Holistic Mathematics - Whole and Part (Multiplication and Division)

Horizontal and Vertical Number Systems
Q We now return to the important issue of how to represent the three fundamental polarities in a precise holistic mathematical fashion. I believe you have some deeply relevant insights regarding the nature of the number system.

PC Yes! the simplest approach is to think in terms of the natural numbers 1, 2, 3, 4 etc.
(We should really confine ourselves to the numbers we have utilised so far such as 0, 1 and 2. However the inclusion of other numbers - though not necessary - may make the exposition a little easier to follow!)

What is often forgotten is that these numbers are implicitly defined with respect to a fixed dimension (i.e. power) of 1.

Thus strictly speaking we should write the natural number system as 11, 21, 31, 41,…

In geometrical terms, a line is literally one-dimensional so that these numbers can be represented by successive points at equal intervals from each other on a horizontal straight line.

So the (horizontal) linear number system is defined with respect to a fixed unitary dimension.

We now return to the binary digits and the basic operations of addition and subtraction with respect to both analytic and holistic interpretation.

Thus in analytic terms 11 + 11 = 21.  Likewise 11 - 11 = 01.

Also in holistic terms 11 + 11 = 21.  Also 11 - 11 = 01.

As regards unitary form, differentiation takes place through the separate positing of opposite polarities. Thus when these (separate) forms are differentiated with respect to the same (fixed) unitary dimension (i.e. within a given level), a dualistic interpretation of the horizontal polarities i.e. exterior and interior (and interior and exterior) arises.

Also with regard to unitary form, integration takes place through the dynamic complementary fusion (and ultimate identity) of opposite polarities. Thus when these (complementary) forms are integrated with respect to the same (fixed) unitary dimension (i.e. within a given level) a nondual interpretation of the horizontal polarities arises.

One interesting aspect of this formulation is that it clearly points to a remaining confusion of form and emptiness (i.e. 01). Thus though integration with respect to (conscious) phenomena entailing Type 1 complementarity can arise at H1 (subtle realm), it still co-exists with a degree of (rigid) form with respect to the general (qualitative) unitary dimension to which it is related.

Put another way, though linear understanding of distinct (conscious) perceptual phenomena in experience gives way to nondual appreciation (0) it tends to initially co-exist with the rigid interpretation of more general dimensional (i.e. conceptual) phenomena.

Vertical System

Q You believe however than another fully coherent vertical number system can be defined which has especially important implications in holistic terms.

PC Yes indeed! I thought long and hard about this issue for many years before it finally became clear. Like most really important ideas, once it is realised it seems very obvious. Furthermore it is intimately connected with the fundamental mathematical operations of multiplication and division (which - by definition - have an equally important holistic mathematical interpretation).

As always we will first deal with this vertical system in analytic terms by considering the fundamental operations of multiplication and division.

Just as we initially sought to add 1 to itself, we now consider the multiplication of 1 by itself.

So 1 * 1 = 1;

Now, when we omit the dimensional characteristic this operation seems very trivial as it results in the same answer (i.e. 1).

Thus whereas the continual addition of 1 to itself (and subsequent totals) results in the (horizontal) natural number system, the alternative operation of continual multiplication seemingly leads to the unchanged result of 1.

The important implication of this is that multiplication (and also division) are conventionally interpreted with respect to the (horizontal) number system, leading to what is strictly - even from an analytic approach - a reduced (merely) quantitative interpretation of numbers.

Q How can this be the case? Surely numbers are quantities!

PC Not quite! However to appreciate this let me ask you a simple question. What is 2 * 2?

Q 4 of course!

PC Well perhaps! But let’s explore a little further!

If we use our more complete representation of numbers then 2 * 2 properly represents 21 * 21.

Now you might remember that when we multiply a number raised to any power (i.e. dimension) by the same number (raised to its respective power) then we express the result as the number raised to its combined addition of powers.

Therefore 21 * 21 can be expressed as 22.

So interestingly the change that has taken place here is with respect to the (qualitative) dimension (while the number as quantity has remained unchanged).

However because conventional mathematics is defined within a linear analytic framework, ultimately the value of numerical operations are expressed with respect to the invariant 1st dimension.

Thus value of 22 - which combines numbers representing both a (quantitative) object and a (qualitative) dimension - is expressed in merely reduced quantitative terms (with respect to the invariant 1st dimension).

Therefore the result of 22 is expressed as 41. However because - in conventional terms - numbers are ultimately interpreted in (quantitative) linear terms the inclusion of the dimensional characteristic is deemed unnecessary.

Therefore, in linear terms, 22 = 4 (as another number in the horizontal system).

Q I can appreciate to a degree what you are getting at. However does this very refined distinction of number as quantity and number as dimension respectively have any practical numerical relevance in analytic terms?

PC Actually it does! But - as so often happens in a reduced approach - its great potential significance is greatly ignored.

The best way of appreciating this is to go back to our simple example and picture it geometrically. Say I want to draw a square with side 2 units. Then the area will be 22.
Though I may well express this answer as 4, it should be obvious that we are now talking about a qualitatively distinctive type of unit. Thus though each side is measured in linear (i.e. one-dimensional) terms, the actual area represents square (i.e. two-dimensional) units.

So properly speaking the area of our square is 4 square units.

Strictly, this distinction applies to all numerical operations involving multiplication.

Therefore 2 * 2 = 22 represents square (i.e. two-dimensional) units. So through multiplication, a qualitative change in the dimensional characteristic always arises. However this critical distinction is - by definition - effectively ignored in a linear (one-dimensional) interpretation of mathematics.

Q Ah! Now I am beginning to appreciate the importance of what you are saying which has implications not only for holistic interpretation but also for analytic interpretation also!

PC Yes! Using Jungian interpretation, there is an important shadow (qualitative) side of analytic mathematical interpretation that has been greatly suppressed. Indeed the recovery of this shadow side is a necessary prerequisite for the provision of an appropriate context for dynamic holistic mathematical interpretation (which entails the interaction of both quantitative and qualitative aspects of understanding).

Q You were about to define an alternative vertical number system that is based on correct understanding of the operations of multiplication and division. With this initial clarification out of the way why not proceed?

PC Let us start again with the multiplication of 1 by itself this time including the important dimensional characteristics.

So we now have 11 * 11 = 12 .

In the horizontal number system we defined the number quantity (which can vary) with respect to an invariant unitary number dimension.

In the vertical number system we define the number quality as a dimension (which can vary) with respect to an invariant unitary number quantity.

Now if we multiply this result by one and continue to multiply each subsequent result in this manner then we again generate the same natural number system 11, 12 , 13,14 , … this time referring to qualitative dimensions (rather than quantitative objects).

So we really have two analytic natural number systems.

In the first system 1, 2, 3, 4, … represent number quantities (as objects) that are expressed with respect to an invariant unitary dimension.

In the second system 1, 2, 3, 4, … represent number qualities (as dimensions) that are expressed with respect to an invariant unitary quantity.

A decisive step in our understanding will come when we show how to express number (expressed as a qualitative dimension) indirectly as a quantitative object (and alternatively how to express number (expressed as a quantitative object) indirectly as a qualitative dimension.

Holistic Multiplication and Division

Q I gather that this is what you mean by a holistic mathematical interpretation of Type 2 complementarity that involves horizontal/vertical and (vertical/horizontal) interactions as between the fundamental polarities. However before doing so, will you explain now what is meant by multiplication and division in a holistic mathematical context.

PC Yes, this is an extremely important topic as it intimately relates to the part and whole (and whole and part) aspects of all relationships.

However let us look first at division in terms of our vertical number system.

Thus 11 / 11 (1 divided by 1) = 11 - 1 = 10.

So just as 1 - 1 = 0 with respect to the horizontal number system, likewise 1 - 1 = 0 with respect to the vertical number system.

What this in effect entails is that we have  defined Type 1 complementarity with respect to both horizontal and vertical polarities (which in geometric terms would be represented by the opposite ends of the horizontal and vertical line diameters of the circle respectively).

Once again differentiation and consequent phenomenal duality, with respect to either horizontal or vertical polarities, arises from positing - in either context - opposite aspects (within relatively independent separate frames of reference).

Integration - and consequent nondual intuitive awareness - arises from complementary recognition of opposite polarities (with respect to both horizontal and vertical polarities).

Thus we can differentiate the horizontal unitary poles i.e. exterior and interior (and interior and exterior) by separating opposite poles thereby treating each one respectively as positive. Likewise we integrate both poles by treating them in a dynamic bi-directional fashion as complementary (both positive and negative) leading to empty (nondual) awareness.

In like manner we differentiate the vertical unitary poles i.e. whole and part (and part and whole) by again separating opposite poles treating each one as positive.

Again we integrate both poles by treating them in a paradoxical bi-directional fashion as complementary (positive and negative) leading likewise to empty (nondual) awareness.

Q Can I raise a relevant point here. You talk about Type 1 complementarity that applies to the integral understanding of opposite poles. However presumably we also have Type 1 separation of poles (which serves as a precondition for such integration)?

PC That’s right. We have both Type I separation and Type 1 complementarity representing both differentiation and integration respectively of opposite poles.

However it is important to remember that Type 1 separation is of a very subtle bi-directional kind (where opposite mirror asymmetrical interpretations are clearly recognised for all linear type sequences).

So for example, earlier we recognised that holism (where holonic relationships are ultimately defined with respect to their whole aspect) has a mirror asymmetrical interpretation as partism (where holonic relationships are defined with respect to their corresponding part aspect).

Therefore Type 1 complementarity (representing in turn Type 1 integration) with respect to the vertical polarities, requires Type 1 separation (representing Type 1 differentiation) where asymmetrical relationships are defined in terms of both holism and partism.

So by including differentiation we can perhaps see more clearly the limitations of Type 0 complementarity. Here separation takes place in one-direction (e.g. merely in terms of holism) leading to a somewhat unambiguous rigid dualistic interpretation of relationships. Therefore - as little paradox is generated - integration is largely reduced to differentiated understanding.

Thus with Type 0 complementarity (i.e. no genuine complementarity) integration is largely confused - in intellectual terms - with differentiated interpretation.

Q So let us return now to the holistic interpretation of multiplication and division.
How does this Type 2 complementarity (and Type 2 separation) of poles take place?

PC Remember that Type 2 complementarity is defined more subtly in terms of the relationship between horizontal and vertical (and equally vertical and horizontal) poles within a given quadrant.

Thus Type 2 complementarity (where vertical and horizontal poles are properly integrated) and of course related Type 2 separation (where vertical and horizontal poles are properly differentiated) arise with respect to each of the four quadrants.

As we have seen the key problem here is that horizontal and vertical poles relate to polar relationships that are not directly comparable in terms of each other.

So if we represent the horizontal direction as representing quantitative object phenomena, then the vertical direction - relatively - represents qualitative dimensional phenomena (though of course in interactive terms these frames of reference can always be reversed).

It might help again to go back to our simple number illustration - using the geometrical representation of the square - where 2 is multiplied by itself.

As we have seen the result here is 4 square units.

So a transformation has thereby taken place in both quantitative and qualitative terms.

Therefore multiplication - by its very nature - entails both a quantitative (horizontal) and qualitative (vertical) transformation.

If one looks up the Oxford Dictionary, to multiply is defined as "to cause to become much, many or more, to make many or manifold".

Now we saw earlier that the relationship of the one and the many (and the many and the one), involves the interaction of a perception with its corresponding concept (and - in reverse fashion - the concept with its corresponding perception).

In terms of our interpretative framework, the actual perception of a unitary phenomenon such as a particular number, operates at the horizontal quantitative level.

However the corresponding concept of number - which provides a general dimension potentially applying to all numbers operates - relatively - at the vertical qualitative level.

However with linear understanding all perceptions are interpreted with respect to a fixed qualitative unitary dimension, so that the distinction between quantitative and qualitative is confused leading to a merely reduced interpretation.

Thus in a linear interpretation, the concept of number is understood in a (reduced) quantitative manner as applying to all actual numbers. So what involves - in dynamic interactive terms - the relationship between both the quantitative (horizontal) and dimensional (vertical) characteristics of number is thereby reduced to its merely quantitative aspect.

In the context we have defined it, the perception relates to the part and the concept to the whole aspect of understanding respectively.

In proper dynamic interactive terms the relation between part and whole (and whole and part) is always subject to Type 2 complementarity where quantitative and qualitative aspects of understanding continually interact with respect to both horizontal and vertical interpretation.

However the reduced linear interpretation of this relationship between part and whole - which characterises conventional analytic understanding - is in (merely) quantitative terms.

We saw this earlier in our discussion of the relationship between the atom and the molecule. In the linear (asymmetrical) interpretation, the atom is viewed as part of a larger (whole) molecule. This is possible because, in this context, both the atom and the molecule are interpreted in quantitative terms. So the atom (as smaller quantity) is thereby seen as part of the molecule (as larger quantity). By the same token - in this asymmetrical quantitative interpretation - the (larger) molecule cannot thereby be part of the (smaller) atom.

However before moving on to deal with the more correct dynamic interpretation, I will point to a fascinating aspect of the relationship between multiplication and division that is evident (even in the linear analytic interpretation).

Let us say that we have a cake divided into two slices.

Now in linear terms the cake represents the whole quantity to which the slices - as parts - are related.

So the whole cake expressed in terms of the number of its quantitative parts = 2. In other words we have been able to multiply the number of slices as parts to 2 (i.e. 1 * 2).

Indeed if we further subdivided each of these slices in half, the whole cake (as expressed in terms of its part slices) would thereby multiply to 4 (i.e. 2 * 2).

Now in reduced linear terms the whole cake expressed in terms of its part slices = 41.

However if we reverse the frame of reference and express each slice as part of the whole cake we thereby divide the cake into its constituent parts.

So in the former case we multiply the quantitative parts (as slices) to obtain the quantitative whole (as cake).

In the latter case we divide the (quantitative) whole as cake to obtain the (quantitative) parts (as slices).

So each slice as part of the cake = ¼.

Alternatively this can be expressed as 4 -1.

So we can see here that whereas multiplication is expressed with respect to the positive direction of the unitary dimension of form, division is expressed with respect to its negative direction.

So the operations of multiplication and division (as regards dimensions) exactly correspond here on a vertical level with the corresponding operations on a horizontal level of addition and subtraction.

So just as the positing and negating of perceptions continually takes place, likewise the positing and negation of concepts takes place. And if we represent perceptions in horizontal terms (as quantities), then - in relative terms - concepts are represented in a vertical manner (as dimensions).

Mathematical Dimensions and Integral Interpretation of Space and Time

Q What you are suggesting here seems to lead to a very different notion of dimensions to what we are currently use. Can you elaborate further?

PC Yes! One of the most exciting features of the holistic mathematical approach is that it leads to a new dynamic interpretation of dimensions which corresponds exactly with mathematical notions.

The intrinsic structure of the dimensions of space and time is therefore mathematical in this holistic dynamic sense.

We will have more to say on this later when we examine the mathematical interpretation of dimensions in greater detail.

However I will just attempt to give a little flavour at this stage.

Object phenomena (in space and time) arise from the continual positing of perceptions in experience. The corresponding negation in these phenomena leads to a certain fusion of opposites (as spiritual emptiness) enabling a switch to the dimensions of space and time, which relate to the positing of corresponding concepts.

Again in the dynamics of experience, concepts in turn are continually negated enabling the reverse switch to perceptions where object phenomena are once again posited.

Thus both object phenomena (as perceptions) and dimensions (as concepts) are continually posited and negated in experience.

Strictly speaking therefore in dynamic terms, objects have both a positive and negative direction. Likewise - which is perhaps even more surprising - the dimensions of space and time have both a positive and negative direction.

So in the dynamic integral view, space and time dimensions are fully symmetrical.

Object phenomena are continually posited in space (as perceptions). The negation of these objects (though corresponding negation of perceptions) causes a switch by which the dimensions of space are posited (as concepts). Again the negation of these dimensions (through negation of concepts) causes a reverse switch by which object phenomena are again posited (in space).

In like manner objects are also continually posited in time (as perceptions). The negation of these objects causes a switch by which the corresponding dimensions of time are posited (as concepts). Once more the negation of these dimensions causes a reverse switch by which object phenomena are again posited (in time).

So space has positive and negative aspects which continually interact both as objects (in space) and the space dimensions (to which these objects are related).

Likewise time has positive and negative aspects again interacting both as between objects (in time) and the time dimensions (to which these objects are related).

Q What is the precise relationship as between space and time?

PC  To appreciate this in a dynamic integral manner we must recognise - from a psychological perspective - the interaction of both conscious and unconscious in experience.

When one reflects on it, unconscious implies not-conscious (i.e. the negation of conscious activity). Likewise conscious implies not-unconscious (i.e. as the corresponding negation of unconscious activity).

Thus conscious and unconscious dynamically interact in experience through the continual positing and negating of one another. The conscious is posited and then negated as unconscious; the unconscious is then (indirectly) posited and negated in turn as conscious.

Now when the role of the unconscious is not properly recognised, this sets severe limits on the quality and flexibility of dynamic interaction that can take place.

Object phenomena are then rigidly experienced and the unconscious - which is not properly recognised - likewise implicitly intervenes in a somewhat rigid (and distorted) manner. So rather than a continual fluid interaction as between object phenomena (and their corresponding dimensions), the dimensions of space and time become largely separated from object phenomena and viewed in absolute terms (as the medium in which all object phenomena are contained).

Thus in the conventional scientific approach, both objects and dimensions are viewed in a merely positive (i.e. conscious) fashion.

In the Newtonian worldview space and time are interpreted in an absolute unchanging fashion.

Though modern physics challenges this perspective interpretation is still somewhat rigid.

For example quantum mechanics would allow – at the level of sub-atomic particles - for both the positing and negating of object phenomena (i.e. matter and anti-matter particles).

However this finding is not equally extended to interpretation of their corresponding dimensions i.e. in an appreciation of dimensions and anti-dimensions (of space and time).

Likewise Relativity Theory allows for a degree of interaction as between space and time.

However space and time are still viewed in non-symmetrical terms (i.e. 3 of space and 1 of time).

By contrast in the fully integral appreciation (which recognises the necessary interaction of conscious and unconscious), both object phenomena and dimensions have positive and negative aspects. Likewise space and time (and time and space) are - ultimately - fully symmetrical with each other.

From an integral perspective, both the psychological and physical (and physical and psychological) aspects of reality are complementary. This therefore entails that science should recognise complementary correspondents for both conscious and unconscious aspects (in terms of the dynamic interactions of physical reality).

However at present it is very lop-sided with material phenomena being interpreted solely as correspondents of conscious understanding. However we equally need recognition of a non-physical (ground of reality) that complements the unconscious.

Then exterior reality would be more readily understood in integral terms as representing the continual interaction of the physical with its non-physical ground, which would correspondingly be interpreted through the interaction of both conscious and unconscious (in psychological terms).

Now the critical question remains as to the precise relationship as between space and time. As we shall see in our next discussion, this can indeed be given a surprising answer in holistic mathematical terms. We will be then able to show how space can be converted into time (and time into space).

What I will say at this stage is that the interaction of space and time is intimately related to the corresponding interaction of the cognitive and affective aspects of understanding.

As always the relationship is circular and relative (and ultimately purely paradoxical).

So both space and time can be associated with the cognitive and affective aspects respectively. However if for example the cognitive aspect is associated with time (and objects in time), then it is the intervention of the affective aspect in this context that enables the switch to experience of space (and objects in space).

Likewise when the cognitive aspect is associated with space (and objects in space), then interaction with the affective aspect enables the switch to corresponding experience of time (and objects in time).

Using an alternative frame of reference we could associate the affective aspect with time, so that now interaction with the cognitive aspect enables a switch to experience of space. Finally when the affective aspect is associated with time, interaction with the cognitive aspect enables a switch to experience of space.

So from a psychological perspective, the two-way continual interaction of space (and objects in space) with time (and objects in time) reflects the corresponding two-way interaction as between the cognitive and affective aspects of experience.

One key implication for an integral scientific appreciation of reality is that the dimensions of space and time (and their associated object phenomena) have both personal and impersonal interpretations, which continually interact in dynamic terms.

We have just commented on one major limitation of conventional scientific understanding in that - at least in formal terms - it ignores the role of the unconscious.

Thus the unconscious is thereby reduced to conscious interpretation leading to the corresponding reduction of qualitative to quantitative interpretation.

Now we can comment on another limitation in that science ignores - again in formal interpretation - the role of the affective aspect of experience. The affective is thereby reduced to the cognitive aspect leading to a merely impersonal interpretation of objects and dimensions.

Now I would accept that by the very nature of science that its formal presentation should rightly rely on cognitive interpretation. We therefore cannot hope to directly translate the nature of affective experience though scientific interpretation. That is why scientific needs to be complemented by artistic type appreciation.

However in an integral scientific interpretation it is necessary to translate the nature of the dynamic interaction of affective and cognitive (and associated personal and impersonal aspects of physical reality) indirectly in an appropriate cognitive manner.

And holistic mathematical appreciation provides a precise means for carrying out this indirect translation.

Dynamics of Type 2 Complementarity

Q So far you have commented on the reduced analytic manner in which the relationship between whole and part is conventionally interpreted (using in turn a reduced notion of the nature of multiplication and division). Can you now explore the dynamics of the holistic integral appreciation (corresponding to Type 2 complementarity)?

PC As appreciation of the appropriate dynamic nature of the relationship between whole and part (and part and whole) is so important we will have more to say about this in our next discussion.

However we can help to clarify the relationship at this stage.

Now the important point to remember is that - in any relative context - the switch from part to whole (or - in reverse fashion - from whole to part) involves the interaction of both quantitative and qualitative (and qualitative and quantitative) aspects that cannot be directly interpreted in terms of each other.

Put another way such interactions always entail the interaction of conscious and unconscious through actual (finite) and potential (infinite) notions. So what we are ultimately looking for here is a satisfactory holistic mathematical way of interpreting the dynamics by which the conscious and unconscious interact in experience.

We will do this in future discussions. However for the moment we will attempt to carefully outline the nature of dynamics that are involved.

So let us start from the perspective of the recognition of a specific perception - say a particular number - which has an actual finite identity (as phenomenal object).

In the language of our holistic mathematical approach, this corresponds to the horizontal level of interpretation as a quantity within a fixed linear (i.e. one-dimensional) framework.

Now in moving from this perception (i.e. of an actual number) to the corresponding recognition of its corresponding concept (i.e. as the general class potentially applying to all numbers), we switch from quantitative appreciation of a particular object to qualitative appreciation of a general dimension, which intrinsically is of a potentially infinite (rather than an actual finite) nature.

This implicitly requires in experience a corresponding movement from linear understanding at the horizontal level (where poles are separated) to circular understanding (where they are complementary).

In other words with respect to the number perception, the implicit fusion of both the positive and negative aspects of understanding at the horizontal level, (unconsciously) provides the spiritual intuition to enable the switch to conceptual appreciation of number (with potentially infinite application).

Thus correctly speaking in dynamic terms, the relationship between any perception and its corresponding concept (in this reference context) involves the relationship of linear to circular understanding.

Put another way, the relationship of quantitative and qualitative always entails - in dynamic terms - both linear (conscious) and circular (spiritual unconscious) understanding.

Finally, using our holistic mathematical terminology the switch from a horizontal (quantitative) object interpretation to a (qualitative) dimensional interpretation always entails the switch from linear to circular understanding.

Therefore - and this is crucial for subsequent interpretation - in the appropriate holistic interpretation, the numbers representing both the (horizontal) quantities and the (vertical) dimensions respectively, relate to distinct systems of interpretation.

Q If I now attempt to summarise, what you are saying is that Type 2 complementarity always entails a dynamic relationship as between part and whole (and whole and part) aspects, with understanding that is - relatively - linear and circular (or alternatively circular and linear) with respect to each aspect.

Thus for example when we interpret a perception in a linear (horizontal) manner (as an actual phenomenal object), its corresponding concept will - relatively - entail circular (vertical) interpretation (as its potential non-phenomenal dimension).

The deeper implication - I take it - is that all numbers can mathematically be given a linear and a circular interpretation in both an analytic and holistic fashion.

Now I eagerly await to see how the nature of this circular interpretation. Presumably - given that you identify the integral understanding with the circular aspect, the holistic interpretation of the circular number system will then assume special significance as a scientific integral approach to development!

PC This is quite true! We will demonstrate in a future discussion how - precisely - this circular number system is derived. Furthermore we will show how it contains all the holistic mathematical notions we have introduced so far and a lot more besides.

Indeed quite simply the circular number system provides the appropriate scientific system for a fully integral scientific TOE.

Q As I understand it, whereas the linear aspect of understanding relates directly to the conscious aspect of understanding, the circular aspect relates directly to the unconscious aspect. Therefore as conscious and unconscious necessarily interact in the dynamics of experience all phenomena - either as objects or dimensions - have both a conscious and unconscious interpretation. Indeed we can see this in ordinary language where a symbol may serve to give either an actual localised or alternatively a potential universal meaning (as archetype). However though this may be recognised in artistic type interpretation it plays no formal role in scientific terms.

In other words science has not yet discovered a way to incorporate the vitally important role of the unconscious in its formal interpretation of reality.

So therefore analytic science gives but a reduced interpretation of reality suited solely to linear differentiated (i.e. conscious) interpretation.

Can you briefly describe this reductionism in the context of Type 2 interactions?

PC We must first recognise that the unconscious necessarily plays a vital role in all scientific understanding. However this remains merely implicit and is not recognised in formal interpretation, which is based solely on the conscious aspect.

I will illustrate this with reference to the standard mathematical way of understanding the relationship between a number perception and its corresponding number concept.

Thus an actual number perception is posited and thereby differentiated in linear terms (representing the conscious aspect of understanding). Then - because in the dynamics of experience - some degree of negation must necessarily occur, this implicitly leads to an unconscious fusion of opposites (in circular terms) where the notion of form gives way to that of emptiness. This emptiness in turn causes a transformation and a switch from the understanding of number as a particular perception to the general concept of number (as dimension) that potentially applies to all numbers.
However because in formal terms only the conscious aspect of number is recognised, the potential (archetypal) concept (representing the unconscious aspect of understanding) is quickly reduced in actual conscious terms. So the interpretation of the dimensional concept of number is thereby reduced as applying to all actual numbers.

Now of course we could interpret this in reverse fashion starting with the actual (conscious) interpretation of the (dimensional) concept. The reverse switch now requires an (unconscious) recognition of each specific number as archetype (i.e. where the potential aspect is made immanent in the specific perception). However again this will be quickly reduced in merely actual (conscious) terms.

So inherent in all standard mathematical interpretation is a basic reduction of infinite (potential) notions to finite (actual) terms.

Now we have already commented on the fact that numerical results in mathematical terms are ultimately expressed in linear (i.e. one-dimensional) terms.

This in turn reflects the fact that mathematical interpretation is formally based on conscious linear understanding (where polar opposites are clearly separated).

Q I see what you are getting at which makes great sense. It would be hard for anyone - even a hard-core mathematician - to seriously deny that the unconscious necessarily interacts with the conscious aspect in understanding. Yet it is patently obvious - as you have shown - that the unconscious is not at all recognised in formal interpretation. Therefore it must also be true as you have demonstrated that conventional mathematical understanding gives a distorted interpretation of reality (even within its own clearly defined terms). Why is this problem not recognised?

PC Good question!

Firstly conventional mathematics has been amazingly successful in analytic terms. For many - perhaps the vast majority - this seems a good enough reason to continue with the recognised approach without addressing fundamental philosophical problems.

Secondly - as I have stated before - the holistic integral approach requires pure intuitive awareness (traditionally associated with mystical development) with the specialised form of cognition (associated with mathematics). This combination rarely goes together. Mathematicians - certainly as regards their own specialised work - do not see a role for mystical type considerations. Likewise, mystics traditionally have rarely shown a keen appetite for applying spiritual intuitive awareness to mathematical type considerations.

However it is indeed a great pity!

Not alone does the appropriate combination of mystical intuition with mathematical type cognition open up vast new possibilities for dynamic holistic type understanding (Holistic Mathematics) that is appropriate for a scientific integral appreciation of reality, but equally it opens up significant new possibilities within the analytic approach to mathematics.

Correctly understood there is a massive "shadow " side to conventional mathematical understanding that has not yet been properly explored. This is due to the misleading reduced manner in which its key notions are interpreted.

Conclusion

Q Let us finally return to the notions of multiplication and division. Can you briefly conclude this discussion by contrasting the (reduced) analytic with the dynamic holistic interpretation?

PC  In the analytic interpretation - based formally on solely conscious recognition - wholes and parts can only be viewed together in either reduced quantitative (or alternatively reduced qualitative) terms.

In other words the dynamic interaction of quantitative with qualitative (and quantitative with qualitative) cannot be satisfactorily dealt with as this requires both conscious and unconscious interpretation.

The (reduced) analytic approach leads to an asymmetrical notion of the relationship between part and whole. So in any asymmetrical context the whole - for example - is essentially viewed as the "bigger" quantity (which includes its parts as "smaller" quantities).

Thus when we refer to our cake of four slices, in this context the cake (as whole) represents the multiplication (by 4) of each of its constituent slices (as parts).

In turn each slice (as part) represents the division of the cake (as whole) into 4 constituent parts (i.e. ¼ of the cake).

So in this reduced analytic context, multiplication and division are viewed in asymmetrical terms as the reciprocal of each other where the fixed unitary dimension alternates between positive and negative. So 41 (i.e. 4 * 1) represents the multiplication of (part) slices as the (whole) cake.

4 –1 (i.e. 1 / 4) represents the division of the (whole) cake as (part) slices.

In the holistic interpretation, multiplication always entails the interaction of quantitative and qualitative aspects that are linear and circular (or circular and linear) with respect to each other. Thus the interaction of an actual (quantitative) perception with its potential (dimensional) concept literally creates the potential to multiply all actual perceptions without limit.

So for example with respect to number any actual number represents just one (finite) example. However the general concept of number (as dimension) provides the potential to multiply actual numbers without limit.

Indeed it is the application of the (potential) concept to an (actual) number that creates the realisation that it is just one of many (actual) numbers.

Likewise in reverse the application of an actual concept to a potential perception (as for example in algebra where we let x be any number) creates the reverse realisation that the actual concept (as set of all finite numbers) can be divided into many constituent numbers.

So to briefly conclude, in analytic terms whole and part (and part and whole) are asymmetrically related to each other in actual terms as "bigger" to "smaller (and "smaller to bigger").

In holistic terms whole and part (and part and whole) are symmetrically related to each other as actual to potential (and potential to actual).