Monday, May 22, 2017

Number and Development (8)

There is an important (unappreciated) paradox with respect to the quantitative definition of any number.

For example, if we take the cardinal number "3"to illustrate, it can be defined in the conventional mathematical manner as,

3 = 1 + 1 + 1.

This represents - what I term - analytic interpretation, whereby the (whole) sum i.e. 3, is treated in an actual quantitative manner as the sum of its independent (part) units.

So here again, each of its three (sub) units is defined in an independent homogeneous manner i.e. without qualitative distinction.

Therefore, from an ordinal perspective, there is no way to distinguish (with respect to dimensions of space and time) which units are 1st, 2nd and 3rd respectively.

So in this ordinal context, each unit can potentially qualify as both 1st, 2nd and 3rd respectively.

In other words in - what I term - holistic interpretation, each (part) unit is treated in a qualitative manner as potentially representing the interdependence of all the ordinal elements of its corresponding group.

Thus from this holistic qualitative perspective, 1st, 2nd and 3rd (as ordinal positions) can equally be identified with each of the individual units of 3.

Therefore when one one appreciates this central paradox i.e. that a number that is defined from an extreme quantitative perspective (in analytic terms), gives rise to the opposite extreme qualitative perspective (in a corresponding holistic manner), then one must accept that the conventional attempt to define number in absolute fixed terms must itself be abandoned.

In other words, to resolve this paradox, one must move to a dynamic interactive interpretation of number, defined in a balanced relative manner, equally containing both quantitative (analytic) and qualitative (holistic) aspects.

So now from this new dynamic perspective, the quantitative aspect of number is viewed analytically in relatively independent manner; in complementary terms, the qualitative aspect is viewed holistically in a relatively interdependent fashion.

Therefore, again from this dynamic perspective - which concurs directly with the human experience of number - quantitative independence (in analytic terms) always implies qualitative interdependence (in a holistic manner); likewise qualitative independence necessarily implies quantitative independence, with both interpreted in a - necessarily - relative manner.


Thus with respect to the number "3" in our illustration, this number is now understood to entail the dynamic interaction of both the quantitative notion of 3 (understood in an analytic manner) and the qualitative notion of 3 i.e. as "threeness" (understood in a corresponding holistic fashion).

Now, when one properly appreciates what is stated here, then it should become apparent that the standard accepted interpretation of number is simply not fit for purpose. It reduces the distinctive qualitative aspect (which can only be properly understood in a holistic manner) in an absolute quantitative manner (that is interpreted in a merely analytic fashion).


I have explained on many occasions how I have sought to remedy this deficiency in the standard interpretation of number by employing a truly dynamic appreciation, which entails the complementary interaction of both Type 1 and Type 2 aspects.

Thus when we interpret "3" - now understood appropriately in a relative manner - with respect to its quantitative characteristics, in Type 1 terms, this is written as 31.

So the dimensional number (i.e. exponent) of 3 is defined with respect to its default status of 1, which implies that we can concentrate in this context on the relative quantitative nature of 3.

Then when we interpret "3" with respect to its corresponding qualitative characteristics, it is now written in Type 2 terms as 13 .

So 3 now directly represents its dimensional status defined with respect to the default base of 1, which implies that we can now concentrate in this alternative context, on the relative qualitative nature of 3.

Thus more simply expressed, when In Type 1 terms we are aware of the quantitative nature of 3 (in analytic terms), then - relatively - in Type 2 terms we are aware of the qualitative nature of 3 (in a holistic manner).

However reference frames continually switch in experience.

Therefore there is also a valid Type 1 sense in which 3 takes on a qualitative meaning (in a holistic manner), with 3 then - relatively - carrying a quantitative Type 2 meaning in analytic terms.

In fact this latter qualitative Type 1 aspect of number is central to proper interpretation of the nature of multiplication, whereas the quantitative Type 2 aspect arises for example in the geometrical appreciation of a cube (with 3 linear dimensions).

Therefore, properly understood, the number "3" - and by extension every number - keeps switching in experiential terms as between its quantitative (analytic) and qualitative (holistic) meanings with respect to both Type 1 and Type 2 aspects.

And once again, the standard interpretation of number (in absolute quantitative terms) distorts these key dynamics in a grossly reduced manner.

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