I have been thinking again in a deeper manner regarding the nature of number representing a dimension (or power).
Let us start with the convenient (default) case of 1.
Now clearly 1 can represent a unit quantity. So implicit therefore in the recognition of any specific object is the number 1 (as an actual finite quantity).
However when used to represent a dimension the number 1 takes on a distinctive holistic meaning (in a potential infinite manner).
So for example if we attempt to represent the number system on a straight line this automatically presumes a linear (1-dimensional) background that is - potentially - infinite.
Therefore though we can use the same symbol 1 to represent a base unit quantity or alternatively the linear dimension (within which such a number is expressed) clearly the meaning is very different in each case.
In the former case 1 represents a specific finite notion that is inherently quantitative in nature; in the latter case it represents a holistic - potentially - infinite notion that is of an inherently qualitative nature.
And as all numbers representing quantitative values must implicitly be expressed with respect to a corresponding number dimension (with the default value = 1), then every number expression - when properly appreciated - necessarily entails a relationship between two aspects which are quantitative and qualitative with respect to each other.
As we have seen, the default dimensional state of a number is 1. And as it is the very nature of linear (1-dimensional) understanding to reduce the qualitative aspect to the quantitative, this means in effect that the qualitative notion of number is effectively always ignored in Conventional (Type 1) Mathematics.
This also causes an important difficulty when dealing with dimensional values (other than 1) which are inevitably treated in a reduced linear manner.
For example 2-dimensional reality would relate to a potentially infinite plane (within which a 2-dimensional object can be placed). However because the qualitative nature of logical understanding remains 1-dimensional, in Type 1 Maths this entails considering the plane as (linearly) extended in two directions that are horizontal and vertical with respect to each other.
However once we depart from 1-dimensional qualitative interpretation, the true nature of dimension is revealed to be of a circular nature.
In fact - when again appropriately understood - this is actually demonstrated in Type 1 Mathematics through the notion of roots.
If we obtain the two roots of unity, they will lie as equidistant points on the circle of unit radius (in the complex plane). Now we can of course in quantitative terms recognise these as + 1 and - 1 respectively. However if we are to give an appropriate 2-dimensional interpretation in qualitative terms (as is appropriate) then we require a logical means of combining + 1 and - 1 as being both true. Now this is done through the paradoxical (both/and) logic of the complementary opposites where each pole like the left and right turns on a road has a merely relative validity.
So we can see here an important inverse relationship as between the 2 quantitative roots of the number 1 and the corresponding 2-dimensional qualitative interpretation (with which they are consistent).
Strictly speaking we do not have 2 roots of 1 i.e 1^1.
- 1 is indeed the (unique) square root of 1. + 1 is however the (unique) square root of 1^2. And 1^1 and 1^2 relate to distinct qualitative numbers (representing dimensions).
Now in a comprehensive appreciation, an even more subtle dynamic interactive understanding is required. Thus when we start with the base - say of 1^2 - here 1 is quantitative and 2 as dimension - relatively - qualitative. However equally the base number 1 can be given a quantitative meaning with the dimensional number 2, thereby in relative terms quantitative. For example all natural logs (representing numbers as powers) clearly have a quantitative meaning as do for example the values of s in the Riemann Zeta Function!
So in relative terms, where the base number has a qualitative meaning this implies that the conceptual nature of the number is highlighted in understanding.
Thus 1 for example can be seen in quantitative terms as a number perception (in relation to the qualitative concept of number). However in reverse terms it can be seen as the number concept (to which 1 relates). So in experiential terms both aspects necessarily interact with all numbers thereby possessing both quantitative (specific) and qualitative (holistic) aspects.
However this poses severe limitations on a mathematical approach that solely recognises the quantitative aspect. And this reduced interpretation is what we misleadingly refer to as Mathematics.
More properly it refers to Type 1 Mathematics. So enormous scope remains for the proper development of Type 2 Mathematics (focussing on the neglected qualitative aspect) and Type 3 Mathematics (where both aspects - quantitative and qualitative - are coherently related).