Note 1 - Numbers and Dimensions

Ultimately the numerical values of all numbers are expressed with respect to the 1st dimension (i.e. are implicitly raised to the power of 1, or alternatively expressed with an exponent = 1).

So if we take a number initially expressed with respect to another dimension (say raised to a higher power) its numerical value will be ultimately expressed with respect to the power of 1.

Thus for example 22 = 4 (i.e. 41).

However what is not properly realised - in terms of conventional interpretation - is that this resultant numerical value (of the conversion of the number to the 1st dimension) is somewhat reduced and in fact ignores the significant qualitative transformation that has taken place.

This can be best appreciated by looking at the matter in geometrical terms.

41 - can be geometrically expressed in terms of a straight line measured off in 4 equal intervals).

However 22 geometrically represents a square with each side a line of 2 (i.e. 21 units).

So strictly speaking the area is 4 square (rather than 4 linear units).

However in conventional interpretation this distinction regarding the qualitative nature of units involved is disregarded and sole attention is paid to the resulting numerical value in merely quantitative terms (which in each case is 4).

Thus - even in analytic terms - numbers have both quantitative and qualitative interpretations.

If the number expressed with respect to a given dimension has a quantitative interpretation, then - strictly speaking - the corresponding dimension (with which it is defined) is qualitative.

So if we take 41, 4 represents in this context the quantitative and 1 (i.e. the power or dimension) its corresponding qualitative aspect respectively.

However as the very nature of linear understanding is - in this context - to reduce the qualitative to the quantitative interpretation, when numbers are defined with respect to the 1st dimension, the qualitative thereby becomes completely reduced to its quantitative aspect.
Thus 41 is in conventional terms - literally - expressed without (qualitative) dimension i.e. 4.

There are remarkable - unsuspected connections - as between numbers as quantities (within a dimension) and numbers as qualities (i.e. the dimension or power to which a number is raised).

As I have indicated, whereas the appreciation of number as quantity leads to the linear interpretation of the number system, the corresponding appreciation of number as (qualitative) dimension leads to circular interpretation (of the number system).

Even more remarkably - though not perhaps obvious at this stage - both linear and circular systems can be converted into each other through an extremely important transformation.

Thus when we attempt to express the circular notion of number (representing qualitative dimensions) indirectly in linear terms, such real numbers become imaginary (in precise mathematical terms).
Likewise when we attempt the reverse conversion of linear to circular terms, such real numbers now become imaginary.
So numbers representing real qualities (i.e. dimensions) - alternatively - represent imaginary quantities.
Likewise numbers representing real quantities - alternatively - represent imaginary qualities (i.e. dimensions).
Far from this being some abstract hypothesis, it is is in fact intimately related to the actual dynamics of reality, explaining precisely the interaction as between object phenomena (as perceptions) and corresponding dimensions (as concepts) respectively.