As I have stated, the 4-dimensional interpretation is especially interesting as it combines both real and imaginary aspects in positive and negative fashion.

And in dynamic terms, these keep switching positions, depending on relative context, so that what is positive (from one perspective) becomes negative from another and what is imaginary (again from one arbitrary perspective) then becomes real!

So any number - when viewed from the 4-dimensional perspective - has 4 relatively distinct meanings, which continually interact with each in dynamic fashion.

Thus we have the real notion of 1 (as an independent entity) with both positive and negative directions. Again when the positive is identified - relatively - with the external (objective) aspect of experience, the negative, by contrast, is identified with the corresponding internal (mental) aspect.

And these can switch, so that the internal may in turn be identified as positive and the external as negative respectively.

Then we have the imaginary notion of 1 (as an interdependent entity) i.e. with the potential capacity to give a related meaning to all separate numbers within its class. This in fact is the dimensional notion of 1 e.g. as a line that is 1-dimensional, that thereby provides a common identity for all - relatively - independent numbers (on that line).

Of course, the general notion of 1 (as 1-dimensional) has a real meaning within its own context. However if we wish to relate, without undue reductionism, the generalised notion of 1 (as representing a dimension) and then the specific notion of 1 as representing an individual number (on the number line), then they should be conceived as "imaginary" and "real" with respect to each other.

And the imaginary notion itself, in vertical terms has positive and negative directions, in that one can switch as between the transcendent notion of 1, as it were, where the dimensional notion of 1 is properly understood in a potential infinite manner (as beyond any actual notion of 1) and the corresponding immanent notion, where the pure infinite notion is reflected through the individual notion of 1 (thereby acting as an archetype).

These four directions (i..e. dimensions) have close complementary parallels in psychological terms.

Once again when we identify the external aspect as positive in a real manner, this is done in a conscious manner (where the number is viewed as a specific object). The internal aspect, is now - relatively - negative in a conscious manner (where the number is viewed as a specific mental perception).

And when we identify the imaginary aspect in a positive fashion, this is identified with the pure intuitive notion of number in potential - rather than actual - terms. This intuitive ability in turn stems from the realisation that positive and negative real polarities are complementary with each other. This thereby generates the realisation of their inherent interdependence (which occurs in an intuitive rather than rational fashion). However, indirectly this interdependence of positive and negative - which seems paradoxical in dualiistic terms - can be indirectly expressed in a circular rational manner. And when this circular understanding is then represented in linear fashion, we have "imaginary" interpretation.

Once again the positive recognition of the imaginary, entails the transcendent appreciation of the imaginary notion (as beyond finite actual appreciation). The - relatively - negative recognition of the imaginary, then entails consequent immanent appreciation of the imaginary notion (as already inherent in each finite phenomenon).

So again with 4-dimensional appreciation, we have four relatively distinct interpretations of any number, which continually interact with each other in a dynamic manner.

However because conventional mathematical interpretation is solely 1-dimensional, these dynamics are grossly reduced in absolute fashion.

Therefore, no distinction is made in conventional terms as between the "real" aspects, i.e. number as objective (in external terms) and number as mental perception (in an internal manner).

Likewise no distinction is made as between "real" and "imaginary" aspects, with the potential (infinite) appreciation of the "imaginary" nature of number reduced - in effect - in "real" terms to its actual (finite) identity.

This problem for example underlines conventional mathematical proof.

The Pythagorean Theorem i.e. that in a right angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides) strictly applies (in a potential manner, to all right angles triangles.

However this does not directly equate with "all "actual triangles (which has an indeterminate meaning in actual terms).

Thus underlining all mathematical proof is a reduction of qualitative to quantitative type meaning, so that the potential infinite meaning of "all" is misleadingly identified in finite actual terms (where "all" has a strictly indeterminate meaning).

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