We return here to the issue of what higher dimensional interpretation of mathematical symbols properly entails.
Once again, Conventional Mathematics is defined by merely 1-dimensional
interpretation (in qualitative terms).
This means is that in any context qualitative is thereby
reduced to quantitative meaning.
More precisely it means that in any context for consideration that merely
one fundamental pole of reference is used (in an absolute type manner).
Again these fundamental poles - which can be summarised as two basic sets -
condition all phenomenal experience.
Therefore in horizontal terms all external (objective) experience of
reality requires an internal (subjective) counterpart;
Likewise in vertical terms all part (quantitative) experience of reality
requires a whole (qualitative) counterpart.
In fact we can simplify this further by referring to just one polar set
(i.e. quantitative and qualitative) with both horizontal (real) and vertical
Thus in this sense 4-dimensional interpretation would be geometrically
represented (on the unit circle in the complex plane) in indirect quantitative terms by the four roots of 1 i.e. +
1, – 1, + i and –
However in direct terms these symbols now take on a distinctive qualitative
So + 1 represents the 1st dimension which in mathematical context entails
the standard positing of symbols in a conscious rational manner.
– 1 represents the corresponding
negation (of what is consciously posited) and is the means by which
(unconscious) intuition is generated in experience.
Thus 2-dimensional interpretation is
already of a dynamic interactive nature, explicitly entailing both reason and
+ i represents the positing of
symbols in an imaginary - rather than real - fashion.
Real in this context refers to
the positing of what is directly of a conscious nature; imaginary then
refers to the indirect positing in a conscious manner of what is primarily of a
holistic unconscious nature.
When this imaginary aspect is not
properly understood, unconscious meaning is blindly projected into
consciousness as projections which are then confused with localised
Now – 1 represents the 2nd
dimension (in the context of 2). So to express this in a 1-dimensional linear
manner, we obtain - what in qualitative terms - is the square root.
Thus + i represents the
indirect linear analytic expression of what is real (in circular holistic
Thus with such imaginary
understanding we recognise symbols (that appear as conscious) as indirect expressions
of a deeper holistic meaning.
– i then represents the
negation - strictly the negation of rigid attachment to - such imaginary
symbols paving the way for an even purer intuitive type realisation.
Now all the "higher"
natural number dimensions can be seen as representing various configurations
with respect to our fundamental polarity sets.
In a way it is a bit like a
compass. We can get fix four points to represent our key directions N, S, E and
However then using these
co-ordinates we can obtain ever more refined notions of direction.
In similar fashion we need 4
dimensions to properly fix our basic polarity positions.
Higher numbered dimensions can then be fruitfully understood as representing ever more refined configurations of reason and intuition (i.e. analytic and holistic type interpretation) in the increasingly dynamic interaction as between conscious and unconscious, where ultimately both dissolve into each other as ineffable.
Now the significance of the primes in this context is that they provide the basic building blocks for all the composite natural number dimensions.
However it is important to recognise that we are here using prime numbers in a completely opposite manner to their normal understanding as "building blocks".
In conventional cardinal number terms, the primes are seen as the most independent of numbers with all composite natural numbers representing a unique combination of primes.
However here in ordinal number terms, the primes are seen as the most interdependent of numbers with all composite natural numbers (as dimensions) representing unique combinations of these prime circles of interdependence.
And this truly reveals the key paradox regarding prime numbers.
From a cardinal perspective (in quantitative terms) they appear as the most independent of all numbers; however from the corresponding ordinal perspective they are the most interdependent.
Thus the mystery of the prime numbers resides in the fact that they combine these two opposite characteristics in their very nature. And the relationship between these complementary aspects can only be properly understood in a dynamic interactive manner.
Thus to recap, the qualitative notion of number as dimension plays a vitally important double role (that is still totally unrecognised within the conventional mathematical perspective).
In psychological terms, each number representing a dimension provides a unique means of configuring the interaction as between (conscious) reason and (unconscious) intuition
From the corresponding physical perspective each number equally provides a unique means of configuring the (quantitative) analytic with the corresponding (qualitative) holistic aspect of reality.
Then the indirect quantitative expressions of these dimensions provides the appropriate means of making ordinal distinctions between numbers (which are unique - except the 1st - for each prime number).
So in this way for example the 5 roots of 1 represent the indirect quantitative expression of the number 5 (as qualitative dimension).
And in this context, these 5 roots - except 1 - provide the appropriate means in a quantitative circular fashion of representing uniquely the ordinal notions of 2nd, 3rd, 4th and 5th!
And though we may still attempt to interpret ordinal distinctions in a merely objective physical manner, properly speaking they necessarily reflect the dynamic interplay of both physical and psychological aspects that are interdependent.