I have stated before on many occasions how - properly understood - there are two aspects of the number system that are in dynamic interaction with each other.
Once again, I refer to these two aspects as Type 1 and Type 2 respectively.
The Type 1 aspect is the standard approach suited directly to the treatment of the cardinal aspect of number where each number is defined with respect to the default dimensional value of 1.
So the natural number system from this perspective is:
1^1, 2^1, 3^1, 4^1,....
The Type 2 approach is the alternative - largely unrecognised - approach, suited directly to the treatment of the ordinal aspect of number where each number (representing a dimensional power) is defined with respect to the default base value of 1.
So the natural number system from this perspective is:
1^1, 1^2, 1^3, 1^4,.....
In the actual experience of number a continual shifting takes place as between each number with respect to both its base and dimensional definition.
So for example the number 2 does not have one unambiguous fixed meaning, but rather continually alternates as between its base number appreciation (as defined in Type 1 terms) and its corresponding dimensional number appreciation (defined in corresponding Type 2 terms).
And quite simply it is this alternative switching of meanings that enables both the appreciation of the cardinal and ordinal aspects of number respectively to take place.
Therefore conventional mathematical interpretation suffers from a gross form of reductionism in the manner in which it attempts to derive ordinal directly from mere cardinal type interpretation!
When one recognises these two aspects of number, i.e. Type 1 and Type 2 as relatively distinct, the question the arises as to their consistent use in terms of each other.
Put another way, we then need to find an indirect means of translating (or converting) the Type 1 aspect in terms of the Type 2, and equally from the opposite perspective the Type 2 in terms of the Type 1.
In fact I have repeatedly stated in these blog entries that ultimately the Riemann Hypothesis serves as the basic requirement for the successful reconciliation, throughout the number system, of both types of meaning.
Up to this I have largely concentrated on the task of converting numbers in the Type 2 system indirectly in Type 1 terms.
So for example to convert the simplest (non-trivial) case i.e. 1^2 in Type 1 terms, we in fact express 1 with respect to its reciprocal i.e. 1^(1/2).
So the solution here is obtained through the equation x^2 = 1, so that x = – 1.
Now this number (indirectly representing a quantitative value) lies on the circle of unit radius in the complex plane.
So the very essence of the Type 2 aspect of number is that it relates inherently to a circular - rather than linear - type understanding.
In other words, whereas the cardinal notion of number is based on the independent identity of each number (which befits linear interpretation), by contrast the ordinal notion relates directly to an interdependent identity i.e. relationship between members of a number group (which befits circular interpretation).
So when use refer to the number 2 for example in a cardinal sense, we give it an (isolated) independent identity. However the corresponding ordinal meaning of 2nd can only have meaning through the interdependent relationship of the members of a number group. So when we have two members in a group the notion of 2nd always implies the relationship with another member (which in this context is designated as 1st).
So circular understanding (of a qualitative nature) in the context of two members, implies the dynamic notion of the complementarity of opposites. So if one member is posited in any context as the 1st, this automatically implies the corresponding negation of the other member which is thereby 2nd.
And the quantitative representation of this complementary type understanding is given through the two roots of 1, i.e. strictly the two roots of 1^2 and 1^1 respectively.
So the 1st member is thereby represented as + 1 , and the 2nd as – 1 respectively.
So we can seen perhaps in this example how the very significance of the (indirect) Type 1 translation of the Type 2 aspect of the number system is that it enables us to express ordinal type notions, expressing qualitative notions of number interdependence, in an indirect quantitative manner (with respect to a circular number scale).
However we also have the opposite problem of successfully converting number defined with respect to the Type 1 aspect, indirectly in a Type 2 manner.
So if a n is a cardinal number in the Type 1 aspect, representing the base (defined with respect to the default dimensional value of 1), then we need to express it indirectly in Type 2 terms (defined with respect to a default base value of 1).
So in general terms n^1 = 1^x
Therefore taking natural logs on both sides
Log n = x * log 1.
Thus x = log n/ log 1
= log n/2iπ.
So therefore once again, in the simplest case, to convert the number 2 i.e. where n = 2 (now defined with respect to the Type 1 aspect) in Type 2 terms as x,
x = log 2/2iπ = – (log 2/2π) i
As we have seen in our earlier conversion (from Type 2 to Type 1) there is a close connection as between the number 2 and its reciprocal 1/2 (in qualitative and quantitative terms).
It is quite similar in terms of this latter conversion (from Type 1 to Type 2).
log 1/2 = – log 2.
Therefore when n = 1/2
x = (log 2/2π) i.
We thus have the fascinating conclusion that - by definition - all numbers, initially defined in Type 1 terms, with respect to the real number line, are correspondingly defined in Type 2 terms as numbers lying on the imaginary number line (both positive and negative directions).