Monday, November 15, 2010

Holistic Mathematical Connections

As we have seen when s is a positive even integer, the value of the Euler Zeta function can be expressed in terms of an expression comprising (Pi^s) * k (where k represents a rational number).

Now what I am concerned with here is to give a holistic mathematical interpretation of the significance of this result.

To do this it is easiest to concentrate initially on the simplest - and best known - case where s = 2.

As we have seen here the value of the Zeta function = (Pi^2)/6

As stated repeatedly, in holistic mathematical terms, standard conventional interpretation is qualitatively of a linear nature (i.e. 1-dimensional).

What this means in effect is that the polarities of experience are inherently separated giving just one positive (unambiguous) direction.

So for example all understanding - including of course mathematical - entails a dynamic interaction of both external (objective) and internal (subjective) aspects.
However in mathematical interpretation these are clearly separated. So mathematical truth is given just one direction i.e. dimension as objective (with which psychological mental constructs are thereby assumed to statically correspond).

So linear logic is decidedly of the unambiguous either/or variety!

However corresponding to all other dimensions is a unique qualitative mode of interpretation and the simplest of these is 2-dimensional.

So with 2-dimensional interpretation, two polar directions are always given (which dynamically interact in a paradoxical manner).

So for example interpretation is not identified with either the external or internal pole of understanding (as separate) but rather with their mutual interdependence. This then leads to an alternative circular mode of both/and logic that is based on the complementarity (and ultimate identity) of polar opposites.

So the qualitative nature of 2-dimensional interpretation corresponds with the structural nature of the two roots of unity.

However whereas the two roots of 1, i.e. + 1 and - 1 are interpreted using either/or (linear) logic in quantitative terms, qualitatively they are interpreted using circular both/and circular logic.

However the use of 2 initially always requires the corresponding use 1-dimensional interpretation. For if we are not able to identify polar opposites as initially separate in experience, then we cannot hope they see how they are complementary!

Thus a key issue that thereby arises is the reconciliation of both the linear and circular modes of understanding.

Now in quantitative terms the constant pi relates to the relationship as between the (circular) circumference and its (linear) diameter.

Likewise in qualitative terms, for correct 2-dimensional interpretation we must reconcile both linear (either/or) and circular (both/and) interpretation.

Thus the very form of the quantitative result for the Euler Zeta function (where s = 2) has its corresponding qualitative holistic mathematical explanation (i.e. as the relationship of linear and circular understanding).

Though in quantitative terms, the Euler function here does indeed have a direct quantitative numerical result, indirectly it also contains a qualitative significance in the form of the result (entailing pi).

Now we will see later, when s takes on negative integer values, that the numerical results arising have but an indirect quantitative significance. Rather in direct terms, the numerical values that result can only correctly be given a qualitative interpretation!

Briefly the holistic mathematical interpretation for all other even integer values of s operates along the same lines. Just as the 2-dimensional case represents the simplest version of the complementarity of opposites, all higher dimensions point to more intricate arrangements of the same complementarity. This corresponds in quantitative terms with the fact that the s roots of 1 can also be arranged in a complementary manner.

In my own work, I have mainly concentrated on interpretation that corresponds to 2, 4 and 8 dimensions respectively (which I believe are the most important). However in principle the basic requirement regarding matching positive and negative directions (now taken in the complex plane) can be easily extended to all even integer values of s.

And once again this constitutes the holistic mathematical explanation as to why the form of all such zeta results can be expressed in the form of pi.

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