Now, once again, when appropriately understood, Type 2 (qualitative) is of equal importance to Type 1 (quantitative) Mathematics.
So from this perspective, I had barely scratched the surface of a uniquely distinctive form of Mathematics, which has a truly unlimited potential for further development.
However I recognised that the most comprehensive form of Mathematics (Type 3) would combine both Type 1 and Type 2 aspects in dynamic interaction with each other.
Indeed ultimately both the Type 1 and Type 2 aspects properly only achieve proper balance with respect to the integrated Type 3 approach (i.e. Radial Mathematics).
Therefore around the turn of the new millennium, I slowly began to attempt some initial attempts at developing a - most preliminary - Type 3 appreciation with respect to some key mathematical issues.
The first problem that I seriously tackled related to the famous Euler Identity, which is generally accepted as perhaps the most remarkable formula in all of Mathematics.
This is usually given as eiπ = – 1, which equally be expressed as,
e2iπ = 1 (which I refer to as the fundamental Euler Identity).
What initially attracted my attention is that the Euler Identity provides a direct connection with the (hidden) Type 2 aspect of the number system!
So once again I had defined the natural number system in terms of two aspects
1) where each (base) number, which varies, is defined in terms of a corresponding default dimensional number (or power) that remains fixed as 1,
11, 21, 31, 41,......
2) where each (dimensional) number, which varies, is defined in terms of a corresponding default base number that remains fixed as 1,
11, 12, 13, 14,......
As I devoted greater attention to the mysterious nature of the Euler Identity, I began to appreciate that it could not be properly understood in - mere - conventional mathematical terms,
Rather each of its symbols required to be given both an analytic (Type 1) and holistic (Type 2) interpretation.
Then when seen in this light, the Euler Identity provided a truly marvellous means of converting between both aspects of the number system.
Now e2iπ = 1 i.e. 11.
Therefore 11, 12, 13, 14,......have corresponding unique expressions in terms of e, as e2iπ, e4iπ, e6iπ, e8iπ, ......
In reduced quantitative terms,
e2iπ= e4iπ= e6iπ= e8iπ= 1
However from a qualitative perspective these represent uniquely distinct expressions relating to varying dimensions (or degrees) of (circular) interdependence.
Now, when we have one independent reference frame (as dimension), interdependence is thereby reduced to independence (as this independent polar frame is necessarily interdependent with itself!).
However with two poles (as dimensions) genuine interdependence arises from the relationship of these opposite poles.
Then as the number of reference frames increases, the corresponding level of interdependence likewise increases.
So the Type 2 aspect of number is properly geared to the holistic notion of natural numbers as representing increasing levels of interdependence (with respect to the corresponding increase of distinct reference frames involved). The reference frames in turn relate to varying interactions with respect to the fundamental poles which condition all experience i.e. external/internal and whole/part).
These are then intimately related to appropriate qualitative interpretation of numbers .
With just one number - say - 3, the qualitative nature of multiplication does not yet properly arise with the result strictly 1-dimensional (i.e. measured in linear units).
However with two numbers to be multiplied - say - 3 * 5 - , a true qualitative issue arises as we can see in geometrical terms that 3 * 5 now represents square (i.e. 2-dimensional) units. So a qualitative change in the nature of units has taken place with the result properly relating to 2-dimensional, rather than 1-dimensional format!
Now if we treat the numbers in solely independent terms then multiplication is not strictly separable from addition (representing just a short hand expression of addition).
So from this perspective 3 * 5 = 5 + 5 + 5 = 15 (i.e. 151).
However, this is not satisfactory, as a dimensional change has also clearly taken place in the units.
So what is involved - treating it geometrically as 3 rows of 5 - is that notions of number interdependence (as well as independence) are now required. Therefore the recognition of 5 items implies a recognition of the common identity of the individual units in each row, with each other (thereby enabling them to be placed in a mutual correspondence).
Thus properly understood, multiplication in this case entails both the notion of independence (as the initial 1st dimension) and the notion of interdependence (in the mutual identity of the 5 items in each row (representing the 2nd dimension).
Now if we extended further - say to the multiplication of 3 numbers - a further layer of interdependence would be involved in the recognition of the mutual identity of 2-dimensional rows and columns with each other (now representing the 3rd dimension).
So each further dimension entails a higher level of interdependence (to which the Type 2 system directly relates).
However the truly incredible feature of the Euler Identity is how it then facilitates (indirect) conversion from Type 2 (holistic) to Type 1 (analytic) interpretation.
Therefore to convert for example the simplest 2-dimensional case in a quantitative manner,
Here we set x2 = 11 and 12 respectively.
For 11 and 12 we use e2iπ and e4iπ.
Through De Moivre's Theorem, e2iπ = cos 2π + i sin 2π
So in the first case x = cos π + i sin π = cos 180 + i sin 180 = – 1
In the second case x = cos 2π + i sin 2π = + 1
Therefore, when expressed indirectly in Type 1 terms, the Type 2 notion of 1 (as representing the 1st dimension) remains unchanged as 1). This again means that that 1-dimensional understanding reduces qualitative to quantitative interpretation.
However the indirect quantitative expression of the 2nd dimension is – 1.
In qualitative terms, – 1 implies the (temporary) negation of posited, i.e. conscious, understanding (which provides the very means of moving from the appreciation of independence to the complementary notion of interdependence).
In fact this new appreciation of the Euler Identity was eventually to provide my appreciation of the vital role of the (unrecognised) Zeta 2 zeros in connection with the Riemann Hypothesis.