Avoiding Contradiction

From the Type 1 mathematical perspective

1) e^(2*i*pi) = 1


2) e^(- 2*1*pi) = 1


3) e^(0) = 1.

One might thereby be attempted to conclude therefore that the dimensional expressions in each case must also be equal

i.e. that 2*i*pi = - 2*i*pi = 0!

This clearly is not permissible from a Type 1 (quantitative) perspective. However it requires inclusion of the Type 2 (qualitative) perspective to properly show why this is the case.

As we have seen in Type 1 terms 1^1 = 1^2 = 1^3 .... = 1^n. So from this merely quantitative perspective each of these terms = 1.

We have already shown that properly distinguishing - 1 (as the root of 1) from + 1 requires a Type 2 explanation. So in qualitative terms 1^1 is recognised as distinct from 1^2. So strictly speaking therefore - 1 is the square root of 1^1, whereas + 1 is the square root of 1^2!

Appropriately distinguishing 2*i*pi, - 2*i*pi and 0 from each other (as dimensional expressions) requires a similar qualitative interpretation!


1) e^(2*i*pi) = 1 (i.e. 1^1)


2) e^(- 2*1*pi) = 1 (i.e. 1^0)

This is easily seen by the fact that 2) is the inverse of 1) so that

e^(- 2*1*pi) = 1/e^(2*i*pi) = 1^1/(1^1) = 1^(1 - 1) = 1^0.


3) e^(0) = 1) * 2) = e^(2*i*pi) * e^(- 2*1*pi) = e^(2*i*pi - 2*i*pi) = e^0.

So to conclude e^0 = 1^1 * 1^0.
Now of course from a merely Type 1 perspective 1^1 * 1^0 = 1^1. However the very point in this context is that we are using a Type 2 (qualitative) interpretation.

Thus in a qualitative context, the three different dimensional expressions 2*i*pi, - 2*i*pi and 0 have a subtly distinct interpretation.

So 2*i*pi = 1^1 corresponds to the linear rational component of non-dimensional understanding (insofar as we can give the the non-dimensional notion a finite actual meaning)!

- 2*i*pi = 1^0 corresponds to the circular intuitive component of non-dimensional understanding (where we are relate to a formless non-phenomenal experience).

0 = 2*i*pi - 2*i*pi combines both rational and intuitive comprehension. So in dynamic experiential terms, our notion of 0 as in the expression e^0, necessarily entails both quantitative and qualitative aspects that are interdependent. However in terms of the the two number systems (quantitative and qualitative) these are necessarily split up.


To conclude the notion of 0 as a dimension literally relates to the concept of a point (which is non-dimensional). This in turn implies the identity of both linear and circular interpretations (in both quantitative and qualitative terms).
Clearly as Type 1 Mathematics is devoid of a circular qualitative dimension, it lacks the means to adequately interpret the Euler Identity. It can indeed provide the quantitative demonstration of its validity, but then lacks the means to convey its deeper significance (which is of a qualitative nature).


The real mystery of the Euler Identity is that it beautifully combines both the quantitative and qualitative meanings of its symbols (in a manner where they become indivisible). However this requires that both Type 1 and Type 2 mathematical interpretations be coherently combined!