Monday, January 23, 2012

Generalising New Approach

As we have seen number can have both quantitative and qualitative characteristics. Therefore in a comprehensive (Type 3) approach a number is longer a number in static terms but rather has twin aspects that are quantitative and qualitative respectively.

So as we saw in the last post 2^2 = 4, 2.

In other words the 1st number here refers to its quantitative characteristic (which defines the conventional approach).
The 2nd number defines its qualitative characteristic (which in this case is 2-dimensional).

Now the quantitative and qualitative characteristics are linear and circular with respect to each other. Thus 4 (as quantity) is a linear number (lying on the real number line).
2 (as quality) is a circular number with its corresponding square root (lying on the circle of unit radius).

In Type 1 terms only the quantitative aspect is recognised.

So 2^2 = 4

More fully this could be written as 2^2 = 4, 1.

So the Type 1 approach is characterised by a situation where the default qualitative (or dimensional) number = 1.

In Type 2 terms only the qualitative aspect is recognised.

So 2^2 = 2.

More fully this would be written as 1, 2

In other words the Type 2 approach is characterised by the situation where the default quantitative number is always 1!

Now one might ask what these qualitative or (dimensional) numbers actually signify!

Well, each number actually defines a unique logical approach with which to interpret mathematical relationships.

Once again default qualitative case of 1 defines the conventional linear rational approach to Mathematics (which is one-dimensional or more precisely one-directional).
For example it is assumed in conventional terms that a proposition is either true or false. However this approach simply defines 1-dimensional logic.

For all other (higher) dimensions a merely relative truth value results. For example in the simplest case of 2-dimensional logic we have the complementarity of opposite polar directions.

For example if truth is defined by the two-way interaction of external and internal poles, then objective truth (which implies only one of these poles) has no strict meaning. Rather truth now represents a dynamic (switching) relationship as between both poles.

All higher dimensions involve a circular relationship as between poles. For example in 4-dimensional terms we have complementary opposites in both real (horizontal) and imaginary (vertical) directions.
So we here have experience switching between positive and negative poles with respect to real (conscious) and imaginary (indirectly conscious i.e. as representation of the unconscious) respectively. So we can perhaps appreciate here that the 4 holistic dimensions here are the qualitative counterpart of the four roots of unity (in quantitative terms).

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