Wednesday, August 7, 2013

The Holistic Nature of the Number System (2)

Once again the root cause of qualitative confusion with respect to Conventional Mathematics relates directly to the manner in which the infinite notion is reduced in finite terms.

Appreciating this in turn helps one to appreciate the two uses of number i.e. as a number quantity (that can be raised to a number dimension) and in reverse a number dimension (to which a given number quantity can be raised).

When one uses 1 in actual terms to represent a quantity, clearly it relates to an actual specific finite notion; however when one uses 1 to represent a dimensional quality strictly it relates to an infinite notion (of a potential nature).

So for example the line is 1-dimensional and this implies that it is potentially unlimited with respect to extension.

Now the direct confusion that is involved is that when we attempt to apply this notion in an actual quantitative sense it is always necessarily limited in a finite manner.

So if I draw a straight line, regardless of how far it is extended it will always necessarily be of a merely finite length!

However because of the quantitative bias of Conventional Mathematics, the attempt is then made to reduce the true potential nature of the infinite notion in an actual finite manner.

Thus the totally misleading impression is then given that the infinite somehow represents the ultimate limit resulting from finite extension.

In common sense terms this could be expressed by saying that if one extends the line far enough, its length will approach infinity.

Now to put it bluntly, this is utter nonsense; and yet it is such a reduced notion of the infinite that pervades mathematical thinking.

Therefore though the infinite properly relates to a qualitative notion, conventionally it is treated in a merely reduced quantitative manner.

In understanding, intuition directly relates to the (qualitative) infinite and reason to the (quantitative) finite aspect respectively.

So intuition strictly relates to the potential infinite aspect that is inherent in all actual finite circumstances.

However once again, because Conventional Mathematics is formally defined in a merely linear rational manner, the potential infinite aspect is necessarily reduced in an actual finite manner.

Therefore the holistic aspect of mathematical understanding - which essentially relates to the (infinite) qualitative aspect of relationships - formally, is completely unrecognised by the profession.

Thus when I repeatedly state that Conventional Mathematics is totally unbalanced, I mean precisely what I say.

Properly understood in dynamic interactive terms, we have two aspects quantitative and qualitative (of equal relevance) that define the nature of all mathematical relationships. Yet, quite incredibly, only one of these is formally recognised.

To use an analogy is it like maintaining that water (that comprises both hydrogen and oxygen molecules) is comprised solely of oxygen. However in truth it is much more serious than that!

Coming back to my original point, the use of number in dynamic interactive terms, as representing base quantities and dimensional exponents respectively, relates to two quite distinctive aspects.

As we saw in yesterday’s blog entry, whereas the cardinal aspect is properly associated with the former, the ordinal aspect directly relates to the latter.

Therefore in actual mathematical experience, we have the continual dynamic interaction of both cardinal and ordinal aspects (which are quantitative and qualitative with respect to each other).

The deepest issue with respect to the number system is therefore the key requirement of achieving consistency as between these two distinctive aspects.

And once again when properly understood, this is what the Riemann Zeta Function (and its associated Riemann Hypothesis) is all about.

However when appreciated in such terms, it is somewhat ludicrous to approach the Riemann Hypothesis from a merely quantitative perspective.

In the truest sense this represents the "reductio ad absurdum" which crucially exposes the limits of the conventional mathematical approach.

Remember again that the Riemann Zeta Function remains uniquely undefined for just one value where s (the dimensional value)  = 1.

Now Conventional Mathematics limits itself entirely to the quantitative interpretation of this statement.

However the corresponding qualitative interpretation is that the Riemann Zeta Function remains uniquely undefined in conventional mathematical terms (defined as it is, in a 1-dimensional manner).

So for all other values of s, a dynamic interaction as between both cardinal (quantitative) and ordinal (qualitative) type meanings necessarily exist.

Therefore through the Functional Equation, we can always match cardinal type interpretation on the RHS of the Function for ζ(s) to a corresponding ordinal type interpretation on the LHS for ζ(1 s).

And so the condition for the mutual coincidence of cardinal and ordinal values is that s = .5.

So the requirement that all the non-trivial zeros lie on an imaginary line drawn through .5 (which is the Riemann Hypothesis) is in fact the condition for ensuring that both the cardinal (quantitative) and ordinal (qualitative) aspects of number are mutually identical.

So again the Riemann Hypothesis - when appropriately understood in dynamic interactive terms - serves as the key requirement for ensuring the subsequent consistency of quantitative and qualitative meaning with respect to all mathematical relationships.
Now clearly such a proposition cannot be proved (or disproved) with reference to a system defined with respect to mere quantitative interpretation!

Indeed the truth of Riemann Hypothesis is already necessarily assumed in the very use of conventional mathematical axioms.  

So the zeta zeros inherently relate to an interdependence with respect to both quantitative and qualitative aspects of mathematical meaning.

Now yesterday, I was at pains to show that Conventional Mathematics necessarily is defined in terms of independent reference frames, in what represents analytic interpretation.

However the very nature of the zeta zeros relates to the interdependence of both quantitative and qualitative aspects.

And the appropriate manner for interpreting such interdependence (of complementary poles) relates to holistic -  as opposed to analytic - understanding.

Once again this creates insuperable problem for Conventional Mathematics (which in formal terms is completely lacking a holistic aspect).

Indeed put simply the (non-trivial) zeta zeros (Zeta 1 and Zeta 2) represent the holistic aspect of the number system, which necessarily underpins (in a dynamic interactive manner) our everyday analytic appreciation of number.  
The seta zeros therefore play an indispensable role in our number system.
However perhaps the most important implication of the nature of these zeros is that Mathematics itself now needs to be completely reformulated in an appropriate dynamic manner. 

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