Wednesday, April 20, 2011

The Trivial Zeros (1)

Precise knowledge of the nature of - even - the first of the trivial zeros in the Riemann Zeta Function is enough to appreciate the true nature of the Riemann Hypothesis (which establishes the key condition required for maintaining consistency as between both the quantitative and qualitative interpretation of mathematical symbols).

In terms of the gradual evolution of my own understanding in this regard the first seeds were set through a boyhood realisation of - what I saw - as a problem with mathematical interpretation.

At that time numbers were illustrated using the geometrical representation of the straight line. So 1 unit could be represented by a straight line segment (representing this magnitude).
So here we would represent the number - literally - in 1-dimensional terms (as a straight line).

If however we now went on to representing 1 in 2-dimensional terms we would get a square (with each of its four equal sides = 1 unit).

Though the answer in both cases, in quantitative terms is the same = 1, a significant qualitative difference is involved with the two answers.

So in the first case with the straight line we are referring to a 1-dimensional format of measurement, whereas in the second with the square we are referring to a qualitatively distinct 2-dimensional format.

Therefore though in reduced quantitative terms the answer remains the same = 1, clearly both answers in qualitative terms are quite distinct.

Again if we moved to measuring in 3 dimensions we could again get an answer = 1 by having all six sides of a cube representing squares (with side = 1 unit).

Conventional Mathematics is therefore - in a very precise manner - based on a linear rational approach. Thus whenever we multiply two numbers (or raise a number to a power other than 1) a qualitative - as well as quantitative - transformation in the numerical value takes place.

However the qualitative aspect of this transformation is then ignored with resulting value expressed in a merely reduced quantitative manner (in terms of the 1st dimension).

So for example in conventional terms 2 * 2 = 4. Though in qualitative terms this numerical operation entails a transformation (from the first to the second dimension), the result is expressed in a merely reduced 1-dimensional quantitative manner as 4 (i.e. 4 linear units).

It has often suggested that the mystery of Riemann Hypothesis relates to a fundamental connection as between addition and multiplication (that is not properly understood).
Well this fundamental connection is ultimately pointing to is the fact that all numbers (and indeed all mathematical symbols) have both quantitative and qualitative interpretations!

However Mathematics - as it is presently construed - is geared solely to the quantitative aspect. By its very nature it is not suited therefore to a proper appreciation – not alone resolution – of the true nature of the Hypothesis.

I became aware again at an early age of another related problem which I felt was glossed over in an unsatisfactory manner in Conventional Mathematics.

Once again - in standard conventional terms - when we square 1, the answer remains unchanged as 1. So the answer here is single valued and unambiguous. However when we now - in inverse terms - obtain the square root, two possible answers arise (in quantitative terms) i.e. + 1 and – 1. So we now - by contrast - have a highly paradoxical two valued solution (where each answer is the negative of the other).

This raises in fact a very deep qualitative issue, which is effectively ignored in Conventional Mathematics (precisely because of its lack of a qualitative dimension).

For example the fundamental notion of proof in Mathematics is based on obtaining a single valued qualitative result. In other words from this perspective a theorem is either true or not true (in unambiguous terms). Indeed the continued fascination with the Riemann Hypothesis is precisely to establish in this context whether it is true or not true. So the positive result i.e. that it is true would thereby - in terms of this logic - rule out the negative result i.e. that it is not true.

Yet in the very simple operation of obtaining the square root of 1 we find that both the positive and negative solutions i.e. + 1 and – 1 can both be true!
So this result, if it was properly appreciated in qualitative terms, would raise fundamental issues regarding the limitations of the standard (linear) logical approach used in Conventional Mathematics!

So to sum up! Even as a child (9- 10 years) two key mathematical issues troubled me.

1) I could vaguely see that every number had both a quantitative and qualitative meaning (with the qualitative meaning simply ignored in conventional terms).

2) I considered - though I would not have used these words at the time - that there was an unsatisfactory asymmetry as between the power of a number on the one hand and the corresponding inverse notion of its root on the other. Whereas again in conventional terms the square of 1, for example, has just one unambiguous result, the inverse operation of obtaining its square root yields two results (which are paradoxical in terms of each other).

However it took some considerable time before I found a way of resolving these issues to my own satisfaction (and realise that they in fact represented two sides of the same coin).

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