Tuesday, July 12, 2016

Riemann Hypothesis: New Perspective (1)

Recently on my companion blog "Spectrum of Development", I have been highlighting a fundamental unrecognised problem with conventional mathematical interpretation, which in fact is the important basis for the proper distinction of both addition and multiplication respectively.

This problem relates to the fact that in conventional terms no distinction is made as between independent and interdependent units respectively.

Now whereas the independent aspect is identified with the quantitative nature of number in cardinal fashion, the corresponding interdependent aspect is then properly identified with its unrecognised qualitative aspect in an ordinal manner.

So to preserve this distinction as between quantitative (i.e. analytic) and qualitative (i.e. holistic) aspects, we must now define number in a dynamic interactive manner (combining notions of both relative independence and relative interdependence). And I refer to these two complementary aspects of the number system as Type 1 and Type 2 respectively.

The natural numbers are then defined in Type 1 terms with respect to the default dimension of 1.

So from a quantitative perspective, the natural numbers - and by extension all real numbers - are viewed as existing on the (1-dimensional) number line.

Once again in Type 1 terms, if we take any natural number, it will appear to be composed of independent units (of a strictly homogeneous nature).

So for example, in quantitative terms, 3 = 1 + 1 + 1. So these independent units (which are completely interchangeable with each other)  thereby lack any qualitative distinction!

However it is all subtly different from a Type 2 perspective.

Here, with respect to each natural number - in inverse terms - the dimensional number varies with respect to a default base number of 1.

And as we shall presently see, this Type 2 aspect now properly defines the true qualitative nature of number (where each number is viewed as being potentially related to all other numbers).

And it is this distinction as between the Type 1 and Type 2 aspects that properly defines the corresponding distinction as between addition and multiplication respectively.

So again in Type 1 terms, 3 is defined with respect to the addition of its 3 "independent" units. It is important in this context to appreciate that "independence" is understood in a relative - rather than absolute - manner!

Thus, 31 = 11 + 11 + 11.

However by contrast in Type 2 terms, 3 is defined with respect to the multiplication of its 3 "independent" units.

Thus 13 = 11 * 11 * 11.

Now we can easily envisage this Type 2 aspect in geometrical terms, by considering a simple cube (with side 1 unit).

So we can readily appreciate that this represents a 3-dimensional object with its (base) side = 1 unit.

However, when one carefully reflects on the matter, the nature of the units comprising 3 in Type 2 is quite distinct from their corresponding nature in Type 1 terms.

Once again whereas these units are relatively independent (and fully interchangeable with each other) in Type 1 terms, this is not the case from the Type 2 perspective (where they are clearly related to each other in an ordered manner). So the 3 units here represent the length, width and height of the cube respectively (which can only be appreciated through their ordered relationship with each other).

So therefore in the Type 1 case, we consider the units in a quantitative manner as relatively independent of each other; then in Type 2 terms we properly consider the units in a qualitative manner with respect to their relative interdependence (i.e. their shared relationship with each other).

We could equally refer to the Type 2 aspect as comprising the ordinal relationship between the units.

So again in Type 1 (cardinal) terms, 3 = 1 + 1 + 1 (in a quantitative manner).

Then in Type 2 (ordinal) terms, 3 = 1st + 2nd + 3rd (in a qualitative manner).

Now once again the (separate) three cardinal units, which are homogeneous in Type 1 terms, thereby lack qualitative distinction; however in reverse fashion the (combined) three ordinal units in Type 2 terms, lack quantitative distinction. This can then be indirectly demonstrated in a quantitative manner through adding the 3 roots of 1 (= 0).

Now what is fascinating to observe here is that what represents multiplication (from a Type 1 perspective), equally represents addition (in Type 2 terms).

So again in Type 1 terms,  11 * 11 * 11 = 13.

However in Type 2 terms, 13 = 11 + 1 + 1.

Therefore what we have really shown here is that there are two distinct forms of addition (which relate to independent and interdependent units respectively).

Thus with reference to a simple example, if we have a cake comprising 3 equal slices, one can readily appreciate that in quantitative (cardinal) terms the cake = the sum of its 3 (independent) slices.

So properly, we are viewing the cake here in quantitative terms as the sum of its 3 parts.

However we can equally view the cake in qualitative (ordinal) terms as the sum of the 3 (interdependent) slices  i.e. as comprising 1st, 2nd and 3rd slices.

So properly we are  now viewing the cake here in true qualitative, i.e. whole terms (as the relationship between its three slices).

Of course, in dynamic experiential terms, both quantitative (part) and qualitative (whole) aspects necessarily interact. So explicit recognition of the quantitative aspect requires corresponding implicit recognition of the qualitative; likewise explicit recognition of the qualitative implies implicit recognition of the quantitative.

However the crucial reductionism in conventional mathematical terms entails that no distinction is possible as between quantitative (part) notions and qualitative (whole) notions.

In other words in every context, the (qualitative) whole is absolutely reduced in terms of its mere (quantitative) parts.

And just as we have two type of addition (relating to Type 1 and Type 2 units respectively), equally we have two types of multiplication.

So in Type 1 terms 11 * 11 * 11 = 13. However in Type 2 terms, 11 *  1 *  1 = 13. However the dimensional number 3 now relates to the sum of (relatively) independent, whreas previously it represented (relatively) interdependent units.

In other words, when we perform multiplication with respect to (relatively) independent units in Type 1 terms, they are thereby transformed to their (relatively) interdependent status (from a Type 2 perspective).

However when we perform multiplication with respect to (relatively) interdependent units in Type 2 terms, they are likewise transformed - in reverse manner - to their (relatively) independent status (from a Type 1 perspective).

Thus we properly have two types of addition with respect to relative independence and relative interdependence respectively (where the status in each context - as independent or interdependent respectively - remains unchanged).

However, equally we properly have two types of multiplication with respect to relative independence and relative interdependence respectively (where the status in each context - as independent or interdependent respectively - is altered).

The huge question then arises as to the consistency with respect to both types of addition and both types of multiplication!

In conventional mathematical terms, because these distinctions (ultimately reflecting the quantitative and qualitative aspects of number) are completely overlooked, in effect this consistency is blindly assumed to exist.

However, when properly understood, the Riemann Hypothesis can be viewed as a fundamental requirement for preserving such consistency (with respect to both addition and multiplication).