(Though this extract is very lengthy, it is perhaps the most important I have yet written. It illustrates with respect to the new Riemann Zeta Function for s = 4), the enlarged three strand approach outlined in the last blog (indicating how Type 1, Type 2 and Type 3 Mathematics are applied in this context).
We will continue on now with our examination of the Zeta Function for real dimensional values of s > 1. And for the moment, we are concentrating on the even integer values of s.
In a previous blog we looked at the Function where s = 2.
So we will know move on to examine the Function with respect to the next even number i.e. 4.
Now in attempting this task, I will deliberately illustrate the three strand approach - outlined in yesterday's blog - that is necessary to give a comprehensive interpretation of what is involved.
So we will look first at the Function with respect to s = 4, from the standard Type 1 perspective based merely on quantitative type interpretation.
We will then look at Function with respect to the same value of s = 4, from the holistic Type 2 perspective based on corresponding qualitative interpretation.
Finally we will look at the Function again with respect to the value s = 4, from the more dynamically interactive Type 3 perspective (where both quantitative and qualitative aspects are inherently combined as interdependent).
The first task is the easiest, as extensive work has already been done with respect to the quantitative aspects of the Function.
So when 4 is used as dimensional number (or exponent) in this context the standard linear notion - in qualitative terms - applies. This then enables the dimensional transformation of the terms of the Zeta Function to be interpreted merely in a (reduced) quantitative fashion i.e. in the way that a power of a number is normally understood in Mathematics.
So as Euler discovered the quantitative value of the Function (when s = 4) is (pi^4)/90.
Of course Euler discovered a lot more and was able to prove that for all (positive) even integer values of s, the quantitative result of the Function can be expressed in terms of an expression of the form k*(pi^s) where k always in the form of a rational number.
I have also drawn attention in this context - though I imagine it is already well known - to the fact that the denominator of k bears a special relationship to the prime numbers. So in brief, when s represents a power of 2, the denominator seemingly will always entail an expression representing the product of all prime numbers up to and including s + 1 (and only these primes).
In all other cases where s is even (but not a power of 2), the denominator will entail an expression representing the product of all prime numbers from 3 up to and including s + 1 (and only these primes).
Now I am using these features here to illustrate the - already accepted - Type 1 strand.
And in conclusion, a key important fact to bear in mind (especially when considering the alternative Type 2 interpretation), is that in qualitative terms, Conventional Type 1 Mathematics represents a linear (1-dimensional) method of interpretation i.e. where s (in qualitative terms) = 1.
We now move on to the more difficult task of providing a corresponding Type 2 interpretation in holistic qualitative terms for each relationship (already dealt with in a Type 1 quantitative manner).
Now, as we have seen in Type 1 terms, when 4 is used as a dimensional number, it is from a merely reduced quantitative perspective (where in qualitative terms 1 as dimension applies).
This can be easily illustrated. To make it simple, let's look at the expression 2^4 (in quantitative terms. Now the answer here is of course 16 (which written more completely is 16^1). This must necessarily be the case with any such quantity, as clearly a different value would result if the dimensional number ≠ 1. So numbers in quantitative terms are ultimately expressed with respect to a default dimension of 1.
So in this quantitative system 1, 2, 3, 4,... are more fully expressed as
1^1, 2^1, 3^1, 4^1....
However when we move on to the true qualitative circular notion of dimension, in reverse terms, the dimensional number is defined with respect to a fixed base quantity.
So in the qualitative system (where numbers represent dimensional qualities),
1, 2, 3, 4.... are more fully expressed as:
1^1, 1^2, 1^3, 1^4,...
As I have stated previously, a complementary inverse relationship exists as between the dimensional numbers and corresponding roots.
I have already explained the key reason for this elsewhere in explaining the fascinating transition from a whole to a part number.
In quantitative terms 4 and 1/4 for example bear an inverse relationship to each other. Now 1/4 can be written as 4^(- 1).
So, in corresponding qualitative terms 4 and 1/4 bear an inverse relationship as dimensional numbers where 4 can be equally written as 4 ^(- 1). Now - 1 in this context relates - literally - to the dynamic negation of (conscious) rational understanding, which results in (unconscious) intuitive appreciation. This then can indirectly be given a circular rational interpretation.
Then in the second system 1/4 is more fully expressed as 1^(1/4) which then gives us a quantitative value, as one of the 4 roots of 1, lying on the circle of unit radius (in the complex plane). So all 4 roots (corresponding to 1^1, 1^2, 1^3 and 1^4), each raised to the power of 1/4, lie as equidistant points on the same circle.
Likewise the four dimensions of 1 in inverse terms lie on the same circle in the complex plane, but now interpreted in a qualitative manner.
Structurally, these four dimensions are therefore identical with their corresponding quantitative roots i.e. 1, - 1, i and - i (but now given a qualitative rather than quantitative interpretation).
So in circular qualitative terms, we express their significance in the Zeta Function for s = 4, by interpreting the qualitative nature of 1, -1, i and - i.
Whereas in the case where s = 2, we could confine ourselves to one complementary pairing with positive (+) and negative (-) polarities, here we have two such complementary pairings, with one relating to real and the other to imaginary poles respectively.
And may I say right away, that interpretation of 4 as dimension is extraordinarily important. After all we are accustomed to defining our world in 4-dimensional terms. In Physics such dimensions are given a mere quantitative explanation! However the qualitative correspondent in truth is equally - if not more - important.
Basically this 4-dimensional perspective, in qualitative holistic terms, relates to the fact that all reality (including of course mathematical reality) is conditioned by two fundamental pairs of polar opposites.
One of these relates to the inevitable fact that all external objects in experience necessarily relate to an internal interpretation (that - relatively - is subjective in nature). So strictly speaking from this perspective, the notion of "abstract objects" in Mathematics is meaningless.
The reason why they do indeed appear abstract is itself very interesting in that it reflects the attempt to view them in absolute terms as if independent of interpretation. Or to put in another way, as repeatedly stated in these blogs, it represents the attempt to view the quantitative as totally separate from the qualitative (which is ultimately a completely untenable position).
The other equally important pairing relates to that as between whole and part.
Again strictly speaking, it is impossible in dynamic experiential terms to have wholes (without parts) or parts (without wholes). Again because of the great lack of this dynamic perspective, the very nature of interdependence is thereby fundamentally misrepresented within Mathematics.
Now true interdependence necessarily requires - at a minimum - two complementary poles. So from this perspective, obviously we cannot have quantitative without qualitative (or qualitative without quantitative). Yet when it comes to the key issue of trying to understand the two-way relationship as between individual primes and their collective behaviour, conventionally in Mathematics, this is merely attempted from within a quantitative framework.
You know, it truly amazes me that such an enormous amount of energy, requiring ever more specialised techniques, has gone into so many attempts to prove the Riemann Hypothesis. However the basic application of a little bit of holistic interpretation would quickly indicate why the Hypothesis is not capable of proof in the first place!
Sadly however, a continued attempt has been made to exclude completely the qualitative dimension - in formal terms - from mathematical research. This unfortunately has led to a situation where specialists continually miss the obvious, due simply to not thinking in a holistic (i.e. qualitative) manner!
So interpretation of 4 (as dimensional number) in wany ways can be seen as an extension of that for 2. We now attempt to relate complementary opposites with respect to both external and internal (and internal and external) polarities and also whole and part (and part and whole) polarities.
It then remains to be explained as to what the imaginary polarities relate.
Well, we have mentioned before that inherent is all conscious recognition is an - often unrecognised - unconscious aspect. Therefore actual experience is always working at two levels, where local objects in many ways serve an unconscious holistic purpose. And of course this is likewise true of mathematical objects when understood in a dynamic manner.
Ultimately this points to the fact that both conscious and unconscious necessarily interact with each other in all mathematical experience.
The conscious rational aspect of such understanding is "real", whereas the unconscious aspect is - relatively - "imaginary". I have indicated before the precise nature of this imaginary notion. So - 1 in qualitative terms, relates to the (unconscious) negation of unitary form (and in dynamic terms combines both positive and negative poles). So to express this indirectly in a unipolar manner, we
need the qualitative equivalent of a square root. So as this corresponds to the square root of - 1, i.e. i, the result is imaginary (in a precise holistic mathematical fashion).
The imaginary in this sense bears a very close relationship with circular understanding, where it is indirectly conveyed through linear type symbols.
And this is precisely what I am doing here. So I am illustrating the second strand of the Riemann Zeta function as an attempt to convey the holistic qualitative significance of mathematical symbols (whhich are customarily used in a linear manner).
So again in very precise holistic mathematical sense Type 2 represents the "imaginary" aspect of mathematical understanding (in qualitative terms).
Now, as we know, the Riemann Zeta Function is defined with respect to the complex plane in quantitative terms (entailing both real and imaginary aspects).
Then Type 1 Mathematics attempts to understand the Function employing merely the real aspect (of qualitative interpretation).
However it requires a complex approach (in qualitative terms) to properly interpret complex relationships (relating to the quantitative perspective). It another words it needs both Type 1 (real) and Type 2 (imaginary) aspects. And once again the Type 2 aspect is entirely missing in the conventional approach!
So I have spent some time in explaining s = 4 (in qualitative terms) though it would take a long book - indeed several long books - to enlarge properly on its great significance.
We now look at the result of the Function for s = 4 i.e. (pi^4)/90.
What is required here in qualitative terms is to give a holistic mathematical explanation as to why the resulting expression has this precise numerical structure.
Now, once again this represents an extension of where s = 2. We saw then that in qualitative terms pi represents the (pure) relationship as between circular appreciation (where the two way interdependence of opposite poles is recognised) and linear understanding (where initially each pole is interpreted within one isolated pole of reference).
So it is is similar here except that a more refined appreciation is required where linear understanding with respect to four separate poles takes place initially before the combined (4-way interdependece) of all poles together can be holistically appreciated in a circular manner.
Thus, in geometric terms we can imagine this in terms of a circle divided into four quadrants with the horizontal line through the centre repesenting the real and the vertical vertical line representing the imaginary poles! Also there is the added requirement here that we can equally recognise that understanding in relative terms, keeps switching as between the "real" poles as representative of conscious recognition on the one hand, and the corresponding "imaginary" poles as the representative of unconscious appreciation (indirectly expressed through conscious symbols).
Now the capacity for such understanding has not been even remotely developed in our culture. It would be characteristic in practice of some who have attained to genuine contemplative awareness, though even here it is highly unlikely that they would translate such experience in a qualitative mathematical manner.
However my true intention is to show - quite literally - that innumerable higher dimensions of understanding have yet to be attained and also that Type 1 Mathematics in still being confined to 1 as dimension, is thereby operating at the "lowest" possible dimension!
With respect to the second strand of Type 2 understanding, we can even suggest a holistic explanation for the interesting prime number pattern in the denominator (which we mentioned in the Type 1 case).
We have commented before on the two two extreme capacities of primes. The individual primes are unique (in that they have no factors). However the relational (collective) aspect of the primes is due to the fact that every natural number can be uniquely expressed as the product of prime factors.
So we could look at the individual aspect as the linear capacity and the relational aspect as the circular aspect of the primes respectively.
I have also mentioned before that when we switch the linear dimensional number i.e. 1 to - 1, we thereby switch from linear to circular format with respect to the primes.
This can even be illustrated simply in quantitative terms. For example if we raise 7 to (- 1) we obtain .142857 where the 6 digits continually recur. This is perhaps the best known of the cyclic primes and possesses many amazing circular type properties.
Now when we look at the at (pi^4)/90, the rational part is 1/90 i.e. 90^(-1).
This would thereby suggest a circular (i.e relational) pattern to the primes and in this case (and in all subsequent cases corresponding to even values of s > 4 that are likewise powers of 2). So indeed here we have the most ordered arrangement possible i.e. where all the primes are included as factors from 2 to s + 1 (and only these primes).
The final strand represents explanation of how the combined interaction of both quantitative and qualitative aspects takes place (in Type 3 terms).
Though we can only scratch the surface as it were here I will try and convey something of what is involved.
Again we start with s = 4. Now we have already explained the nature of this dimensional number - in a relative separate manner - with respect to both its Type 1 (quantitative) and Type 2 (qualitative) significance.
The implication of Type 3 understanding is that we now simultaneously can recognise the number with respect to both its quantitative and qualitative aspects (thereby enabling ready switching as between both types of meaning). So at one moment, we recognise its quantitative significance (in Type 1 terms) and then where holistic appreciation is required, its alternative qualitative significance (in Type 2 terms).
Now with mental and spiritual capacities well developed, one could indeed understand mathematical relationships in a comprehensive (radial) type manner, thereby combining considerable quantitative rigour with a greatly enhanced quality of holistic intuitive awareness.
And as I have repeatedly stated, Mathematics as a speciality is defined in an extremely narrow manner by the profession. So, I would see its manifest lack of holistic awareness in tackling problems (such as the Riemann Hypothesis) as its greatest limitation.
The result of the Function for s = 4 i.e. (pi^4)/90 would also be interpreted in a distinctive manner from the more advanced Type 3 perspective.
The expression now is not seen in either a quantitative or qualitative manner (as separate) but rather as combining both aspects in a dynamically interactive manner.
And as already suggested this leads to the need for a considerable refinement with respect to both quantitative (and qualitative) interpretation.
When one looks at (pi^4)/90 from a Type 1 perspective, its value is interpreted in a merely quantitative manner.
So, in maintaining that the expression represents a constant value, one understands this to unambiguously mean its quantitative value.
However in the light of Type 3 interpretation, this is now seen to be strictly in error.
The constant value here relates to a combined interaction (of both quantitative and qualitative aspects) which is ineffable. In other words its constant nature cannot be expressed in either a quantitative or qualitative manner (as separate). And as all phenomenal understanding requires a degree of separation of both aspects, this entails that its true (constant) value must necessarily remain ineffable.
Thus when we try to actualise the value of the expression (pi^4)/90 i.e. 1.082323233..., an indeterminacy necessarily operates (with respect to its true unknowable value). So as we keep attempting to achieve greater accuracy in our finite approximation, the number (from this relative quantitative perspective) necessarily keeps changing!
So the Uncertainty Principle here strictly applies with respect to the number expression (pi^4)/90 in an actual finite manner. So to use an analogy that may be helpful, we are here dealing with the quantum mechanical properties of number!
Now we already witnessed a similar situation with respect to (pi/2)/6 where the number likewise keeps changing.
Also, the qualitative aspect of interpretation keeps changing. In general terms, the reason for such quantitative uncertainty still relates to the dynamic interaction of quantitative and qualitative aspects. These can be appreciated as representing in more refined terms both aspects (understood now in both a real and qualitative manner).
Finally even with respect to the denominator of (pi/2)/90, the two aspects can now be seen in a more synchronous manner i.e. with respect to the quantitative nature of prime numbers involved and the qualitative appreciation of why they are related in this manner.
So there is a lot to digest here. It is not necessary to understand every detail. What is important at this stage is the general appreciation that this represents (in embryonic form) the nature of an altogether more comprehensive approach to Mathematics, which has the intention of combining both the quantitative and qualitative aspects of Mathematics in a balanced harmonious fashion.
I have no doubt that such an integrated approach has the capacity in time to completely transform the world we live in.