Now as I
have expressed repeatedly in these blog entries the deeper basis of this (1-dimensional)
paradigm is rooted in an approach whereby mathematical interpretation is
conducted absolutely in terms of just one polar reference frame. This in turn
leads (i) to the notion of mathematical “objects” that are unaffected through
(subjective) mental interaction and (ii) an exclusive focus on the quantitative
aspect of mathematical relationships (with no reference to a qualitative
context of meaning).

Of course
momentary reflection on the matter will quickly show that our actual experience
of Mathematics is inherently of a dynamic relative nature entailing the
continual interaction of both internal and external aspects and likewise the
continual interaction of part (quantitative) and whole (qualitative) notions.

In
psychological terms this equally entails both conscious and
unconscious aspects of understanding through the corresponding interaction of
rational (conscious) and intuitive (unconscious) processes.

Momentary
reflection on the matter should also show that the accepted method adopted by
Conventional Mathematics reduces the truly dynamic nature of actual
mathematical experience in a greatly reduced absolute static manner.

So the
necessary dynamic interaction of opposite poles of experience is formally expressed
statically in terms of just one pole!

Likewise
the dynamic interaction psychologically of (conscious) reason and (unconscious)
intuition is rigidly expressed solely in terms of (conscious) reason.

It is
therefore hardly surprising in this context that our present understanding of
the number system should be fatally flawed.

Certainly
from my perspective it has long been obvious that - when appropriately
understood i.e. in accordance with actual experience - that the number system
is necessarily of a dynamic interactive nature.

And of
course this equally implies that it should be interpreted in a relative -
rather than absolute - manner.

So we
cannot therefore ultimately divorce the (apparent) objective nature of number
from the subjective mental means of its interpretation.

Therefore
in this relative sense when we change the nature of interpretation the
objective nature of number objects likewise changes!

Likewise -
and perhaps even more tellingly - we cannot ultimately divorce the (apparent)
quantitative aspect of number from its corresponding relational context (which
is of a qualitative dimensional nature).

As I
expressed in yesterday’s entry, once we accept that numbers can indeed be
related to each other then this implies that their independent status (in
specific cardinal terms) is necessarily of a relative - rather than absolute -
nature.

So in this
dynamic context the quantitative nature of number relates to its cardinal
status as relatively independent.

In other
words, without an implicit acceptance that to have meaning, numbers must be
placed in an ordered manner with respect to other numbers, it would be
impossible to locate cardinals on the number scale! So the quantitative
definition of number thereby implicitly implies a distinctive qualitative
aspect.

Also, in
this dynamic context, the qualitative nature of number relates to its ordinal
status as relatively interdependent (with respect to other numbers).

So once
again the very notion of ordinal requires placing a number in a group context
(with respect to other numbers) with its identity thereby defined with respect
to the group members involved.

Thus for
example to define 2

^{nd}I must place the number 2 in relation to other group members. The simplest case would involve a group of 2 members. Therefore in this context, I can unambiguously define a 2^{nd}member. However the definition of 2^{nd}clearly changes when the number of group members increases. So 2^{nd}in the context of 200 members is clearly distinct from 2^{nd}in the context of 2.
So the
ordinal identity of a number (reflecting its qualitative identity) springs
from a relationship of interdependence with a wider group of numbers.

However,
once again we must define interdependence in a merely relative sense here, as
we start with the cardinal notion before we can establish its ordinal identity.

So the
notion of 2

^{nd}(as ordinal) already dynamically implies 2 (as cardinal). Likewise the notion of 2 (as cardinal) already dynamically implies 2^{nd}(as ordinal).
However
dynamic interaction necessarily implies a phenomenal context of space and time.
Though we may appreciate from what has been said, that ultimately cardinal and
ordinal identity must be identical, clearly this cannot be the case in a
phenomenal context (which necessarily implies a degree of relative separation).
Thus total unification of both cardinal and ordinal aspects of the number
system points to an ineffable state.

Now this is
all deeply relevant for appreciation of the true nature of the Riemann
Hypothesis (which postulates the very condition for this ultimate identity)!

The vital
fact to grasp is that cardinal and ordinal aspects require two distinctive
interpretations respectively.

Thus great
confusion pervades Conventional Mathematics. Because it is 1-dimensional in
nature, it must necessarily attempt to interpret both cardinal and ordinal
aspects from the same quantitative perspective.

Therefore
though the ordinal aspect properly relates to the qualitative aspect of number,
it is mistakenly dealt with in a merely quantitative manner.

This is a
key problem of the very first magnitude with respect to the number system, which
is at present just totally ignored!

As I was
stating yesterday the quantitative (cardinal) aspect directly relates to the
analytic interpretation of number; however the qualitative (ordinal) aspect
properly relates to the (as yet unrecognised) holistic interpretation.

Whereas the
cardinal notion relates to the number line, the ordinal notion - by contrast
- relates to the circle.

So cardinal
and ordinal notions are linear and circular with respect to each other.

Solving the
ordinal problem (i.e. relating to the qualitative nature of ordinal numbers) comes through the Type 2 number system.

Therefore
the qualitative i.e. dimensional notion of 2 is expressed as 1

^{2}.
As we have
seen the corresponding cardinal notion is expressed in terms of the Type 1
number system as 2

^{1}.
However to
give indirect quantitative expression to the qualitative notion of number we
convert to a circular format through obtaining the corresponding 2 roots of
1.

So + 1 and – 1 are these 2 roots which again - in this context - provide the indirect
quantitative means of expressing the two ordinal members of a group of 2.

However with
holistic understanding both quantitative and qualitative appreciation must be
balanced with each other.

So the true
qualitative appreciation requires intuitively being able to see + 1 and – 1 as interdependent with each other.

Put another
way, this is the appreciation of ordinal rankings as purely relative (depending
on context).

I have
explained before how implicitly this is what one does for example in
recognising that turns at crossroads can be either left or right (depending on
context).

So if
approaching to crossroads from one direction we label the 1

^{st}turn + 1 (a left turn) and the 2^{nd}, – 1 (i.e. not a left turn) then clearly if the crossroads is approached from the opposite direction the 2^{nd}will now be labelled ,+ 1 (a left turn) and the 1^{st}, – 1 (i.e. not a left turn).
So each
turn can be both left and right (depending on context).

This understanding
therefore illustrates the very nature of interdependence which requires the
ability to see simultaneously from - at a minimum - two opposite reference frames.

Now clearly
we cannot make such paradoxical connections in the context of just one polar
reference frame.

However
Conventional Mathematics is formally defined in terms of just one such frame (where
relationships appear unambiguous and linear).

Therefore
by its very nature, Conventional Mathematics is not equipped to deal with the key
notion of interdependence (except in a misleading reduced sense).

And as
ordinal meaning directly relates to number interdependence, Conventional
Mathematics cannot properly deal with the ordinal notion.

It is quite
remarkable. Though the use of the circular number system (defined in the
complex plane with circle of unit radius) provides the appropriate means for
defining the ordinal nature of number, I have yet to see it mentioned anywhere!

We have
something simple, right under our noses as it were and we fail to see it! This
is directly due to the pronounced linear bias of Conventional Mathematics (based on mere analytic
type understanding).

Now of course the unit circle is well recognised in Conventional Mathematics. However appreciation of its true role remains greatly limited due to the inevitable attempt to understand its nature in a merely linear manner. However to properly understand the circle, we require circular type interpretation!

Thus to
appreciate the nature of ordinal interpretation we must adopt holistic
understanding that requires at a minimum 2-dimensional interpretation (entailing two interacting polar reference frames).

And again
such holistic understanding is totally missing from what is formally accepted
as Mathematics!

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