An intimate mathematical relationship exists as between wholes and parts in both linear and circular logical terms respectively.

Now it is the very essence of the linear approach to give a merely reduced rational interpretation of wholes and parts (where the whole is viewed solely in quantitative terms as the sum of its constituent parts).

From a conventional (linear) perspective a cake for example might comprise four (equal) slices. Thus the whole cake in this context is interpreted as the sum of the four (4) part slices.

Then in reverse manner each individual slice is interpreted - again in quantitative terms - as a quarter (1/4) of the whole cake.

Now in (Type 1) mathematical terms the whole cake (containing the four slices) = 4^1.

Meanwhile the reverse relationship of each part to this whole = 4^(- 1) = 1/4.

Thus, fascinatingly, the very ability to switch in experience from whole to part recognition entails the corresponding ability to implicitly negate the 1st dimension. What this means is that understanding implicitly must switch from rational to intuitive mode before this important transition can be made. However when qualitative interpretation is absolute, this switch from positive to negative sign is ignored so that it formally remains at a merly rational level!

Thus it requires Type 2 mathematical understanding to provide this important insight of how intuition insight is necessarily involved in the understanding of the true relationship as between whole and part..

Therefore once again Type 1 interpretation - which is conducted in merely reduced rational terms - cannot explain the decisive qualitative means of switching from whole to part (and part to whole). In order words in confining itself to a merely (reduced) quantitative (Type 1) interpretation of its symbols, it ignores completely the equally important qualitative (Type 2) interpretation of the same symbols.

Thus the minus sign (-) has a well-defined meaning in quantitative (Type 1) terms. However the same sign has a largely unrecognised qualitative (Type 2) meaning as the negation of what is (consciously) posited in rational terms. In dynamic interactive terms, this implies the (unconscious) intuitive aspect of recognition. Thus from a Type 2 perspective, in transferring from whole to part recognition, linear (1-dimensional) interpretation of a rational kind must be (temporarily) negated in a qualitative form of holistic recognition. Thus though the part can then be likewise interpreted in a quantitative manner, the decisive switch from whole to part recognition requires Type 2 qualitative recognition.

However equally we have a circular number system that allows for a whole range of differing qualitative interpretations in each case entailing a unique configuration of both (conscious) rational and (unconscious) intuitive type appreciation. From this perspective the (whole) dimension given by the number 4, is qualitatively defined in Type 2 terms (as 1^4).

Then 1/4 i.e. 1 ^(1/4) represents the reverse quantitative relationship in part terms for this system.

In circular quantitative terms this is given as i (which lies on the unit circle in the complex plane).

Thus in reverse terms i refers to the qualitative meaning of 4 (as a whole dimension).

This clearly indicates again therefore that the very switch from whole to part recognition always entails a corresponding switch from qualitative to quantitative. However understanding here is much more refined allowing for imaginary (rather than real) appreciation.

So 4-dimensional interpretation from a qualitative mathematical perspective is of an imaginary (rather than real) rational nature, as the indirect expression of unconscious meaning.

One important physical application of this principle relates to quantum mechanics where the "higher" psychological interpretation of a qualitative nature is always necessary to properly interpret "lower" sub-atomic phenomena (of a physical nature). In other words if the manner of the "higher" intuitive recognition in qualitative terms is n-dimensional, the corresponding "lower" physical recognition with respect to physical "parts" in nature is 1/n. So once again their is a reciprocal qualitative to quantitative (and quantitative to qualitative) relationship as between whole and part.

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