We have seen that the ordinal nature of number cannot be appropriately expressed within a merely quantitative (cardinal) approach.
And as cardinal, relating to independent and ordinal, relating to interdependent aspects respectively, dynamically interact in experience, the cardinal approach cannot likewise be appropriately expressed without implicit recognition of the qualitatively distinct ordinal aspect.
So we need therefore to redefine the number system in dynamic fashion, with two aspects (Type 1 and Type 2) that are quantitative and qualitative with respect to each other. These two aspects in turn thereby can give appropriate expression to the cardinal notion of number (as relatively independent of other numbers) and the corresponding ordinal notion (based on the relative interdependence of such numbers) respectively.
We have already looked briefly at the Type 1 aspect geared to cardinal interpretation in quantitative terms.
We have also addressed the crucial limitation with such interpretation as to why it cannot provide an adequate basis (without resolving to gross reductionism) regarding an ordinal ranking of numbers.
We have also looked at an important problem with respect to raising a number to another number (representing a dimensional power).
So once again for example, when we attempt to obtain the value of a simple number expression such as 2^2, strictly both a qualitative as well as quantitative change is involved.
Therefore when we express 2^2 = 4 (i.e. 4^1) we are simply reducing its result in a quantitative manner.
Thus to isolate the merely qualitative nature of number transformation, we thereby need to define a new Type 2 aspect (that is the direct inverse of the Type 1 cardinal variety).
With Type 1, when we refer to the natural numbers 1, 2, 3, 4,.... this implies that all these numbers are implicitly defined with respect to the dimensional power of 1.
So the Type 1 system (allowing for the relationship of two numbers which are quantitative and qualitative with respect to each other) is defined again as,
1^1, 2^1, 3^1, 4^1,.......
So in this case - properly understood in dynamic terms - if the base number is defined in quantitative terms, the correspoding dimensional number is of a qualitative nature.
However in Conventional Mathematics where this Type 1 aspect is treated in a static absolute fashion, both base and dimensional numbers are misleadingly defined in a merely quantitative manner.
Therefore the result of any expression relating a base number with its dimensional power is interpreted in a merely reduced quantitative manner.
So once again 2^2 in the conventional approach = 4 (i.e. 4^1) with the result interpreted in quantitative terms.
Therefore when we view the conventional approach (with its absolute number interpretation) from the more comprehensive dynamic perspective (entailing the dynamic interaction of two complementary number aspects), this conventional treatment reveals itself as the limiting special case, with 1 is the default dimension (representing the qualitative nature of number).
And the very basis of this conventional approach is that it is qualitatively defined - literally - in a linear (1-dimensional) fashion.
So Conventional Mathematics is accurately defined, from a holistic qualitative perspective, by this approach. And once again its very essence is that the qualitative aspect of number is thereby directly reduced in quantitative terms!
Now the Type 2 aspect (of the more comprehensive mathematical treatment) represents the direct complementary inverse of the Type 1 aspect.
So with respect to the natural numbers we again have,
1, 2, 3, 4,.....
However these now represent dimensional numbers, which vary with respect to a fixed number quantity as base i.e. 1.
Thus the natural number system is defined in the Type 2 approach as,
1^1, 1^2, 1^3, 1^4,....
Now, one can immediately note that from the (cardinal) quantitative perspective, such a number system seems trivial as the quantitative value of each number expression = 1.
However the very point of this second aspect is to indicate the ordinal (qualitative) rather than the cardinal (quantitative) nature of the number system.
So, to appreciate its ordinal nature, we need to convert from a linear (which befits the Type 1 aspect) to a corresponding circular approach to number, which serves as the true home of ordinal understanding.
Remember the qualitative aspect relates to a quality of interdependence as between numbers (thereby enabling a consistent set of ordinal rankings)!
Now in popular language the expression "a circle of friends" is commonly used. We also speak of political circles, sporting circles - even mathematical circles - as representative of communities of like-minded individuals that share a common interest.
Thus the very notion of a circle in popular language implies an interdependence of individuals thereby defining in an appropriate context an identifiable community with a shared interest .
Not surprisingly it is similar also in the context of number, where ordinal notions are approached in a circular fashion (thereby enabling their interdependent nature to be appreciated).
Quite simply to define this circular system with respect to any dimensional number power of 1 (i.e. n) we obtain the corresponding n roots of 1.
Therefore to define the qualitative nature of 2 (as a dimensional number) we thereby obtain the 2 roots of 1 (in quantitative terms).
Now the key to understanding this system is that the qualitative nature of the number (as dimension) and its corresponding roots (in quantitative terms) are dynamically related.
So for example in this case the qualitative nature of 2 (as a number) is intimately related to the two roots of 1 (i.e. + 1 and - 1) in quantitative terms. And in geometrical terms these are represented as two equidistant points on the unit circle in the complex plane!
However, the crucial point to appreciate is that whereas quantitative interpretation is in accordance with a linear (either/or) logic , the corresponding qualitative interpretation is in accordance with a circular (both/and) logic.
Thus whereas in linear terms + 1 and - 1 are understood as separate number quantities, from a circular perspective + 1 and - 1 are understood as complementary opposite poles of understanding (in qualitative terms).
We will develop this further in future blog entries.
However for the moment let us return to the problem that we faced with in the cardinal approach (according to Type 1 understanding).
So in Type 1 terms
2 = 1 + 1 (understood in a quantitative manner).
This left us with no means of making a qualitative distinction.
However now in ordinal terms the same two terms
= (+) 1 - 1.
Now interestingly when we add these two numbers the result is zero.
This directly implies that the ordinal (interdependent) nature of the relationship between terms strictly has no quantitative meaning!
And if take the n roots of 1 (where n is any natural number other than 1) again the sum = 0.
The one exception of course is where the dimension = 1
So when take the 1 root of 1 we obtain 1 (which does indeed have a cardinal meaning).
Once again this represents another way of stating that Conventional Mathematics is defined in qualitative terms by a linear (1-dimensional) logical approach, where qualitative notions are reduced to quantitative.
However once we appreciate that each number, representing a dimension, possesses a unique circular expression (through obtaining the corresponding roots of 1), this directly implies that potentially an unlimited set of qualitative interpretations can be given to mathematical symbols.
Thus the special case where the dimensional number = 1, accurately defines the approach of Conventional Mathematics (where qualitative is reduced to quantitative meaning).
Sadly - and so misleadingly - this one limited interpretation is widely assumed in our culture as synonymous with valid Mathematics.
However in truth a potentially unlimited set of other interpretations – all with a valid applicability in relative terms - exist.
And in all of these cases, quantitative is related to qualitative understanding in a dynamic interactive manner.
When the truth of this gradually begins to dawn, which I believe is inevitable at some stage, then we will begin the greatest revolution yet in mathematical history (with potentially vast ramifications for all of the sciences).