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!

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.

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.

## No comments:

## Post a Comment