Thursday, December 15, 2011

The Harmonic Series Again!

As I have repeatedly stated Conventional (Type 1) Mathematics is formally based on a linear i.e. 1-dimensional rational approach (in qualitative terms).
However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.

Thus the conventional approach simply reduces the intuitive aspect in rational terms. Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.

In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.
Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.

Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.

I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.

Thus in quantitative terms 4 is more fully represented as 4^1.
In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).

However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).

So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.

However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1. So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).

Many indications of what I am saying here are provided by simple mathematical results.

Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.

Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).

So 7^1 is truly linear in this regard with no factors.

4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.

However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.

So 7^(- 1) = 1/7 = .142857..
Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).

So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).

Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +..... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.

So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.

However when we raise these numbers same numbers to - 1

i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +.... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + .....

Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!
So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).

However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.

Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact. So to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).

Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.

So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).

And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!

Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).

So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.

So moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.

So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).
And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.

This circular nature of the harmonic series can be demonstrated in yet another striking manner.

Now the harmonic series represents one important example of the Zeta function
i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + .... where s = 1.

As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi. And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).

For example when s = 2, the sum of the series = {(pi)^2}/6.

The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.

So 1 + 1/2 + 1/3 + 1/4 + ........

= 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ......}

And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!

It is again notable that the harmonic series is intimately associated with musical sound. So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.

And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.