interpretation of number (where powers or dimensions are considered redundant) is with respect to 1. And in qualitative terms this corresponds directly with linear (1-dimensional) understanding.

However the reciprocal 1/4 can be represented as 4^(- 1). So from a qualitative perspective the very manner through which we switch from the cardinal to the ordinal notion of number implies the negation of the 1st dimension.

As we have seen before, this entails switching from rational to intuitive type understanding.

At a deeper level this implies that the proper understanding of number requires an approach that combines both rational (specific) and intuitive (holistic) type interpretation (relating to quantitative and qualitative aspects respectively).

In Type 1 quantitative terms, the absolute value of a number arises when the sign is ignored. So in absolute terms - 1 is thereby indistinguishable from + 1.

Likewise in Type 2 qualitative terms, absolute interpretation arises when we ignore the sign (i.e. direction) of understanding. So linear understanding ( + 1) is thereby indistinguishable from its negation i.e. - 1.

Put another way with absolute type interpretation (as befits Type 1 Mathematics), quantitative cannot be clearly distinguished from qualitative meaning!