Tuesday, July 12, 2011

Reformulating Basic Notions

We have seen how the notion of a root is problematic from a Type 1 mathematical perspective.

The reason for this is that such Mathematics attempts to view relationships from a merely (reduced) quantitative perspective.

However whenever powers (or roots) or multiplication (and division) are involved both qualitative and quantitative transformations in the nature of the variables take place.

We illustrated this point at length with respect to the simple expression 1^2 showing that here 2 (as the dimensional number) relates in qualitative terms to a new holistic manner of qualitatively interpreting relationships in logical terms.
This logical system - which has a direct relevance to interpreting quantum mechanical relationships - is commonly referred to as the complementarity of opposites and is based therefore on the dynamic interaction of opposite polarities (each of which has an arbitrary definition depending on context).

When one attempts to view relationships from a Type 1 perspective, recognising merely the quantitative aspect of numbers, it is easy to show that this leads quickly to logical confusion whereby for example we maintain that + 1 as a result is the same as either + 1 or - 1.

In practice in Type 1 calculations this problem is avoided by using only principle roots. So even though the square root of 1 from a Type 1 perspective can be + 1 or - 1, if we confine ourselves to the principle root, then it is unambiguously + 1. And of course this is the value that will be provided for the square root of 1 on any calculator!

However if we are to properly solve the logical inconsistency raised by Type 1 Mathematics, then we need to use Type 2 understanding.

So we have already explained at length why the use of 2 (as dimension) leads to a uniquely distinct manner of logically interpreting relationships (as opposed to the default dimension of 1 as used in Type 1 Mathematics).

With Type 2 Mathematics, a unique result is associated with each number (when used as a dimensional power).

Therefore from this perspective whereas 1^1 = + 1, 1^2 by contrast = - 1 (from a qualitative perspective).

What this entails is that, whereas the former relates to the conscious positing of unitary form as in linear rational understanding, the latter relates to the corresponding dynamic negation of such form (which is the very basis through which unconscious type intuition is generated).

Then in corresponding inverse quantitative terms 1^1 = + 1 and likewise 1^(1/2) = - 1.

Thus from a Type 2 perspective, it is strictly inaccurate to refer to the square root of 1 as 1. This is but a reduced interpretation (where the corresponding inverse dimension is likewise reduced to 1).

Remarkably there is direct support for this latter Type 2 approach given by use of the modified Euler identity.

As is well known e^(2*pi*i) = 1.

This expression therefore provides the means for obtaining the value of any root of 1.

So therefore to obtain the square root of 1 we raise the RHS to the power of 1/2 which results in the well known result

e^(pi*i) = - 1.

So using the (modified) Euler Identity we can obtain an unambiguous answer in quantitative terms that is associated with any fractional power of 1 (as dimension).

Likewise in corresponding inverse terms we equally have an unambiguous qualitative interpretation associated with each dimension as number.

Thus for example e^(4*pi*i) = 1^2.

Therefore once again the qualitative interpretation associated here with the use of 2 (as dimension) corresponds in direct inverse terms with the quantitative result associated with 1/2 (as dimension).

So in qualitative terms + 1 (linear reason) is directly associated with the 1st dimension.

By contrast - 1 (circular reason i.e. as the indirect expression of intuition) is directly associated with the 2nd dimension.

However 2-dimensional interpretation then necessarily requires combining both the 1st and 2nd dimensions (in this context).

So in a combined 2-dimensional logical approach we employ both linear (either/or) logic and circular (both/and) logic.

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