Thus in
standard linear terms, where its sub-units are considered in an absolute
independent manner, 2 is given an analytic quantitative identity (in cardinal
terms).

However in
paradoxical circular terms, where its sub-units are now considered as
relatively interdependent - and thereby fully interchangeable with each other -
2 is given a holistic qualitative identity (in an ordinal manner).

Though the
ordinal nature of number is of course recognised in conventional mathematical
terms, it is invariably reduced in a merely quantitative manner (whereby each
position is given a fixed identity).

Thus using
a physical analogy from quantum physics, every number can manifest itself in
both a particle and wave-like manner, and these two aspects necessarily keep
switching with each other in the dynamics of understanding.

However the
conventional mathematical approach grossly misrepresents the true nature of
number by attempting to (formally) interpret it in a static fixed manner.

Now, I have
frequently referred to the two aspect of number as Type 1 and Type 2
respectively!

Type 1
corresponds with the linear conventional interpretation of number in analytic quantitative
terms.

The natural
numbers from this perspective (1, 2, 3, 4, 5, …) are more fully represented
as

1

^{1}, 2^{1}, 3^{1}, 4^{1}, 5^{1}, …
Type 2
corresponds (indirectly) with the circular paradoxical interpretation of
numbers in a holistic qualitative manner

The natural
numbers from this perspective are more fully represented as

1

^{1}, 1^{2}, 1^{3}, 1^{4}, 1^{5}, …
Now these
have no distinctive meaning in quantitative terms. However from a holistic
qualitative perspective 1

^{2}represents the interdependence of two related units.
Then
indirectly this interdependence can be expressed in a holistic circular manner
as + 1 and – 1, where both units
are interchangeable.

(Of course + 1 and – 1 can equally be expressed - as in
conventional mathematical terms - in an absolute manner where they are
clearly separated in an analytic manner).

However it is the continued failure to recognise the true
holistic aspect of number, which I am addressing in these blogs.

So mathematical understanding necessarily contains both analytic
and holistic aspects, which in direct terms are represented by reason and
intuition respectively.

For example I have just read this quote from Poincare.

“It is by logic, we prove, by intuition we invent”

Logic - in the sense that Poincare intends
- represents the analytic aspect of understanding which is of an unambiguous
rational nature.

However intuition represents the corresponding holistic
aspect, which inherently is of a paradoxical circular nature.

Therefore though analytic reason and holistic intuition are
clearly distinct from each other, in conventional mathematical terms the
holistic aspect is inevitably reduced in formal terms to the analytic.

In this sense formal mathematical interpretation is of a
grossly reduced nature (and ultimately not fit for purpose).

But before this crucial point can be properly grasped, the
distinctive holistic aspect of all mathematical understanding must be properly
recognised.

Now if we go back to the original simple expression i.e. x =
1, we can perhaps now better appreciate what happens when both sides are raised
to the power of n.

So x

^{n}= 1^{n}.
Therefore when n = 2, x

^{2}= 1^{2}.
Therefore, the number on the right side of the equation
properly belong to the Type 2 aspect of the number system.

However in conventional mathematical terms, as 1

^{2 }has no distinctive quantitative value, it is inevitably reduced to 1 (i.e. 1^{1}) where it is interpreted as the first natural number in the Type 1 system.
Thus a number that should be treated in a holistic manner -
and indirectly expressed in a circular interdependent fashion - is now treated
in analytic terms.

And this is what then creates the conflict with earlier
analytic type interpretation.

So once again when x = 1, x – 1 = 0.

Therefore squaring both sides (x – 1)

^{2}= 0.
And this equation in Type 1 terms has two “linear” solutions
i.e. + 1 and + 1 (as independent).

However equally when x = 1, x

^{2}= 1^{2 }.
And this equation in Type 2 terms has two “circular”
solutions i.e. + 1 and – 1 (as interdependent).

Thus we obtain two “different” answers because both correspond
respectively to different notions of dimensions (that are analytic and holistic
with respect to each other).

However again, these two sets of answers cannot be reconciled
satisfactorily through conventional mathematical interpretation (that solely
recognises the analytic aspect of number as quantitative).

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