In yesterday’s blog entry, I emphasised how all experience - including mathematical - is conditioned by two fundamental polarity sets that are (i) external and internal and (ii) quantitative and qualitative with respect to each other.
These polarities in fact are the basis for the alternative Type 2 aspect of the ordinal number system, where numbers are represented as equidistant points on the circle of unit radius (drawn in the complex plane).
So the first set of external and internal poles - initially with respect to conscious understanding - are represented on the (horizontal) real axis as + 1 and – 1 respectively.
Once again the cardinal (Type 1) aspect of the number system treats a number e.g. “2”, as a collective unit in quantitative terms. So if we attempt to sub-divide it as the sum of number parts, we must represent these in a homogenous manner (i.e. without qualitative distinction) as 1 + 1.
The ordinal (Type 2 aspect) then treats the number 2 in complementary fashion with respect to its distinctive individual components in a qualitative manner.
So from this qualitative perspective, 2 is composed of a 1st and 2nd member that are uniquely distinct in a relative manner. Now we can represent these 1st and 2nd members, indirectly in a quantitative manner, through obtaining the corresponding two roots of 1 i.e. + 1 and – 1 respectively.
However the true qualitative recognition of these two polarities as interdependent (which defines the qualitative aspect) comes from simultaneously combining both directions.
So this would be represented as + 1 – 1 (in indirect quantitative terms) = 0 (from a direct qualitative perspective).
Now the recognition of 0 in this context is directly of an intuitive rather than rational nature representing the Type 2 appreciation of the number “2”.
Thus in summary the Type 1 aspect is - directly - of a (linear) rational nature geared to the quantitative interpretation of “2”.
The Type 2 aspect by contrast is - indirectly - of a (circular) rational nature, i.e. paradoxical, culminating in direct intuitive recognition. This provides the corresponding qualitative interpretation of “2”.
Now with respect to our experiential understanding, both of these aspects necessarily interact in continual fashion.
However the qualitative aspect is then completely edited out in terms of accepted formal mathematical interpretation.
Therefore though the true understanding of number is thereby inherently dynamic in nature, Conventional Mathematics is built on a significantly reduced - and thereby greatly distorted - interpretation (i.e. that recognises merely the quantitative aspect).
Put another way the Type 2 aspect is inherently geared to appreciation of the manner in which the fundamental polarities (underlining all experience) interact.
Therefore associated with each number from this perspective is a corresponding set of individual ordinal members (as distinctive directions with respect to the two basic sets of polar co-ordinates).
So for example the number “4” is associated with four ordinal members represented by the four equidistant points on the unit circle. So once again along the real axis we again have + 1 and – 1 and now two additional points along the vertical axis i.e. + i and – i respectively.
Now + 1 and – 1 along the real axis are identified with understanding of a direct conscious nature. + 1 literally relates to the unitary direction of experience whereby phenomena are posited in conscious manner (which in our present scientific framework are thereby identified as “real”).
– 1 then relates to the (unconscious) negation of such phenomena which serves as the very means by which we are thereby enabled to switch polar direction (e.g. from external to internal) in experience.
When such dynamic switching takes place in a flexible manner, significant amounts of intuitive energy are generated (through the complementary interaction of both poles) which thereby enables understanding of a creative nature.
However when little dynamic switching occurs, understanding becomes ever more rigid in nature whereby existing assumptions are constantly re-affirmed.
This is why I would expect considerable resistance to the views that I am expressing.
Once again conventional mathematical interpretation is strongly 1-dimensional in formal terms. This means therefore that the conscious rational direction (+ 1) is solely recognised. Though implicitly, a degree of unconscious intuition informally takes place, it operates solely within the accepted paradigm.
However there will always be some - not necessarily professional mathematicians – operating at the margins, that perhaps suspect a fundamental problem may indeed exist with present Mathematics. And it this audience that I am mainly addressing!
Just as the notion of “real” can be given - according to the Type 2 aspect - a holistic (qualitative) mathematical meaning (i.e. as corresponding to linear rational interpretation) the notion of “imaginary” can be given a vitally important holistic interpretation.
As we know important national and religious symbols can convey a holistic significance whereby they embody an unconscious desire for meaning.
If for example we compare national flags there is little to distinguish one from another (from a mere rational perspective). However when we accept that a flag can embody deep notions of identity, we can then perhaps recognise that the significance is more of an unconscious than conscious origin.
If we generalise, then all local symbols of a conscious kind necessarily also embody projections of an unconscious universal nature.
Now in a precise mathematical manner, the very notion of “imaginary” relates to the indirect linear rational attempt to convey meaning that is properly of an (unconscious) holistic nature.
And as the Type 2 aspect of the number system is indeed properly of such a holistic nature, one could accurately express this in qualitative terms as the “imaginary” aspect of the number system.
In quantitative (Type 1) terms i is expressed as the square root of – 1.
It is similar in qualitative (Type 2) terms. – 1 here represents the negation of (conscious) understanding. A dynamic fusion thereby results through interaction with the existing positive direction leading to the generation of spiritual intuitive energy (that is inherently 2-dimensional in nature).
The resulting attempt to explain such holistic understanding indirectly (in a linear fashion) entails the notion of a square root (in a qualitative manner).
So again Type 2 represents the “imaginary” counterpart to the recognised Type 1 aspect of mathematical understanding.
Therefore we can perhaps now appreciate that just as we can define both real and imaginary aspect to numbers in quantitative terms, equally we can define real and imaginary aspects in a qualitative manner.
So a comprehensive paradigm for Mathematics is necessarily of a complex rational nature (with real and imaginary aspects).
The great limitation of Conventional Mathematics is that it is solely interpreted in a real rational manner!
So these two imaginary directions (+ i and – i) represent – in Jungian terms – the archetypal nature of number (as embodying a holistic qualitative element) now indirectly expressed in a rational manner. And once again this precisely defines the nature of the Type 2 aspect.
So we can only posit the qualitative aspect of number in an indirect conscious manner, as the true nature of holistic interdependence is unconscious in origin.
And as this qualitative nature continually alternates between the whole (in relation to the parts) and the parts (in relation to the whole) negating as well as positing with respect to number must continually take place.
Therefore the Type 2 nature of “4” relates to this more refined interaction as between both its real and imaginary co-ordinates (that are positive and negative respectively).
In principle any number “n” can be indirectly defined in Type 2 terms with respect to its n individual roots (the full combination of which represents its true interdependent appreciation)
And the sum of the n roots of 1 (except 1) = 0. So this circular interdependence of the all the ordinal members of n represents the Type 2 interpretation of number (in its pure qualitative appreciation).
In physical terms as nature becomes ever more dynamic at sub-atomic levels, an increasing number of directions (i.e. dimensions) is involved with respect to polar interactions.
Likewise in psycho spiritual terms as contemplation becomes ever more refined, appreciation with respect to a growing multiple of directions can be explicitly brought into conscious awareness.
In fact what we are talking about here - in Type 2 terms - is the direct appreciation of each number as representing a pure energy state.
And - as always - we have complementary directions in both physical and psychological terms.
Thus in Type 2 terms, every number - in principle - has a direct physical (or more correctly psychophysical) relevance as a pure energy state.
Likewise every number has a direct psycho spiritual relevance as a pure (intuitive) energy state.
Therefore in the dynamics of experience, intuition and reason implicitly interact enabling one to appreciate (to some degree) both cardinal and ordinal aspects with respect to number .
However because explicitly our subsequent formal interpretation is merely rational, we misleadingly identify the ordinal with the cardinal aspect.
Therefore we think that 1 (as cardinal) implies 1st (as ordinal), 2 (as cardinal) 2nd (as ordinal), 3 (as cardinal) 3rd (as ordinal) and so on! In fact the very process enabling us to make these connections entails the whole mystery of how the primes are related to the natural numbers (and the natural numbers to the primes) which entails two sets of zeta zeros (as complementary shadow systems).
So in this important respect our understanding of number still remains greatly confused.