As we have
seen number can be defined in two ways (which dynamically interact in
mathematical experience).

Now to see
this clearly, numbers must be defined with respect to base and dimensional
numbers (which properly in a dynamic manner, are quantitative and qualitative
with respect to each other).

So again to
take the simplest (non-trivial) case of the number 2, this can be defined in
two distinct ways.

In the
conventional quantitative approach - related to (pure) addition

2 = 1 + 1

i.e.
expressed more fully

2

^{1 }= 1^{1 }+ 1^{1}^{ }

So the
essence of this quantitative relationship is that default dimension to which
the (base) number quantity (2) is expressed remains fixed as 1.

And as I
have repeatedly stated, Conventional Mathematics, geared exclusively to the
quantitative interpretation of relationships is precisely defined, from a
qualitative perspective, in terms of its linear (i.e. 1-dimensional) mode of
interpretation.

So in
Conventional Mathematics with respect to multiplication a merely reduced (i.e.
quantitative) interpretation is possible.

So in this
context if we multiply 2 by 2 i.e. 2

^{2}, the answer will be given as 4 i.e. 4^{1}.
Thus the
answer in reduced quantitative terms is expressed with respect to the default
dimensional value of 1 (which again illustrates well the 1-dimensional nature
of mathematical interpretation).

However if
we think about it for a moment, clearly a dimensional transformation (of a
qualitative nature) is likewise involved.

Thus in
simple geometrical terms, 2 * 2 would be represented by a square (with side 2
units).

Therefore
the area of this square is properly expressed in square (i.e. 2-dimensional
units).

Thus a
dimensional change of a qualitative nature is clearly involved through this
simple multiplication process.

However in
conventional mathematical terms the qualitative nature of this transformation
is simply edited completely out of the process with the result expressed
thereby in a reduced i.e. merely quantitative manner.

And such
reductionism universally characterises the nature of multiplication from a
conventional mathematical perspective.

It is
hardly surprising therefore that mathematicians eventually realise that there
is something fundamentally missing from their appreciation of the relationship
as between addition and multiplication (as multiplication is inherently
interpreted in a reduced merely quantitative manner).

This issue
struck me so strongly at the age of 9 or 10, that I already realised then that
there was - literally - a fundamental dimension with respect to
conventional mathematical understanding that effectively was overlooked. In other words in the mere
quantitative interpretation of number, its qualitative dimensional aspect is
simply ignored.

So the
conventional quantitative approach to number - where number is defined in a
cardinal manner - I refer to as the Type 1 aspect of the number system.

So again in
Type 1 terms,

2

^{1 }= 1^{1 }+ 1^{1}
However in
the corresponding Type 2 aspect, the number 2 is expressed in terms of a (pure)
multiplication process.

So in Type
2 terms,

2 = 1 * 1

In other
words,

1

^{2 }= 1^{1 }* 1^{1}
Now 2
in our definition of number has been inverted, with 2 representing the dimensional
qualitative nature of number, which is defined with respect to the default
(base) quantity of 1.

So again if
we think of this in geometrical terms, when we square 1, we obtain a
2-dimensional figure (with area 1 sq. units)

Therefore, though
in quantitative terms nothing has changed (with the 2-dimensional area the same
as its 1-dimesnional side), clearly a qualitative change in the nature of units
has been involved.

Thus, the
very purpose of the Type 2 approach is to isolate the qualitative aspect of
number transformation, which is related directly to the pure nature of
multiplication.

As I
have repeatedly stated in my blog entries, the mystery of the relationship of
addition and multiplication is the same mystery as the relationship of the
quantitative and qualitative aspects of number.

Once again there are insuperable difficulties in attempting to understand this
relationship in a mere quantitative manner!

However
having isolated the qualitative aspect of number transformation (through its
Type 2 aspect), the next problem is in attempting to give expression in an
indirect quantitative manner to this aspect.

The deeper
reason for this is that when one accepts that there are necessarily two
distinctive aspects to number (which are quantitative and qualitative) with
respect to each other, then the ultimate consistency of both aspects becomes
the indisputable key mathematical requirement.

So
demonstrating such consistency - which clearly cannot be proved within the reduced
conventional mathematical perspective - requires the ability to (i) indirectly
convert from the qualitative to the quantitative aspect and (ii)
the ability in reverse manner to convert from the quantitative to the
qualitative aspect, while establishing complete harmony between both aspects.

Put another
way it requires establishing the interdependence of both quantitative and
qualitative aspects (from two complementary perspectives).

Now once
again, by its very nature, this is not strictly possible within the
conventional mathematical perspective (defined as it is in a mere quantitative
manner).

Such
interdependence relates directly to a holistic - rather than analytic - type
appreciation of mathematical relationships.

It is
central to appreciation of the true nature of the number system and yet in
formal mathematical terms currently does not even exist.

To sum up
this blog entry, properly understood in dynamic interactive terms, there are
two distinctive ways of interpreting every number.

The Type
1 relates to its quantitative aspect, directly associated with the (pure) notion
of addition.

The Type 2
relates to its qualitative aspect directly associated with the (pure) nature of
multiplication.

Both of
these continually interact in a - necessarily - relative manner in experience
and are directly associated with the cardinal and ordinal aspects of number
interpretation respectively.

Conventional
mathematical interpretation therefore represents but a highly reduced - and
greatly confused - interpretation of the number system.

We will
have more to say about the - much misunderstood - ordinal nature of number in
the next entry.

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