^{rd}(higher) band on the Spectrum often arises through an existential crisis leading to fundamental questioning of the dualistic nature of conventional activity.

This in turn can lead to a substantial deepening with respect to the holistic (unconscious) aspect of the personality.

Once again
the middle stages - especially in a scientific and mathematical context - are
characterised by the 1-dimensional approach to understanding, which again simply
means interpretation that is made with respect to independent poles of
reference.

So to consider quantitative meaning in abstraction from its
complementary qualitative pole,
represents 1-dimensional (linear) interpretation; likewise to consider
objective meaning as independent of subjective, again precisely represents
1-dimensional interpretation, as I define it!

And the
acceptance that such interpretation is possible defines science and mathematics
as currently understood!

So during this transitional stage, conscious (dualistic) phenomena
i.e. in accordance with 1-dimensional interpretation, are dynamically negated in
an unconscious (nondual) manner (though initially there are inevitably
substantial limitations on the extent to which this process can take place).

Quite
literally therefore, from a holistic mathematical perspective, such experience
takes place qualitatively in the negative 1

^{st}dimension.
We have
seen already that implicitly this negative dimension is necessarily involved in
the experiential process of moving from whole to part recognition.

So if I am
aware of 2 in (cardinal) terms this can be expressed more fully as 2

^{1}.
Now the
corresponding part recognition of its reciprocal ½ can be more fully expressed
as 2

^{– 1}.
So the very process of switching from whole to part recognition in
experience implicitly requires the intervention of the unconscious, which
directly facilitates this dynamic switch to occur.

Again -
quite literally - in holistic mathematical terms, this entails
experience of the negative 1

^{st}dimension. Now when we interpret the actual result of the switch i.e. ½, it is again interpreted in a conscious dualistic manner.
However
what is completely missing from Conventional Mathematics is recognition of this vital
qualitative (holistic) dimension, which literally enables such dynamic
switching between different polar reference frames to take place.

Now this part
- as opposed to whole - notion of number in the reciprocal of a given number,
is closely associated with its ordinal identity.

This for
example can be more clearly appreciated in the case of the reciprocal of 3 i.e.
1/3. We would express this in ordinary language as a third. Likewise we could
express 1/4 , 1/5. 1/6 etc. as a fourth (or
a quarter), a fifth, a sixth etc.

This in a
sense is fascinating, for though all these represent number quantities
(as conventionally understood) yet we use ordinal language - which properly
is of a qualitative nature - to express such fractions!

And as
we have seen, this qualitative aspect is always implicitly involved in experience, enabling one
once more to switch polar reference frames i.e. from whole to part recognition.

This
unconscious intuitive recognition, as I say, remains merely implicit in
conventional mathematical terms. So explicitly, number integers and their
reciprocals are formally interpreted in a conscious rational manner.

Now, during this transition stage, unconscious recognition moves from
an (unrecognised) implicit to a - now - increasingly recognised explicit
state.

The
implications of this for mathematical and scientific interpretation are of the
very first magnitude. It entails that we can no longer hope to properly
understand relationships in a merely static conscious manner. Rather we must likewise incorporate the unconscious, thereby enabling switching between opposite polar reference
frames to take place.

An even
deeper connection can be made!

If we go back
to the - apparently – simple questions of the two roots of 1, in standard quantitative
terms these are given as + 1 and –
1 respectively.

So + 1 here
represents the 1

^{st}root of 1 and – 1 the 2^{nd}root!
Now fully
explained + 1 represents the square root of 1

^{2 }while – 1 represents the square root of 1^{1}.
So the 2
roots in this case (in quantitative terms) are directly related to the
corresponding 2 dimensions of 2 (i.e. 1

^{st}and 2^{nd}) in qualitative terms.
This
implies that when we incorporate both the positive and negative expressions with
respect to the 1

^{st}dimension, we move in fact to 2-dimensional (rather than 1-dimensional) interpretation.
So the 1

^{st}dimension (of 2) is concerned with the (analytic) positing of conscious phenomena in experience, whereas the 2^{nd}is concerned with the (holistic) negation of such phenomena (in an unconscious manner).
Put another
way, whereas 1-dimensional interpretation in mathematical terms is merely
quantitative in nature, 2-dimensional interpretation combines both quantitative
and qualitative aspects.

I have
developed this point at some length here because of its great importance.

We have
been accustomed to considering contemplative spiritual type development as
having no direct relevance for mathematical interpretation.

However
this position in truth is completely untenable.

What we
define as Mathematics, simply reflects the reduced kind of
understanding (i.e. 1-dimensional) that characterises the 2

^{nd}(middle) band on the spectrum of development.
However
once we move to higher bands, a new more refined kind of interpretation is necessary
with respect to all mathematical (and of course scientific) relationships.

In fact the
3

^{rd}band opens up an entirely new qualitative aspect of mathematical understanding, that I refer to as Type 2 Mathematics (i.e. Holistic Mathematics)
So I have
been concentrating briefly here on the profound mathematical implications of the
emergence of the 3rd band!

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