Friday, December 3, 2010

True Significance of Riemann Hypothesis (3)

Though again not properly realised in conventional mathematical interpretation, the nature of the roots of a polynomial equation indirectly points to the true multidimensional nature of mathematical interpretation.

Indeed an enormous amount can be gleaned from consideration of the simplest polynomial equation of degree 2 i.e. where x^2 = 1.

It is the very nature of original mathematical discovery to see a fundamental problem with an explanation (which every one else accepts without apparent question).

As I have so often stated Conventional Mathematics adopts a linear logical approach that is literally 1-dimensional from a qualitative perspective. Now one of the key characteristics of this approach is that it unambiguous in nature.

So crucially for example if one proves for example that a theorem is true, then this rules out the possibility of any other alternative (especially the polar opposite case of being false).

Thus if we designate the outcome that a proposition is true as positive, then the opposite case of it being false is thereby negative. In this sense therefore a proposition cannot be given both a positive and negative truth value.

Now there is a direct correspondence here as between qualitative and quantitative for in the equation where the value of x is raised to 1, its answer is entirely unambiguous.

So, as we see for example in the simplest case where x = 1, the equation has only one correct value.

So this represents the one-dimensional case where x ^1 = 1.

Once again, here the qualitative interpretation of the result (as 1-dimensional and unambiguous) corresponds directly with the quantitative nature of the result that is likewise 1-dimensional and unambiguous).

However in the case where the dimensional number (power or exponent) is now 2, a fundamental problem arises.

We will demonstrate this first by looking carefully at the quantitative aspect. Now again in conventional mathematical interpretation, the value of x in this equation (representing the two roots of 1) can be given as either + 1 or - 1.
So what is remarkable is that two answers - which are polar opposites of each other - are deemed as a correct solution.
Now this state of affairs points to an inherent - unresolved - ambiguity with respect to mathematical interpretation.

So let us probe more deeply where in fact Conventional Mathematics falls short in this regard.

Now the key point that I am making is that the use of a number as dimension directly relates to a unique mode of qualitative interpretation of mathematical symbols.

However because the qualitative mode of Conventional Mathematics is linear (i.e. 1-dimensional) when we raise 1 to a dimension (other than 1) the resulting quantitative result is given in reduced terms as 1 (i.e. 1^1).

So therefore from this perspective (i.e. in reduced quantitative terms) 1^2 is indistinguishable from 1^1! So here the dimensional number 2 is given a reduced qualitative interpretation as 1.

So the key to unlocking the apparent ambiguity attached to the two roots of 1 is to recognise that 1^1 and 1^2 are actually distinct from each other (in qualitative terms).

Thus, from this newly defined perspective, there is only one unambiguous root of 1 (i.e. - 1) which when squared = 1^1. The other (supposed) root (+ 1) when squared gives the result 1^2.

So 1^1 (i.e. + 1) is not properly the square of 1 but rather (1^2).

However, having redefined this relationship in quantitative terms, we must now complete the more difficult task of redefining it correctly also in corresponding qualitative terms.

And it as this stage that Holistic Mathematics properly starts.

So once again in qualitative terms 1-dimensional interpretation corresponds with the unambiguous linear logical approach (based on either/or distinctions).

The key to the qualitative interpretation of a dimensional number is the recognition that it is structurally identical with its corresponding root.

And as the second root of 1 is, as we have established, - 1 this entails that 2-dimensional understanding literally entails the qualitative negation therefore of linear (rational) type interpretation.

The very word "unconscious" that we use in psychological terms implies the negation of "conscious". So the clue to the nature of 2-dimensional understanding is that it is of a direct unconscious nature arising from the dynamic negation of what is considered positive (and thereby true) at a rational linear level.

Such understanding is holistic and intuitive . However intuitive understanding can be indirectly expressed in a rational manner in circular both/and logical terms (as the complementarity of opposite polarities).

Therefore, the significance of recognition of both 1-dimensional and 2-dimensional interpretation in qualitative terms is that intuition must be formally included with reason in mathematical understanding.

Briefly all "higher" dimensional interpretations can be expressed structurally in terms of their corresponding quantitative roots.

So for example if we wished to understand the true nature of 4-dimensional interpretation, in qualitative terms, we look at the structural nature of the 4th root of 1) which is i.

Therefore in qualitative holistic mathematical terms this requires explaining the precise philosophical meaning of what is meant by "imaginary" interpretation.

I have spent most of my adult life elaborating exactly such issues. However it is sufficient to state here that what we refer to as "imaginary", in qualitative terms, relates to the rational means through which circular both/and logic can be indirectly represented in a rational linear manner.

So just as in quantitative terms the comprehensive number system is complex (with real and imaginary components), likewise a comprehensive interpretation of mathematical reality includes both real and imaginary aspects (relating to two different logical systems).

And as all other roots of 1 entail both real and imaginary parts in quantitative terms, corresponding multidimensional interpretation of these roots entails unique configurations of both real and imaginary interpretation (ultimately relating to a precise relationship of reason to intuition in understanding).

And the truly wonderful - yet totally mysterious - correspondence as between quantitative and qualitative interpretations, that enables all this meaning to unfold, once again is directly implied by the Riemann Hypothesis.

So 1^(1/x) in quantitative terms corresponds directly with 1^x in a qualitative structural manner. (Again this key relationship as between quantitative values and corresponding qualitative interpretation - to which the Riemann Hypothesis relates - is completely missed from a conventional mathematical perspective. For when x = 1, both sides are identical so that qualitative interpretation is thereby reduced in merely quantitative terms)!

However in remains to be shown why such a correspondence is especially relevant to prime numbers (which we will do in the next contribution).

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