And this
inherent feature can in principle be extended to all numbers.

So far from
representing abstract independent entities with an absolute interpretation in
quantitative terms, the very nature of number is inherently dynamic and
relative, representing the two-way interaction of both its quantitative and
qualitative aspects.

And the
fundamental nature of addition and multiplication is revealed through this new
understanding, where these two operations likewise operate in a dynamic
complementary fashion enabling understanding to ceaselessly switch as between
the quantitative and qualitative type aspects of number.

In a way I
find it utterly surprising that this fact is not more widely appreciated for I
have been keenly aware from childhood of an obvious problem (that is
steadfastly avoided in conventional mathematical interpretation).

Thus in the
most basic sense, there is a valid sense in which we can
quantitatively distinguish (as independent members) the individual units of
number.

However
equally there is a valid sense in which we must likewise qualitatively relate
(as shared members) these same units.

And to put
it bluntly in conventional mathematical terms this latter qualitative aspect of
number is inevitably reduced in a merely quantitative manner.

In other
words from this limited perspective, no clear distinction can be made as
between the twin notions of number independence (where units have a separate
identity) and number interdependence (where units have a shared identity)
respectively.

And it is
this fundamental problem that lies at the root of the problem of properly
relating the operations of addition and multiplication.

In an
attempt to clarify this point further let us look at the simple example where
we multiply the first two primes i.e. 2 and 3.

So in
conventional terms 2 * 3 = 6. Now expressed more fully we could represent this

as 2

^{1}* 3^{1}= 6^{1}.with the three numbers represented as points on the same real line.
However, as
well as a quantitative transformation through multiplication (i.e. to 6) a
qualitative (dimensional) change in the nature of the units also takes place.

So if we
imagine a rectangular table of width 2 and length 3 metres respectively, then
the area of the table will be expressed in square metres.

So we move
from 1-dimensional to 2-dimensional units. However in the standard treatment of
multiplication, this qualitative transformation is then reduced in a
quantitative manner, with the result i.e. 6, represented in 1-dimensional terms
(as a point on the real number line).

Though in
dynamic relative terms, base and dimensional numbers are quantitative and
qualitative with respect to each other, both are understood however through
this geometrical representation in a somewhat linear manner.

So we
switch in other words from the analytic use of number representing (actual)
finite quantities to the corresponding analytic use of number representing
(actual) finite dimensions.

However in
dynamic interactive terms, the switch is always of a truly complementary nature
(i.e. from analytic to holistic and holistic to analytic aspect, respectively).

Thus there
is a hugely important holistic aspect to the interpretation of multiplication
(without which it has no strict meaning).

To more
easily illustrate this point, imagine that we have 3 objects (say coins) placed
in two rectangular rows!

Using
addition in standard analytic fashion, we could add up the 3 independent coins
in row 1, = 3 and then proceed to add up the 3 independent coins in row 2, = 3.

And as both
rows would likewise be considered as independent, the total number of coins
represents the sum of the 2 rows i.e. 3 + 3 = 6.

From the
conventional mathematical perspective, multiplication can then be used to
short-circuit the process of laboriously adding separate rows, through the
recognition that a common similarity exists as between each row. So now, the
realisation that we have two similar rows leads to the use of the operator 2,
which is then multiplied by the number of objects in each row.

So from a
multiplicative perspective, the total number of objects is 2 * 3 = 6.

Thus in
conventional mathematical terms, multiplication is represented as a form of
short-hand addition.

So, 2 * 3 =
3 + 3

Of course,
with just 2 similar rows, multiplication does not offer any real benefit over
addition. However, say with 100 similar rows, multiplication would then offer a
much simpler way of expressing the total no. of objects (than the successive
addition of many rows).

However
there is a hugely important - though largely unrecognised - flaw to this
interpretation of multiplication.

Again, the
conventional mathematical approach to cardinal number is based on the
assumption that unit members are independent of each other in an absolute
fashion.

However,
the very process of multiplication entails the recognition - as in the example
above - of the common similarity as between the various rows (or alternatively
various columns) and likewise the common similarity as between members of each
row (and column).

Thus in our
example, the use of 2 as an operator depends on recognition that the two rows
are similar in number terms, i.e. share a mutual interdependence.

And from an
ordinal perspective, it does not matter which row is identified as 1

^{st}or 2^{nd}, for by definition they are mutually interchangeable.
Therefore,
in this example, the recognition of the similarity of the two rows implies the
holistic - rather than the analytic - interpretation of 2, where the mutual
interdependence of both unit rows is recognised.

Expressed
in an equivalent manner the recognition of the similarity of the two rows,
whereby they are recognised as mutually interdependent, entails the qualitative
aspect of 2 (as “twoness”). By contrast the recognition of the independent
nature of each row entails the quantitative aspect of 2 (as two).

This
recognition of similarity as between the two rows equally implies recognition
of the similarity i.e. shared interdependence of the 3 items within each row
(where 1

^{st}, 2^{nd}and 3^{rd}units are potentially interchangeable with each other). Equally it implies recognition of the similarity of the 2 items within each column (where 1^{st}and 2^{nd}units are potentially interchangeable with each other).
So all in
all, the operation of multiplication implies recognition of both the analytic
independence (where units are viewed as separate) and the holistic
interdependence of number (where units are viewed as similar).

And this
operation can only be properly understood in a dynamic interactive manner,
where both the analytic (quantitative) aspects of number as independent and the
holistic (qualitative) aspects of number as interdependent, are viewed as
complementary.

This
likewise applies to addition.

It is all
very well treating numbers as independent in an analytic manner, but the consequent
result from addition requires a transformation to a new holistic identity (not
apparent in the independent units).

So, for
example, we can start by viewing 1 + 1 as independent units in individual
quantitative terms. However the very act of recognition, enabling the new whole
collective identity of 2, requires corresponding realisation of the
interdependence of unique units (as interchangeable). So addition and
multiplication properly entail both analytic and holistic aspects.

Thus the
crucial point here is that addition and multiplication are analytic
(quantitative) and holistic (qualitative) with respect to each other with both
operations complementary in dynamic interactive terms.

This
implies that when - as is customarily the case - addition is identified
directly with the quantitative transformation of number (in default
1-dimensional terms), multiplication is then - relatively - identified with its
qualitative transformation (where the dimensional aspect changes) and vice
versa.

Thus, in the
simplest case, from the former perspective of addition, 1

^{1 }+ 1^{1 }= 2^{1}. So here we have a quantitative change in number in linear (1-dimensional) terms. Thus the base aspect of number changes while the dimensional aspect (implicitly) remains fixed.
However,
from the latter perspective of multiplication, we have 1

^{1 }* 1^{1 }= 1^{2}. Here in inverse relative fashion, a qualitative change in number occurs. So in a complementary manner, the dimensional aspect of number now changes while the base aspect remains (implicitly) fixed.
In
conventional mathematical terms, it is customary to represent the natural
numbers (without reference to base or dimensional characteristics), once again
reflecting the reduction of qualitative to quantitative characteristics.

Therefore
the natural numbers are listed, representing solely number quantities, as 1, 2,
3, 4, …

However
from the enhanced interactive perspective, number possesses two distinct
aspects (base and dimension respectively) both of which alternate as between
analytic (quantitative) and holistic (qualitative) meanings in the dynamics of
understanding.

Alternatively,
we could say, using a very close physical analogy, that number keeps switching
as between its particle (independent) and wave (interdependent) aspects, with
respect to both base and dimensional characteristics.

So again
rather than the natural numbers being given in absolute fashion as 1, 2, 3, 4,

^{ }…, we now have two twin interacting aspects,
1

^{1}, 2^{1}, 3^{1}, 4^{1}, … and 1^{1}, 1^{2}, 1^{3}, 1^{4}, …, which I customarily refer to - in relative terms - as the Type 1 and Type 2 aspects of the number system respectively.
And when
understood in the appropriate dynamic manner, experience of these two aspects
keep dynamically switching in complementary fashion, as between analytic
(quantitative) and holistic (qualitative) aspects with respect to both base and
dimensional characteristics, depending on which reference frame is
employed.

So, if the
base aspect of number is interpreted in a quantitative manner, then the
dimensional aspect, relatively, is qualitative; however if the base aspect is
interpreted in a qualitative manner, then in this context, relatively, the
dimensional number is now quantitative.

Thus all
numbers can be given - depending on relative context - analytic and holistic
interpretations with respect to both base and dimensional characteristics.

Both Type 1
and Type 2 aspects, in this interactive appreciation, are intimately related to
the key distinction as between addition and multiplication as complementary
opposites.

When, for
example, we multiply 1

^{1}* 1^{1}to obtain 1^{2}, a qualitative change in the nature of number as dimension takes place. So 2 in this context, represents the holistic notion of dimension, relating to the interdependence of both unit members. Though in direct terms the transformation applies to dimensional units, implicitly this likewise entails the unit base numbers.
Then, when
we add 1

^{1}+ 1^{1}to obtain 2^{1}, a quantitative change, relatively, in the nature of the base units takes place. So 2, by contrast, now represents the analytic notion of number (as base), relating to the independence of both unit members. However, though in direct terms the transformation now applies to the base units, implicitly this likewise entails the dimensional notion of 2.
Therefore
the remarkable fact remains that whenever we use any number, with respect to
base or dimensional aspect, in the standard analytic quantitative manner, this
likewise implicitly requires the corresponding holistic notion of that same
number, which has now become implicitly embodied, as it were, in the number.

So from a
psychological perspective, the conscious rational interpretation of number
always carries the distant echo of its unconscious holistic counterpart.

However, in
the main, we remain completely deaf to this unconscious echo.

So we
cannot explicitly define addition (without implicitly requiring
multiplication); and we cannot explicitly define multiplication (without
implicitly requiring addition).

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