It might helpful to clarify here in greater detail the precise distinction as between the quantitative and qualitative interpretation of numbers respectively.
Our starting point - though ultimately it is not quite so simple - is to consider with respect to any expression (comprising both base and dimensional numbers) the base number as quantitative and the dimensional number as qualitative respectively.
So in this context therefore, when we take the number expression 3^2 for example, the (base) number 3 refers to a quantitative and the (dimensional) number 2 - relatively - to a qualitative meaning.
Now this interpretation of quantitative and qualitative bears a very close relationship in turn with the corresponding interpretation of cardinal and ordinal.
So when we consider 3 as a number quantity, we have the cardinal use of number in mind. 3 here conveys the notion of multiple homogeneous units. So when we say 1 + 1 + 1 = 3, the very point is that no qualitative distinction is made as between the units, thereby enabling them to be treated in a merely quantitative manner.
However when we consider by contrast 3 as a (dimensional) number quality, we have the ordinal use of number in mind. So for example, I spent some time in the context of the Riemann Zeta Function, clarifying the qualitative nature of the 2nd dimension, and then later the 4th dimension. Therefore the nos. 2 and 4 in this ordinal sense i.e. as representing the 2nd and 4th dimensions respectively, clearly should not be confused with their cardinal use.
And then it becomes quickly apparent that ordinal always to a degree implies cardinal (and cardinal, ordinal) suggesting strongly therefore, that numbers in dynamic interactive terms, necessarily keep switching as between cardinal and ordinal meaning.
In truth a high degree of subtlety is required to properly clarify, in any relevant context, the precise relationship as between quantitative (cardinal) and qualitative (ordinal) interpretation. And this is what defines the Type 3 mathematical approach.
However, again because of the explicit lack of any qualitative dimension, Conventional (Type 1) Mathematics must necessarily confuse ordinal with cardinal meaning.
I was already aware of an important example of this key issue from childhood. Even then it struck me as distinctly odd that the two roots of 1 could be expressed as + 1 and - 1.
I tended to always look at Mathematics in a holistic manner. So for example I reckoned that from a qualitative perspective, mathematicians would not accept that the truth value of a proposition could be true and false at the same time. For this would be tantamount for example to accepting that the Riemann Hypothesis could be proved true one day and then equally proved false the next.
So Conventional Mathematics is strongly based on the premise as regards proof that the positive pole necessarily excludes its negative (and vice versa)!
However yet in the parallel world of quantitative truth, in the simple case of the square root of 1, the positive includes the negative (and the negative the positive).
So again even from a young age, I could see that here was a fundamental issue of great importance that pointed to a glaring inconsistency. For given that Mathematics is defined in a merely quantitative manner, there should thereby be an automatic correspondence regarding the logical approach with respect to the behaviour of specific mathematical quantities on the one hand and generalised mathematical proof on the other!
Now the key to unlocking this problem is the recognition that when we refer, for example, to the two roots of 1, that both the cardinal and ordinal use of number are necessarily involved.
So the 2 roots (in cardinal terms) relate to both the 1st and 2nd roots (from an ordinal perspective).
Therefore the key to dealing with the matter consistently is to recognise that the qualitative aspect of the number system is also required and that the 1st and 2nd roots (understood in quantitative manner) have complementary qualitative interpretations as the 1st and 2nd dimensions of 1.
So the 1st dimension, is written as 1^1 and the 2nd dimension as 1^2.
Then the 1st answer + 1 (for the two roots) properly represents the square root of 1^2 (i.e. 1^1) whereas the second root - 1, properly represents the square root of 1^1 i.e. 1^(1/2)!
So we can now see clearly that an important qualitative distinction actually applies to the two roots, with the positive root defined with respect to the 2nd dimension and the negative with respect to the 1st.
We could of course express these roots as simple equations.
+ 1 is the solution to x2 (i.e. x squared) = 1^2.
- 1 is the solution to x2 = 1^1.
So the first root results directly in the customary linear notion of 1 (i.e. the 1st root of 1). Not surprisingly this is taken as the principle root in Conventional Mathematics.
However from a qualitative perspective the true square root of 1 relates to its second root i.e. - 1.
And though these roots do indeed have a quantitative meaning, properly they cannot be divorced from corresponding qualitative interpretation. And to obtain the appropriate qualitative dimension that can provide such interpretation, we find the reciprocal of the dimensional number used to obtain the root.
Now for the 1st root this is trivial as the reciprocal of 1 is 1. So the positive root in quantitative terms, corresponds directly with linear (1-dimensional) interpretation.
However the 2nd root, represented by 1/2 (as dimensional power of 1 in quantitative terms) is thereby correctly interpreted in a qualitative manner as 1^2. This implies that the 2nd dimension - in an ordinal sense - is required to properly interpret any expression with the dimensional power of 1/2 (in quantitative terms). And all of this is deeply central to proper interpretation of the Riemann Zeta Function (where s as a dimensional number, can assume any value in the complex plane)!
There are however subtle difficulties regarding the precise definition of an ordinal number.
So when I defined the ordinal dimension 2 and then later the ordinal dimension 4, these were tied up with the cardinal numbers 2 and 4 respectively.
In other words, we cannot define the ordinal number 2 in the absence of the cardinal number 2. So we require 2 in cardinal terms to give a 2nd unit meaning (from an ordinal perspective).
Likewise we cannot define the ordinal number 4 (which we also dealt with in earlier blog entries), in the absence of the cardinal number 4. So the 1st time we encounter 2 in an ordinal sense is with respect to 2 (in cardinal terms) and the first time again we encounter 4 in an ordinal sense is with respect to the cardinal number 4!
And this is precisely the context in which we thereby give meaning to 2 and 4 (as ordinal numbers).
However, we also could give a potentially unlimited meaning to the 4th dimension with respect to cardinal numbers 5, 6, 7....and do on. However these would not be written as integer values.
To illustrate let us briefly deal with the 4 roots of 1
i relates to the 4th root of 1^1 i.e. 1^(1/4) i.e. the true 4th root.
- 1 relates to the 4th root of 1^2 i.e. 1^(2/4) = 1^(1/2)
- i relates to the 4th root of 1^3 i.e. 1^(3/4)
+ 1 relates to the 4th root of 1^4 i.e 1^(4/4) = 1^1.
Now the corresponding dimensional numbers with which these are qualitatively associated, are 4/1 = 4, 4/2 = 2, 4/3, and 4/4 = 1.
So interestingly when we switch from quantitative to qualitative interpretation, the order sequence is reversed, so for example the 1st root is associated with the last dimension!
However the 3rd dimension in the context of 4 (as cardinal number) is not an integer! And this is because the integer value only applies when both cardinal number and ordinal ranking coincide!
Then, as we have been repeatedly saying with Type 3 interpretation, it can keep alternating as between both the quantitative and qualitative meanings associated with each number!
All of this is potentially extremely rich territory for further investigation. And again because Type 1 Mathematics has no qualitative dimension, not alone can these issues not be properly interpreted, they cannot even be properly approached from that limited perspective!.
The very clue to the interaction of quantitative and qualitative with respect to the roots of unity, is provided through their geometrical representation. Now, quantitative always implies linear and qualitative circular, respectively!
And this is perfectly mirrored in geometrical terms. Here the coordinates of each root of 1 (in the complex plane) simultaneously lie on the extremity of a radial line (drawn from the centre of the circle) which also lie on the circular circumference. So at this point of intersection, both line and circular circumference coincide!
In like manner, corresponding qualitative interpretation of such roots (where the root bears an inverse relationship with its dimensional number), always entails the intersection of linear (quantitative) with circular (qualitative) type interpretation where both ultimately coincide. And this ultimately can only be dealt with satisfactorily in a comprehensive Type 3 mathematical approach.
An appropriate Type 1 approach deals with the quantitative aspect in relative isolation; an appropriate Type 2 approach then deals with the qualitative aspect in relative isolation; finally the comprehensive Type 3 approach deals with both aspects, as relatively independent and interdependent respectively, in a dynamic interactive manner.
In this context, not alone is the conventional mathematical approach confined to the Type 1 approach, but unfortunately in a manner that greatly hinders access to the other two approaches. This is due to the absolute nature of its assumptions (which do not accord with authentic experience of mathematical reality).
So even with respect to specialisation in Type 1 Mathematics, the appropriate explanatory manner properly should recognise the important role of the other approaches. And this patently is not the case at present!