Monday, September 30, 2013

Where Science and Art Coincide (3)

Again the general finite expression for the Zeta 2 equation is given as

1 + s+ s+ s+….. + st – 1  (where t is prime)

The simplest of the Zeta 2 zeros (which acts as the important template for all other Zeta 2 zeros) arises in the case where t = 2,

i.e.  1 + s= 0,  so that  s=  – 1. 

Now t in this context refers to the qualitative notion of number (as representing a dimension).

And this is defined by its two ordinal members i.e. 1st and 2nd respectively.

However, as we have seen the 1st dimension is always 1 (and thereby indistinguishable from its cardinal definition). Also as this is necessarily always one of the t roots of 1, in this sense it is not unique.

Therefore when 2 represents a dimensional number, its one unique (i.e. non-trivial) root relates to its 2nd dimension which is given as – 1.

Thus there is an important complementary connection as between the notion of an ordinal dimension and its corresponding root.

So in this case the 2nd dimension (which has a qualitative meaning) is closely associated with the 2nd root of 1 (in quantitative terms).

Thus in quantitative terms – 1 is recognised in Conventional Mathematics as the 2nd root of 1 which geometrically can be represented as a point on the unit circle (in the complex plane).

However what is not at all realised is that – 1 has also a distinct qualitative meaning (which provides the true interpretation of the ordinal notion of 2nd in this context).

It cannot be stressed strongly enough that this qualitative meaning simply cannot be understood in conventional mathematical terms.

Once again Conventional Mathematics is defined in a linear (1-dimensional) rational manner. This implies in the case of fundamental polarities (such as quantitative and qualitative) that just one isolated pole is taken as the exclusive frame of reference.

So in Conventional Mathematics number is solely interpreted with respect to its quantitative aspect. Therefore we cannot incorporate qualitative meaning - except in a grossly reduced fashion - in terms of such 1-dimensional interpretation.

However once we move to 2-dimensional interpretation, it is indeed possible to define both quantitative and qualitative aspects of meaning to number (without reducing one in terms of the other). 

So in the case of 2 (as dimension) the 1st provides us with the quantitative aspect of interpretation. However the 2nd dimension now relates to the true qualitative meaning (implied by the ordinal notion of 2nd).

As we have seen this is given as – 1. Though this indeed does have an indirect analytic quantitative meaning (now in a circular rather than linear fashion) in direct terms this relates holistically to the negation of the form (that is implied through conscious rational understanding).

In other words the true qualitative (dimensional) meaning of – 1 relates to the (literal) negation of 1-dimensional  interpretation (based as it is on just one isolated pole of reference).

Implicitly,  the very ability to successfully negate in this manner, implies the experiential nondual recognition that at an unconscious level both positive and negative poles necessarily co-exist (as identical). Thus the very recognition of the qualitative requires thereby the (temporary) negation of the quantitative aspect (which hitherto had been given an absolute dominance).

Thus – 1 represents the unconscious negation of the quantitative pole (1-dimensional interpretation) which had been hitherto absolutely posited in a merely conscious rational manner.

So once we move to 2-dimensional interpretation, we must explicitly include both conscious (analytic) and unconscious (holistic) type interpretation, representing the interaction of both quantitative and qualitative type meaning.

Now once again  let us be very clear about this! There is no way this process can be short-circuited so as to be understood in a conventional mathematical manner (as this would once again just represent the reduction of qualitative to quantitative meaning)!

And this is the all important message that I am trying to get across here. The very relationship of numbers with each other implies a qualitative aspect (which cannot be properly interpreted in a mere quantitative manner).

So the present interpretation of our treasured number system (and indeed all mathematical relationships) is strictly speaking quite untenable.

Once again we have developed but a highly reduced - and thereby distorted - understanding, which admittedly as a limited extreme case has proved incredibly useful.

However, look at it this way! If this pale reduced version has proven so useful, imagine how much greater the role of Mathematics will be when we begin to understand it properly (without such distortion)!

So once again from a holistic mathematical perspective, to posit (+) simply entails to make conscious.

Corresponding negation (–) then implies to make unconscious.

And just as matter and anti-matter particles will fuse in the generation of physical energy, in like fashion when the positive (conscious) direction of understanding is negated (in an unconscious manner) this leads to the generation of psycho spiritual energy (i.e. intuition).

So just as we can say that 2-dimensional understanding implies the dynamic interaction of both conscious and unconscious aspects (both explicitly recognised), equally we can say that it entails the dynamic interaction of both reason and intuition.

And it is vital to appreciate that in this interaction, that reason and intuition play uniquely distinctive roles.

Basically the rational aspect provides us with (analytic) quantitative recognition (in a finite manner).
The intuitive aspect provides us with (holistic) qualitative recognition (in an infinite manner).

So inevitably just as the qualitative aspect is grossly reduced to the quantitative in conventional mathematical terms, equally the infinite aspect is likewise grossly reduced in a finite manner i.e. where it is misleadingly portrayed as a linear extension of finite notions.

So once again the qualitative dimensional notion of 2nd (in the context of 2) is represented as – 1. Though this can be given indirect quantitative expression (in a circular manner) its direct meaning is qualitative (in a holistic manner).

Furthermore there is always a vertical complementary link as between the qualitative notion of a dimension and its indirect quantitative expression as its corresponding root.

For example the 4th (in the context of 4 as qualitative dimension) has an indirect quantitative expression through the 4th root of 1 as i (again lying on the circle of unit radius). 

However, its direct qualitative meaning is of a  holistic nature.

In fact the great importance of i, in a holistic sense, is that in enables circular holistic meaning (of a paradoxical nature) to be indirectly expressed in a linear rational manner! 

Finally to conclude this entry we can see that the only root of 1 which involves no change (implying a distinct qualitative meaning) is of course the 1st root of 1 which is also 1.

This represents just another way of demonstrating that it is only with 1 as dimension, that qualitative is reduced to quantitative  meaning. For all other numbers as dimensions, a distinctive root value will result (implying likewise a distinctive qualitative meaning).

The great significance of this is that the only value for which the Riemann Zeta function is undefined is where s (the dimensional value) = 1.

From a 2-dimensional mathematical perspective, as the Riemann Zeta Function now entails the mapping as between both quantitative (cardinal) and qualitative (ordinal) type values, this simply becomes meaningless in 1-dimensional terms.

So once again strictly speaking the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted in conventional (1-dimensional) mathematical terms.

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