Mathematical Notions as Primary Archetypes

In previous blog entries I have mentioned the fundamental importance of mathematical notions esp. with respect to number and the operations of addition (and subtraction) and multiplication (and division). These serve as the essential interface as between the (spiritual) world of emptiness and the (material) world of form, thereby in a crucial sense embodying both aspects, though not being capable of direct identification with either aspect separately.

An equivalent way of expressing this point is that such mathematical notions represent the original archetypes from which all phenomenal reality is necessarily derived.

So I have already referred to the somewhat paradoxical conundrum that whereas neither emptiness nor form can be directly identified with mathematical notions in their separate analytic and holistic aspects yet, what we identify as phenomenal creation (which represents the interpenetration of emptiness and form) simply results from the dynamic interaction of both the analytic and holistic aspects of these mathematical notions.

I would have a problem however with more conventional notions of archetypes which for me represent secondary rather than primary expressions.

Plato does indeed refer to mathematical archetypes in his identification of the 5 solids i.e. the tetrahedron (4 faces), the cube (6 faces), the octahedron (8 faces), the dodecahedron (12 faces) and the icosahedron (20 faces).  These preserve through rotation a perfectly symmetrical structure and are identifiable as fixed forms. However - as we have seen - the most fundamental mathematical notions, while necessarily are already inherent, cannot be directly identified with these forms.

So in truth they are somewhat more elusive existing as the mysterious interface as between the spiritual domain of emptiness and the material world of form.

Neither can the well-known Jungian view of archetypes by viewed as primary.

For Jung the archetypes comprise images existing in the collective unconscious of mankind that play a universal role in subsequent human behaviour.

Now he identified the four major archetypes as the Self, the Shadow, the Anima (and Animus) and the Persona.

Again, these really are secondary, as primary mathematical notions are implicitly contained in these archetypes.

Implicitly contained in the Self, in the unity of both conscious and unconscious, are the holistic notions of 1 and 0, Then the Shadow as the hidden (unconscious) counterpart to the ego implicitly entails the holistic notion of subtraction as negation (–).  The Anima (animus) entails the holistic mathematical notion of the imaginary (in contrast to the real notion of the ego). The Persona then as the conscious projection of an ideal self thereby entails the holistic notion of addition as positing (+). 

However the issue I chiefly wish to address here is that these primary archetypes (as fundamental mathematical notions such as number and operation) have two equally valid complementary expressions (which ultimately are identical in an ineffable manner).

Indeed one could validly refer to these archetypes as transcendent and immanent (or alternatively transpersonal and prepersonal). And with respect to both of these, confused and mature expressions exist.

And remember we already have identified that normally when archetypes are used, it is their secondary rather than primary nature that is involved!

For example many of the myths to which Jung refers represent their secondary confused expression in an immanent fashion (as inherited potential for future growth).

They are secondary because they do not directly entail mathematical notions (that I have already identified as primary). They are confused because the phenomenal forms entailed in such myths are given a somewhat literal meaning. Finally they are immanent as they are viewed as inherited psychic potential already available at birth.

On the other hand many of the doctrines of the great religious traditions typically represent secondary confused expressions in a transcendent fashion.

Again they are secondary because they do not directly relate to the fundamental mathematical notions (which are necessarily inherent in all phenomenal understanding).

They are then to a degree often confused when the spiritual truth which they attempt to convey e.g. when the Trinity in the Christian tradition, is interpreted in a literal fashion.

They are transcendent in that appreciation of such truths is seen as the ultimate goal of the spiritual life.

A key point that I have repeatedly addressed in my recent blogs is that the process in development, which leads to transcendence, is quite distinct - though ultimately complementary - to the corresponding process that leads to immanence.

Speaking from personal experience, the zenith in terms of mastery of the analytic aspect of mathematical understanding (where symbols are understood in a rigid absolute manner) earlier occurred during the formal operational stage (of Band 2). I remember for example at about the age of 15 completing an algebra paper for one of the State exams in less than half the time allocated, as it came easily to me at that time in my life.

However I was somewhat  reluctant to subsequently accept that this specialised analytic understanding had already peaked at that age and was set to steadily decline in future years.

The decline of the analytic coincided with a new emerging holistic mathematical ability. However as there was no formal recognition in the culture for such ability, I was destined to travel thereafter an often lonely and unfamiliar route.

The first substantial breakthrough of sustained holistic appreciation occurred at Band 3 (Level 3) with respect to the transcendent nature of mathematical symbols.

This then represented the opposite extreme to the absolute understanding of Band 2, where, by contrast, such symbols were understood - at Band 3 - in terms of their purely relative nature. Here they now represented the elusive - and highly dynamic - frontier between form and emptiness before melting entirely into that emptiness.

I have illustrated this point many times before with respect to the simple example of

1 – 1 = 0.

From an analytic perspective this seems a trivial mathematical relationship.

Here 1 and 0 are considered as absolutely independent each other. Then when 1 is subtracted from the same number the result is nothing in phenomenal terms i.e. 0.

However 1 – 1 = 0 also has an important holistic meaning, where the RHS represents the unity (of all form) and the LHS emptiness (with respect to all form).

And in holistic terms these are fully interdependent with each other in a purely relative manner.

So with transcendent spiritual development, as one’s intuitive capacity becomes increasingly refined, one gradually moves beyond any rigid notion of form to clearly see in spiritual  terms the true interdependent nature of all such form.

And the essence of such interdependence lies ultimately beyond any phenomenal identification of its nature. So the last frontier as it were of this increasingly intuitive ability relates to pure holistic appreciation of mathematical notions such as number and the operations connecting number.

And these notions thereby relate to the primary archetypes.

This transcendent appreciation is of an increasingly general nature where the whole (in its purely spiritual nature) is understood as beyond the collection of all phenomenal parts.

As already stated this is associated - in my terminology - with the most advanced of the contemplative stages of the ascent i.e. Band 3 (Level 3).

I also refer to this as the point level. From the more traditional Eastern perspective it is referred to as the causal realm. So with this level one approaches the stage where phenomenal form in its most elusive fundamental structure (as mathematical) finally gives way to pure emptiness.

Now strictly the mathematical archetypes are most keenly recognised during the transition from Band 3 to Band 4 with the latter Band leading to the attempted specialisation of pure contemplative awareness. 

However inevitably a conflict will emerge as between the transcendent aspect (as spirit beyond form) and the corresponding immanent aspect (as spirit already inherent within form).

So in my approach, properly addressing this issue requires a lengthy further period of development entailing the three levels of Band 5.

This in turn requires a switch from attempted top-down integration of the psyche (where lower levels are integrated with respect to the highest level yet experienced) to bottom-up integration (where higher levels are integrated with respect to the lowest level yet experienced in a mature fashion).

In other words continual revisiting of lower levels is required before one can properly uncover the earliest forms of repression that block free expression of primitive instinctive behaviour.

So this latter recognition of the need for counterbalancing bottom-up integration leads to a descent to the earliest levels of instinctive response, to free it from the excessive control of the “higher” rational self.  

And this journey approaches completion at Band 5 (Level 3) where now an ever more refined experience of the primary archetypal nature of mathematical notions is attained.

Not only are these notions seen to transcend all secondary phenomena of form in collective terms but they are also seen as being necessarily already inherent in each individual phenomenon in a unique manner.

So the transition between Band 5 and Band 6 represents the most complete experience of fundamental mathematical notions (as representing the primary archetypes) with respect to both their transcendent and immanent aspects. These then become identical as ineffable spirit (that represents in evolutionary terms both the ever present goal and source of all phenomenal form). 

For me, holistic appreciation of the mathematical operations of addition and subtraction occurred during the transcendent ascent (of Band 3).

However true holistic appreciation of both multiplication and division did not properly take place until the immanent descent (of Band 5).

So I will end here by providing a more refined holistic explanation of the fundamental relationship as between form and emptiness (with respect to both transcendent and immanent directions).

Now in conventional linear terms, all numbers are defined with respect to the default dimension (i.e. power) of 1.

So in this context, 3 for example can more fully be expressed as 3¹.    

Bearing this in mind the basic analytic relationship connecting 1 and 0 can be written

1¹  – 1¹  = 0¹.

The corresponding holistic interpretation entails that the complete negation of form (with respect to its interdependent collective identity) leads to the experience of spiritual emptiness (0) that implicitly represents potential for the union of all form (1).

Then in standard analytic terms 1¹/1¹    =  1^{1 – 1}   = 1⁰

So in the first case, where 1¹  – 1¹  = 0¹, now from a holistic perspective, achieving emptiness (as pure spiritual interdependence) requires removing any distinct independent phenomena; in the second case, where 1¹/1¹    =  1^{1 – 1}   = 1⁰, this experience is inverted in that achieving pure unity as pure spiritual independence (where spirit can be seen to be contained uniquely in each individual part of creation) requires removing all dimensional phenomena i.e. phenomenal notions of interdependence.

However this latter experience of unity (in immanent terms) implicitly involves emptiness (from a dimensional perspective).

So from both the transcendent and immanent perspectives emptiness (0) and union (1) and union (1) and emptiness (0) mutually imply each other.

Thus one cannot enjoy a pure transcendent experience of spirit without it equally being immanent; and one cannot enjoy a pure immanent experience of spirit without it equally being transcendent.

For in the full balanced experience of spiritual reality, transcendence implies immanence and immanence implies transcendence.