Once again we have shown how in Type 1 Mathematics numbers when used as dimensions - though inherently of a qualitative nature relative to the (quantitative) number to which they are raised - are given but a reduced quantitative meaning.
Therefore from this perspective 1^2 is indistinguishable from 1, i.e. 1^1 (in quantitative terms).
However this assumption subsequently gives way to basic logical inconsistency when we attempt to find the 2nd (square) root of each expression.
Thus in the former case the square root of 1^2 (which is obtained through raising to the power of 1/2 = 1, 1^1. However the square root in the latter case = 1^(1/2) which cane be given two equally valid answers in Type 1 terms, i.e. + 1 and - 1.
So in the first case we obtain just one answer which is unambiguous. however in the second case we obtain two possible answers that are directly paradoxical in quantitative terms!
This problem points directly to the missing (qualitative) dimensional aspect of interpretation which is developed through Type 2 Mathematics.
So from a Type 2 perspective 1^2 ≠ 1^1 (in qualitative terms). In this context each number as dimension refers to a unique logical manner of holistically interpreting mathematical relationships. And the structure of this logical system is inversely related to the corresponding root number of 1.
In the case of 1, the qualitative problem does not even arise as clearly the 1st root of 1 - if we are to conceive of such a thing is identical to 1 (raised to the power of ).
However for all other dimensional numbers ≠ 1, a unique qualitative logical system (for that number) will arise.
So where the dimensional number = 2 (as in our example) the structure of the logical system involved will be inversely related to the two roots of unity which are + 1 and - 1 respectively.
However whereas in quantitative terms, these roots are understood in linear (either/or) terms, in corresponding qualitative fashion they are given a circular (both/and) interpretation.
So the logical system of interpretation associated with the number 2 (as dimension) is based on the dynamic complementarity of opposites, i.e that involves opposite polarities of (unitary) form that are positive and negative with respect to each other.
Now when we try and explain the nature of such circular complementary interpretation in reduced linear terms, the opposite polarities split in either/or unambiguous terms. Therefore if one root is designated as + 1, then other root is - 1.
To see more clearly what is involved here I will illustrate with respect to the frequently used example of turns on a road.
As we know whether a turn on a road is designated as left or right, depends on the direction from which it is approached. So if a turn is designated as left, walking up the road, then it will appear right when approached from the opposite direction (walking down the road).
So if we designate a situation where a road can be approached from both and up and down directions, clearly any turn will thereby be both left and right. And this is precisely what the 2-dimensional logical system implies! However if we now unambiguously fix movement with just one direction, then each turn can be given an unambiguous answer. However in this situation there are two independent directions that can be taken i.e. either up or down the road.
So if we designate left as positive and a designated turn is indeed left when approached from walking up the road, then it corresponds with the positive root; however this inevitably implies that when taking the opposite direction down the road that we will encounter the negative of left (i.e. a right turn).
Thus in this reduced linear (1-dimensional) situation where we can take a direction up or down as independent, either a positive or negative answer can apply with respect to a specific turn.
However in the former circular (2-dimensional) situation where both up and down are simultaneously seen as interdependent, any turn - by definition - is both positive and negative.
Therefore it is only by introducing the neglected qualitative aspect of Mathematics (Type 2), can we resolve a fundamental logical inconsistency with respect to Type 1 Mathematics in showing clearly the relationship between a quantitative result (the two roots of 1) and its corresponding qualitative interpretation (based on 2 as dimensional number).