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 1st 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 21.
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 1st 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 1st root of 1 and – 1 the 2nd root!
Now fully explained + 1 represents the square root of 12 while – 1 represents the square root of 11.
So the 2 roots in this case (in quantitative terms) are directly related to the corresponding 2 dimensions of 2 (i.e. 1st and 2nd) in qualitative terms.
This implies that when we incorporate both the positive and negative expressions with respect to the 1st dimension, we move in fact to 2-dimensional (rather than 1-dimensional) interpretation.
So the 1st dimension (of 2) is concerned with the (analytic) positing of conscious phenomena in experience, whereas the 2nd 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 2nd (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 3rd 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!