In the Riemann Zeta Function, values of the function are provided for s (i.e. the power or dimensional values to which the function is defined) even though such values are often meaningless from a conventional perspective.

For example when the value of s = 0 the Riemann Zeta Function = 1 + 1 + 1 + ... which of course diverges to infinity (as conventionally understood).

However it is possible to give the function a value in the following manner.

The Zeta Function is defined as

1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......

Now if we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms

1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......

Now through dividing each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.

This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta = Eta Function divided by {1 -1/{2^(s - 1)}}

When s = 0 the Eta function results in the alternating sequence of terms

1 - 1 + 1 - 1 + 1 - ...

The sum of this sequence does not properly converge in conventional terms.

When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can derive a single answer = 1/2.

And then from this Eta value the corresponding Zeta value can be easily calculated = - 1/2.

However the qualitative problem of explaining why the Zeta Function now has a simple finite value, when in conventional terms it diverges to infinity, needs to be explained.

In general terms a key problem generally involved with domain stretching is that finite and infinite notions are mixed indiscriminately. This is properly associated therefore with a qualitative change in the nature of interpretation involved (which however due to the reduced nature of Type 1 Mathematics is overlooked).

Once again in conventional terms numerical understanding is based on 1-dimensional linear interpretation.

Now I already have defined 2-dimensional logical interpretation as involving the complementarity of opposites which is defined in holistic terms as + 1 - 1 (taken as a complementary pair).

In corresponding quantitative fashion when we take the terms in our sequence as complementary pairs we obtain the sum of 0.

However when we take the sum of terms in a single fashion (thereby using an odd number) we obtain the sum of 1.

Therefore in qualitative terms we would explain the resulting average of 1/2 in qualitative terms as resulting from the balanced mix of both 1-dimensional (linear) and 2-dimensional (circular) interpretation.

Now remember once again in conventional terms quantitative calculations that seem to make intuitive sense are always defined by a merely linear qualitative interpretation.

So the key factor in now explaining why we can come up with this non-intuitive value for the Riemann Zeta Function where s = 0 is precisely because it actually involves in qualitative terms both 1-dimensional and 2-dimensional interpretation.

And as the Riemann Transformation formula establishes important links as between values of the Function with conventional and non-conventional numerical values, we cannot possibly hope to understand the proper nature of the Function in the absence of corresponding qualitative interpretation.

Indeed ultimately this is what the Riemann Hypothesis is all about i.e. establishing a key condition for consistency with respect to both quantitative and qualitative type mathematical interpretation.

In other words it establishes the key condition for consistency as between both Type 1 and Type 2 Mathematics which can be seen therefore as the fundamental axiom necessary for Type 3 Mathematics (which is the most comprehensive of all Types where both quantitative and qualitative aspects dynamically interact).

A fascinating further example of this qualitative issue can be given with reference to Fibonacci type sequences.

For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.

What we do here is to start with 0 and 1 and then combine the second term (* 1) with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.

Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).

Now the equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.

Noe this corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.

So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....

What is interesting here is that we cannot approximate the value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.

However we can obtain the value directly through concentrating on the ratios of terms (occurring as each second term in sequence). In this we get either 1/1 or 0/0.

Now the first would give us the conventional rational quantitative interpretation. However the second actually corresponds to the qualitative holistic relationship.

So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Once again because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.

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