There is an additional point following on from yesterday's blog contribution.
I had mentioned - again from the Type 1 conventional mathematical perspective (which defines its quantitative nature) that Euler had proved that for all even values of s > 1, that the numerical result for the Zeta Function can be expressed in the form of k *(pi^s) where k represents a rational number (< 1).
So the remaining task is to demonstrate why this is equally true from a Type 2 qualitative perspective.
As we have seen there is an inverse relationship as between the the qualitative number s (representing dimensions) and its corresponding s roots (in quantitative terms).
Now where we have an even number of roots these can always be paired off in a complementary manner.
Geometrically this could be illustrated by plotting the s roots on the circle of unit radius (in the complex plane). Now, if from each point we draw a a diameter line it will connect with an opposite point on the circle. So for example in the case of the 6 roots of unity we would have three diameter lines with the coordinate points at the end of each line representing the same numbers with opposite signs (with respect to both real and imaginary parts).
So though the numbers can now be complex, they still are arranged in a complementary manner (with positive and negative poles).
Thus, they therefore represent an extension of the basic relationship of the circumference to its line diameter (with for each even value of s, s/2 complementary pairs existing).
Having said this, a special significance attaches to the complementary pairings of s (where s represents a power of 2). And with respect to such powers, 2, 4 and 8 are the most important.
So in the case of 2 we get positive and negative real poles.
With 4 we get positive and negative poles (in both real and imaginary directions). Then with 8 we get positive and negative poles in real, imaginary and complex directions (where the 4 complex cases have the special property of representing null lines).
So likewise in an inverse qualitative fashion (representing dimensions) when s is even, we again can arrange opposite poles in the complex plane in a complementary fashion (so that they all literally lie on diameter lines through the centre of the circle).
Thus the same basic relationship of circular to linear is maintained (which represents the qualitative definition of pi).
Therefore we can have a 2-dimensional interpretation of pi in qualitative terms (representing one pair with 2 opposite poles), a 4-dimensional interpretation of pi (representing two pairs, each with two poles), a 6-dimensional (representing 3 pairs. each with 2 poles) and so on.
But for those dimensions relating to powers of 2, a special significance attaches, with again 2, 4 and 8 the most important, as these fundamentally dictate the dynamic nature of all phenomenal relationships.
In this qualitative context, 2 represents the complementarity of opposite real polarities (such as external and internal); 4 represents the combined complementarity of both external and internal and also whole and part that are real and imaginary with respect to each other; 8 represents the more intricate combined complementarity of external and internal, whole and part, and form and emptiness in ways that are - relatively - real, imaginary and complex (as null lines) with respect to each other.
And once again remember we are now using all these mathematical notions in a Type 2 qualitative - rather than quantitative - sense.
Again the special significance of even dimensional values s (that are powers of 2) is demonstrated through the denominator of the rational value by which the relevant expression in pi is multiplied.
As we have seen in such cases the denominator can be expressed as the product (where the same value can be used more than once) of all primes from 2 to s + 1 (and only these primes). For all other even values the denominator can be expressed as the product of all primes from 3 to s + 1 (and only these primes).
So in qualitative terms a stronger relational aspect with respect to the primes attaches to denominator values associated with s (representing a power of 2).
I have mentioned before that Jungian Psychology eespecially lends itself to mathematical interpretation of a qualitative nature.
Jung drew attention to the role of mandalas (pictorial symbols) serving as key archetypes of integration (i.e. with a strong relational value).
Now, the most frequently used mandalas are based on 4-fold or 8-fold ornate pictorial patterns, which are based on the geometrical structure of 4 and 8 (as dimensions) resprectively.
Both 4 and 8 are rich enough to include both real and imaginary aspects (which in qualitative terms would represent both conscious and unconscious). A 2-fold pattern however would be confined to real aspects (which therefore would not adequately include the unconscious).
So the mandalas therefore can be seen in psychological terms as expressing the harmonious integration of conscious and unconscious aspects of the personality. And the most popular forms are based on the use of 4 and 8 as numbers (understood in a qualitative dimensional context).
When correctly interpreted from a Type 3 mathematical perspective, all numbers are subject to the Uncertainty Principle.
We demonstrated this already in relation to pi. One can perhaps appreciate, because of the nature of pi as a transcendental number, how such undertainty might apply.
However, strictly speaking it is in fact true of all numbers (in Type 3 terms). So even with the individual prime numbers, a necessary uncertainty attaches to their nature in quantitative terms.
This is due - as always - to the fact that at the Type 3 level, every mathematical symbol represents a dynamic interaction of both quantitative and qualitative aspects in its inherent identity.
So again a number e.g. an individual prime is properly defined in terms of such a 2-way interaction. Therefore when we attempt to abstract the quantitative aspect it ncessarily has an approximate - merely relative - status.
Thus to illustrate, the (absolute) identity of 5 (as representative of an individual prime number) properly relates in Type 3 terms, to the dynamic interaction of both its quantitative and qualitative aspects.
So when we try to look at 5 in merely quantitative terms, we meet the old problem that the finite number system can never be fully determined in a actual manner. Once again the potential infinite nature of number properly relates to the qualitative aspect, which cannot be reduced in a finite actual manner!
Now an individual number properly only has meaning in relation to the overall concept of number as infinite (which is always to a degree indeterminate in finite actual terms).
So in this sense as we are understanding 5 in relation to what is ultimately of a finite indeterminate nature, its value is likewise approximate. Or to put it another way, though we define 5 as a number, we cannot properly define what number is (in an actual - merely - quantitative manner).
Once again there are great similarities here with the nature of Quantum Mechanics. Though quantum mechanical efefcts are most easily demonstrated at the level of sub-atomic matter, strictly speaking - as is now accepted in Physics - all objects are subject to the same effects.
However because these effects are scarcely noticable at the macro level of everyday experience, they can be ignored for all practical purposes (where the unambiguous Newtonian viewpoint serves as an excellent approximation).
And it is the very same with number (and indeed all mathematical relationships). The apparent absolute nature of a number such as 5, in strict terms represents an approximation (which however can be effectively ignored in the context of Type 1 Mathematics).
So the relationship of Type 1 to Type 3 Mathematics is somewhat similar to the corresponding relationship as between Newtonian Physics and Quantum Mechanics.
In truth the approximate nature of quantum mechanical relationships is itself rooted in the more fundamental uncertainty that attaches to all mathematical relationships.