Things are
not always what they seem!

I mentioned
this yesterday with reference to the standard analytic treatment of number 1 + 1 = 2.

Indeed
based on our common-sense intuitions of reality this statement seems so obvious
that it is frequently used as a metaphor for self-evident truth.

In fact
however it reveals the reduced nature of accepted mathematical understanding,
which through millennia now of unchallenged usage has become so deeply
ingrained in our thinking that we have become completely blind as to its
limitations.

The
conventional understanding of this simple number relationship is based on mere analytic
appreciation (where the “whole” number 2, represents the quantitative sum of
its part unit members).

So the unit
numbers here are viewed as absolutely independent of each other (in a
homogeneous manner). In other words, as there is no way to distinguish as
between the two units, they thereby lack qualitative distinction.

However if
the individual units were indeed absolutely independent in such a manner, there
would be no way of combining them to obtain the new collective identity of
2!

Therefore
the very ability to relate the two unit numbers presupposes a distinct
qualitative identity of relational interdependence.

In the
conventional treatment of addition, this key fact is completely overlooked.

In other
words the qualitative aspect of number (i.e. relational interdependence) is
simply reduced in an absolute quantitative manner.

This then
enables the convenient fiction to be maintained that numbers can be successfully combined
with each other (while preserving their absolute quantitative identity).

Thus from
this perspective 1 + 1 = 2 (understood
in absolute quantitative terms).

However
when one addresses the gross inconsistency entailed here, then the very nature of
number is thereby radically changed.

For when we
accept that all numbers necessarily possess a qualitative aspect of relational interdependence,
then the corresponding quantitative aspect (of distinct identity) can only be
properly conceived in a relatively independent fashion.

Thus all
numbers necessarily possess both an analytic aspect (of quantitative independence)
and a holistic aspect (of qualitative interdependence) respectively, which are
dynamically related to each other in complementary fashion.

So let us
now return to our simple identity, 1 + 1 = 2.

Again, in
terms of appropriate dynamic appreciation, each unit is relatively independent
in a quantitative manner, while equally possessing a qualitative identity
of shared relational interdependence.

Whereas the
quantitative aspect is directly related to the cardinal, the qualitative aspect
is directly related to the corresponding ordinal nature of number respectively.

So again
from the quantitative perspective, the two units are given a separate individual
identity (in actual terms); however from the qualitative perspective, they are
given a shared collective identity (in a potential manner).

So with
respect to number recognition, each individual unit must be quantitatively
distinguished (in an actual manner).

However
equally, the two units must be given a qualitative identity of shared
relational interdependence (in potential terms). From this perspective ordinal notions
of 1

^{st}and 2^{nd}with respect to the two units are viewed as fully interchangeable with each other.
At a deeper
psychological level, whereas the analytic (quantitative) aspect of number relates
directly to rational (conscious) recognition, the holistic (qualitative) aspect
relates by contrast directly to intuitive (unconscious) appreciation.

So in
conventional mathematical practice, the reduction of the holistic (qualitative)
aspect of number recognition, in analytic (quantitative) terms, parallels
precisely the corresponding psychological reduction of intuitive (unconscious)
appreciation in a rational (conscious) manner.

Properly
therefore, a comprehensive understanding of number entails both conscious and
unconscious aspects of appreciation where both need to be explicitly recognised.

However once more, in
conventional mathematical terms, the holistic (unconscious) aspect is formally
reduced in an analytic (conscious) manner.

Therefore
the true nature of number is inherently dynamic with quantitative notions of
relative independence continually interacting with qualitative notions of
relative interdependence respectively.

And in this
new dynamic context, the two relative notions of number independence and
interdependence are in turn intimately linked with the mathematical operations
of addition and multiplication respectively.

Thus when
we start with the quantitative aspect using the operation of addition,

1 + 1 = 2.

More fully
we can express this relationship as,

1

^{1}+ 1^{1}= 2^{1}.
And using
the Peano postulates all the natural numbers can ultimately be derived from
successively adding 1 to a previous natural number (starting with 1).

So
to
combine numbers in a quantitative manner (through the operation of
addition) we
must explicitly define them with respect to the default dimension of 1
(which in this context remains implicit in understanding). In this way
all natural numbers are viewed as points on the same real number line.

Thus in
this context, 2 (i.e. 2

^{1}) represents the quantitative sum of the two base units.
However
when we now, in relative terms, combine the two units through multiplication, a
different qualitative notion of number emerges, which can be expressed as follows,

1

^{1 }* 1^{1 }= 1^{2}.
So in
dynamic complementary fashion the focus is explicitly on the dimensional
units (with the corresponding base units now remaining implicit).

Thus in
this dynamically related context 2 (i.e. 1

^{2}) represents the qualitative product of the two dimensional units).
What this
means in effect is that these two units are now understood in holistic ordinal
terms (as relatively interdependent) whereby 1

^{st}and 2^{nd}positions are freely interchangeable with each other.
So just as
we recognise that what are designated right and left turns at a crossroads can
switch with each other depending on the direction from which the crossroads is approached,
we likewise recognise that the two dimensional units can likewise switch
ordinal positions (depending on context).

However as
always in dynamic interactive terms, reference frames can switch.

Thus there
is a valid sense in which addition can likewise be identified in qualitative
terms with the interdependence of its unit members.

So 1

^{1}+ 1^{1}= 2^{1}. So here we are explicitly recognising in a holistic manner the qualitative interdependence of the two base units (with quantitative recognition now implicit).
And finally
there is equally a valid sense in complementary fashion, through which
multiplication can be likewise identified with the quantitative independence of
its dimensional units (i.e. where each dimensional unit is recognised as separate).

Thus 1

^{1 }* 1^{1 }= 1^{2}.
Therefore
what we have demonstrated here is the following:

1)
The
number 2 - and by extension all natural numbers - can be given both
analytic (quantitative) and holistic (qualitative)
interpretations with respect to both base and dimensional usage. Whereas
the analytic aspect corresponds directly with rational, the holistic
aspect by contrast corresponds directly with intuitive type
recognition.

2) The
relationship between both is of a dynamic interactive nature with quantitative
and qualitative aspects understood in complementary fashion.

So when the
analytic (quantitative) aspect of number is made explicit, the corresponding holistic (qualitative) aspects remains, in
this context implicit.

However,
when the holistic (qualitative) aspect of number is now in turn made explicit,
the corresponding analytic (quantitative) aspect is now implicit.

3) What is
extremely important to bear in mind is that in this new dynamic framework, addition and
multiplication become the very means by which the switching from quantitative to qualitative (and in turn qualitative to quantitative)
recognition takes place.

So addition
and multiplication operate as dynamic complementary partners in the very
recognition of number.

Thus if we
return to the number 2 for a moment, there is of course a valid sense in which
we can recognise its two units as quantitatively independent of each other;
however there is an equally valid sense in which we can recognise that these
two units are likewise interdependent (and thereby interchangeable with each
other) in an ordinal manner. So in this context 2 necessarily contains unique 1

^{st}and 2^{nd}units.
And both of
these interpretations are necessarily related in two-way fashion with each
other.

Thus we
cannot explicitly recognise the quantitative independence of the two units
(without implicit recognition of their corresponding qualitative interdependence).

And we
cannot explicit recognise the qualitative interdependence of the two units as, interchangeably,
1

^{st}and 2^{nd}respectively (without implicit recognition of their quantitative independence).
Thus
properly understood, the number 2 - and by extension every other number - is necessarily
of a dynamic interactive nature, comprising analytic (quantitative) and
holistic (qualitative) aspects in a complementary manner.

## No comments:

## Post a Comment