Wholes and Parts (7)

The holistic nature of number, which we have just encountered, is directly relevant for the coherent interpretation of its ordinal aspect.

Once again, in conventional mathematical interpretation, the ordinal nature of number - though inherently of a qualitative nature - is directly reduced in an absolute quantitative manner.

So for example if we take the simple case of "2" to illustrate, in cardinal terms this can be absolutely expressed as the sum of its quantitative part units.

Thus 1 + 1 = 2.

However if one was to express this in the conventional ordinal manner,  we would say that
with respect to the two units,

1st + 2nd = 2.

Thus implicit in this interpretation is the identification of 1st and 2nd with 1 (unit) in each case.

Therefore to spell it more fully this implies that 1st (unit) = 1 and 2nd (unit) = 1 respectively.

So we thereby have a direct reduction of ordinal meaning in a quantitative cardinal manner!

However, once we move to the relative notion of number - where both quantitative (analytic) and qualitative (holistic) aspects necessarily interact in a dynamic interactive manner - we require a new interpretation of the true ordinal nature of number.


In fact, when one reflects a little on the matter, the merely relative nature of ordinal identity should become quickly apparent!

For example, if one was to state that a horse came in 1st (in a one-horse race) this would not be considered as a worthwhile achievement. However if one was then to say that in another race, the horse came in 1st (with 40 horses participating) this would indeed appear much more impressive.

Thus the relative meaning of an ordinal ranking depends on the cardinal size of the group (to which it is related).

Thus we have an unlimited range of possible relative interpretations of 1st, 2nd, 3rd, etc. depending on the cardinal size of the group involved.

And once again, to define these various holistic meanings, we simply obtain the n roots of 1 (where n is the cardinal size of the group in questions) and then interpret these (roots) in the appropriate holistic manner.

Now, as we know, one of these roots will always = 1. What this means in effect is that the notion of interdependence must necessarily always start from the corresponding notion of independence (as in like manner the notion of independence always implies the corresponding starting notion of interdependence).

This means that one of the solutions (for the n roots of 1) will always reduce down to 1.
And in fact it is this default solution that is the only root considered in the conventional treatment of ordinal rankings (where holistic qualitative meaning is in turn reduced in an analytic quantitative manner).

I will illustrate this now a little further with respect to the simple case of the 2 roots of 1.

These therefore provide - when appropriately interpreted in holistic manner - the true qualitative meaning of 1st and 2nd (in the context of 2 members).

Now these would be represented as 1^1/2  and  1^2/2, which gives – 1 and + 1 respectively.

Therefore 2nd in the context of 2 is + 1.

So the default interpretation of ordinal rankings - where they are effectively reduced in a quantitative manner - implies that we always define the nth ordinal ranking (in the context of a cardinal group of n).

However clearly we can define the nth ordinal ranking likewise in terms of any group > n!

For example instead of considering 2nd (in the context of 2), we could consider 2nd (in the context of 3, 4, 5,.....) where these all now acquire a true holistic identity. So an unlimited number of holistic interpretations are available for 2nd (and indeed by extension for any ordinal number).

Thus 2nd (in the context of 3) would be represented as 1^2/3 , which correct to 3 decimal places is
  – .5 + .866i. Then this numerical measurement would be given the appropriate holistic interpretation (as representing a certain unique configuration with respect to the two fundamental polar pairings (i.e. internal/external and whole/part respectively).   

The key to appreciating the holistic interpretation represents an inescapable paradox (from the standard 1-dimensional dualistic perspective).

Now in relative terms 1st (in the context of 1) creates no paradox and reduces to standard linear interpretation.

However 1st and 2nd (in the context of 2) creates this inescapable paradox, as either unit (of the cardinal group) can potentially be 1st or 2nd.

In fact this type of potential recognition where an ordinal position can holistically range over the entire group of cardinal unit members), necessarily informs our common sense recognition (at an implicit unconscious level).

For example, say one is ranking cars as to size (with largest ranked 1st) and we have two models - a Fiat Panda and standard Mercedes - the Mercedes (in this context) will be ranked 1st and the Fiat 2nd. However let's say we switch to ranking by age (with newest ranked 1st) and that the Fiat is registered in 2015 and the Mercedes in 2010. Then the Fiat (in this new context) will be ranked 1st and the Mercedes 2nd.

So before any ranking takes place, implicitly we must be able to accept the notion that 1st and 2nd have a merely arbitrary meaning (depending on context). In other words what can be 1st in one context can be 2nd in another and vice versa. And this is paradoxical in terms of standard linear logic (which is unambiguous in a dualistic fashion).

Of course in any actual context (framed by just one polar reference frame), linear logic will apply.
However implicitly, all possible rankings must apply, before actual rankings (in any given context) can be explicitly made.

Thus once again the deeper significance of all this is that the unconscious level of understanding directly underlines true holistic - as opposed to analytic - interpretation.

Sooner or later, Mathematics as a discipline will have to face the deeply uncomfortable fact that a truly coherent meaning (with respect to any of its concepts) implies the proper integration of holistic (unconscious) with analytic (conscious) interpretation. 
       
If readers can at least clearly grasp the supreme importance of this one fundamental fact (at present completely denied by the Mathematics profession) this blog will have not been in vain.     

Wholes and Parts (6)

We dealt yesterday with holistic meaning of "4".

This relates to the simultaneous recognition of complementary opposite polarities in both "real" and "imaginary" terms.

Now just as in analytic terms the complex plane built around (horizontal) real and (vertical) imaginary axes provides the basis for the definition of all (finite) numbers, in like fashion from a holistic perspective, the real and imaginary axes (defined with respect to the unit circle) can be used to provide the holistic meaning for all numbers.

However, in this context we are simply concerned with the holistic meaning of the natural numbers.

Now, in an indirect quantitative manner manner, n, in holistic terms, can be expressed with respect to the n roots of 1.

However the direct holistic appreciation requires that these roots roots be then simultaneously intuited with respect to their qualitative nature.

I explained the qualitative nature of each of these roots again in the last blog entry. And  this procedure can be extended for any number of roots.

What this means in effect, is that the holistic meaning of each natural number relates to a certain unique manner by which the fundamental polarities of experience (i.e. internal/external and whole/part) interact. Put another way, this relates to a unique configuration with respect to the dynamic manner in which holons (whole/parts) and onhols (part/wholes) interact in physical and psychological terms.

The deeper significance of this holistic type understanding is that it provides a ready means for recognition of the relative independent identity of each unit member (of the number group) while preserving recognition of the pure qualitative interdependence of the combined group of members.

Again to appreciate this better, let us return to the simple example of the holistic nature of "2".

We have seen that the roots of this number are + 1 and  – 1. So in the context of 2 unit members (of this number group) we can give each member a relative independent identity as + 1 and  – 1. respectively. With reference to our earlier crossroads example, this would imply that we understand left and right turns at the crossroads, as - relatively - opposite to one another, so that if the left for example is denoted as + 1, the right thereby (in this context) is  – 1.

However the simultaneous recognition of both turns, as left and right, requires the combination of these two - relatively - separate individual identities. 

So we express this as + 1 – 1 = 0 (which implies the pure qualitative appreciation of the combined relationship of interdependence).

Another important implication of this holistic appreciation is that - because of its inherent qualitative nature - each number is now seen to possess a unique personality.

Now in popular terms, in fact we often have recognition of this fact e.g. where a person might have a "lucky" number (such as "7").

However we now perhaps can begin to appreciate the deeper holistic mathematical rationale of "number personality".

Indeed it is quite clear why this aspect of number has been completely discarded in conventional mathematical treatment.

Once again ,conventional treatment is of a merely analytic quantitative nature (where wholes are reduced to parts). 

Thus from this perspective, each number has a merely impersonal identity (in common with every other number).

However, once we allow for its true holistic nature, each number thereby assumes a unique qualitative identity (through the interaction of its individual unit members).

So number is now seen as having a (personal) holistic as well as (impersonal) analytic identity. 

Therefore we can no longer hope to maintain the misleading absolute interpretation of numbers as somehow abstractly existing independent of nature!

Rather properly understood, number is now seen as the deepest inherent encoding of all phenomenal reality (in both physical and psychological terms).

So just as the multi-facted aspects of nature acquire distinct personalities (though their unique qualitative features), numbers likewise possess distinct personalities (as the most fundamental encoded nature of this reality).

Wholes and Parts (5)

As I have stated, the 4-dimensional interpretation is especially interesting as it combines both real and imaginary aspects in positive and negative fashion.

And in dynamic terms, these keep switching positions, depending on relative context, so that what is positive (from one perspective) becomes negative from another and what is imaginary (again from one arbitrary perspective) then becomes real!

So any number - when viewed from the 4-dimensional perspective - has 4 relatively distinct meanings, which continually interact with each in dynamic fashion.

Thus we have the real notion of 1 (as an independent entity) with both positive and negative directions. Again when the positive is identified - relatively - with the external (objective) aspect of experience, the negative, by contrast, is identified with the corresponding internal (mental) aspect.
And these can switch, so that the internal may in turn be identified as positive and the external as negative respectively.

Then we have the imaginary notion of 1 (as an interdependent entity) i.e. with the potential capacity to give a related meaning to all separate numbers within its class. This in fact is the dimensional notion of 1 e.g. as a line that is 1-dimensional, that thereby provides a common identity for all - relatively - independent numbers (on that line).

Of course, the general notion of 1 (as 1-dimensional) has a real meaning within its own context. However if we wish to relate, without undue reductionism, the generalised notion of 1 (as representing a dimension) and  then the specific notion of 1 as representing an individual number (on the number line), then they should be conceived as "imaginary" and "real" with respect to each other.

And the imaginary notion itself, in vertical terms has positive and negative directions, in that one can switch as between the transcendent notion of 1, as it were, where the dimensional notion of 1 is properly understood in a potential infinite manner (as beyond any actual  notion of 1) and  the corresponding immanent notion, where the pure infinite notion is reflected through the individual notion of 1 (thereby acting as an archetype).

These four directions (i..e. dimensions) have close complementary parallels in psychological terms.

Once again when we identify the external aspect as positive in a real manner, this is done in a conscious manner (where the number is viewed as a specific object). The internal aspect, is now - relatively - negative in a conscious manner (where the number is viewed as a specific mental perception).

And when we identify the imaginary aspect in a positive fashion, this is identified with the pure intuitive notion of number in potential - rather than actual - terms. This intuitive ability in turn stems from the realisation that positive and negative real polarities are complementary with each other. This thereby generates the realisation of their inherent interdependence (which occurs in an intuitive rather than rational fashion). However, indirectly this interdependence of positive and negative  - which seems paradoxical in dualiistic terms - can be indirectly expressed in a circular rational manner. And when this circular understanding is then represented in  linear fashion, we have "imaginary" interpretation.

Once again the positive recognition of the imaginary, entails the transcendent appreciation of the imaginary notion (as beyond finite actual appreciation). The - relatively - negative recognition of the imaginary, then entails consequent immanent appreciation of the  imaginary notion (as already inherent in each finite phenomenon).

So again with 4-dimensional  appreciation, we have four relatively distinct interpretations of any number, which continually interact with each other  in a dynamic manner.

However because conventional mathematical interpretation is solely 1-dimensional, these dynamics are grossly reduced in absolute fashion.

Therefore, no distinction is made in conventional terms as between the "real" aspects, i.e. number as objective (in external terms) and number as mental perception (in an internal manner).

Likewise no distinction is made as between "real" and "imaginary" aspects, with the potential (infinite) appreciation of the "imaginary" nature of number reduced - in effect - in "real" terms to its actual (finite) identity.

This problem for example underlines conventional mathematical proof.

The Pythagorean Theorem i.e. that in a right angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides) strictly applies (in a potential manner, to all right angles triangles.
However this does not directly equate with "all "actual  triangles (which has an indeterminate meaning in actual terms).
Thus underlining all mathematical proof is a reduction of qualitative to quantitative type meaning, so that the potential  infinite meaning of "all" is misleadingly identified in finite actual terms (where "all" has a strictly indeterminate meaning).

Wholes and Parts (4)

We saw yesterday how the two key polar pairings can be holistically interpreted in terms of the coordinates of the both the real and imaginary axes (in positive and negative directions) in the complex plane.

In particular the unit circle (drawn in the complex) plane will have coordinates on the (horizontal) real axis (x) of + 1 and – 1 respectively and on the (vertical) imaginary axis (y) of + i and – i respectively.

And once again, the former reflect the holistic mathematical interpretation of the external and internal polarities and the latter the corresponding interpretation of the two directions with respect to the qualitative aspect (relating to interdependence). And these are both "imaginary" with respect to the quantitative aspect as "real" (relating to independence).

The dramatic importance of this new holistic mathematical mapping is that all the various roots of 1 can now be expressed as representing unique dynamic combinations with respect to the interaction of whole and part in both physical and psychological terms.

And the importance of these roots in turn is that they enable us to uniquely express the qualitative holistic nature of each number (indirectly in a 1-dimensional manner). And remember again that such 1-dimensional interpretation informs the normal dualistic nature of rational discourse!      

So I will illustrate such holistic interpretation with respect to one of the the simplest - and many ways most important - numbers i.e. "2".

Thus once more in an indirect linear (1-dimensional) manner, we can express the holistic qualitative nature of "2" through obtaining the 2 roots of 1 (and interpreting the results in the corresponding holistic manner).

Now the roots of 1 are + 1 and – 1. Therefore these directly relate to the complementary nature of external and internal polarities. 

Once again conventional dualistic understanding is strictly 1-dimensional based on just one positive pole of understanding (where the qualitative holistic aspect of understanding is reduced in a quantitative manner).  Therefore conventionally the two roots of 1 (i.e. + 1 and – 1 ) are interpreted in a merely quantitative unambiguous fashion (as separate opposites in absolute terms). 

However the essence of 2-dimensional understanding is that we now understand relationships more subtly as necessarily entailing the interaction of both external and internal polarities, that are - relatively - positive (+) and  negative  (– 1) with respect to each other. Therefore they are independent in only a relative sense. This implies that the recognition of their complementary nature implies the new appreciation of qualitative holistic interdependence. 

So now, the two poles are understood as relatively independent to a degree (implying quantitative appreciation) and also relatively interdependent (implying corresponding qualitative appreciation).


This can easily be illustrated with reference to the common place example of a crossroads.

If one is heading N towards a crossroads both left and right turns can be given an unambiguous meaning (i.e. in 1-dimensional terms).

In one now from the opposite direction heads S towards the crossroads, again left and right turns can be given an unambiguous meaning (i.e. in 1-dimensional terms).

However, if one now tries to understand the two turns at the crossroads, when simultaneously combining N and S directions, then the notion of direction is rendered paradoxical (i.e. circular). For what is a left turn (heading N) is right (heading S). And what is right (heading N) is left (heading S). 

So this latter paradoxical appreciation implies 2-dimensional interpretation (where 2 polar reference frames are simultaneously combined).

And this quite simply represents the qualitative holistic appreciation of 2!

Therefore, by extension the qualitative holistic appreciation of 3 would imply the ability to simultaneously combine 3 reference frames (as represented by the 3 roots of 1).

And the qualitative holistic appreciation of n, would imply the ability to simultaneously combine n reference frames (as represented by the n roots of 1).

This is fairly easy to state, but the implications are truly enormous. I will return to this in the next entry.

Wholes and Parts (3)

As we have seen, properly understood every number keeps switching as between whole and part aspects in a dynamic manner.

This reflects the fact that the understanding of Mathematics always entails an interactive experience where number keeps switching as between these two aspects (depending on context).

And again, properly understood Mathematics has no strict meaning apart from its corresponding understanding.

Unfortunately, for millenia now we have become increasingly conditioned to the untenable notion that Mathematics has an independent abstract existence (apart from the enquiring mind).

And this has led to - what I refer to as - the 1-dimensional approach which in fact defines all Conventional Mathematics.

As this is so important, I will briefly clarify what the 1-dimensional approach precisely entails.


All phenomenal experience - including of course mathematical - is conditioned by fundamental polarity pairings.

Two of these pairings are especially important.

The first relates to external (objective) and internal (subjective) polarities.

Fro example when one experiences a number, both of these polarities are necessarily involved. So a number that is viewed as objective, existing in external space, has no strict meaning in the absence of a corresponding mental perception, which - relatively - is internal in nature.

So rather than number having a static absolute existence, in truth number represents a dynamic interaction pattern as between two polar aspects that are - relatively - external and internal with respect to each other.

And this, by extension, applies to all mathematical constructs.

We could truthfully say therefore that in mathematical terms, objective truth has no meaning in the absence of corresponding mental interpretation.

So what happens in Conventional Mathematic is that an attempt is made to totally freeze this interaction as between external and internal, so that mental interpretation is viewed to be in absolute correspondence with the objective situation (which is then given an abstract independent existence).

In this way we can see how conventional mathematical "truth" takes place within just one isolated polar reference frame (i.e. as objective in an absolute manner).


The second key polarity pairing relates to the relationship - which we have been directly looking at - as between whole and part. This could also be referred to as the relationship (in any context) as between general and particular, (or individual and collective), qualitative and quantitative etc.

Again, the experience of number (and indeed all mathematical relationships) entails the dynamic interaction of whole and part aspects, which keep switching, depending on context.

Once more, conventional mathematical interpretation attempts to absolutely freeze this interaction by reducing the qualitative aspect in a mere quantitative manner.
So once again, we can see how such mathematical "truth" takes place within just one isolated polar reference frame.

So 1-dimensional interpretation refers therefore to interpretation that is explicitly conducted in an absolute manner within an isolated polar reference frame.

And the very essence of such interpretation is that dynamic interaction cannot be recognised to take place as between opposite poles (though implicitly some unconscious interaction must necessarily take place).


Many years ago, when I first recognised the all-embracing importance of these two fundamental polar pairings, I slowly began to see that they concurred exactly with a new holistic mathematical manner of interpreting mathematical symbols.

Now, basically when we become conscious with respect to a phenomenon in an independent rational manner, we thereby posit in a conscious manner. So here we have the holistic meaning of the plus sign as used in addition (i.e. +).

Then to switch as between opposite poles e.g. from the objective to the mental recognition of the object, we must implicitly negate the external pole (in an unconscious manner). So here we have the corresponding holistic mathematical  meaning of the negative sign as used in subtraction (i.e. – ).

Now the extent to which such unconscious negation is involved, determines the degree to which recognition of the interdependence of opposite polarities takes place.
This occurs directly in an intuitive manner, whereby psychic energy is generated. In fact it parallels very much the manner in which matter and anti-matter particles annihilate each other creating physical energy. So the holistic intuitive realisation of interdependence entails the direct coincidence of both positive (+) and negative (–) poles. 

Note here how the holistic interpretation is paradoxical with reference to the corresponding analytic (1-dimensional) interpretation (where poles are separated in an absolute dualistic manner)!

Though the direct intuitive realisation of the interdependence is nondual in nature, indirectly it can be expressed in a circular manner through paradoxical reason. 

This contrasts heavily therefore with corresponding analytic interpretation that is expressed in the standard linear manner through the unambiguous use of reason!

Put simply, all analytic interpretation of mathematical symbols is 1-dimensional in nature. Once again this is the only interpretation that is formally recognised within present Mathematics.

However all holistic appreciation (of an authentic nature) entails "higher" dimensional interpretation of which the simplest is 2-dimensional.   

This basically relates to appreciation of the interaction of the first pairing of polarities - which I refer to as the horizontal polarities - i.e. internal and external.
And once again in holistic mathematical terms, these are positive (+) and negative (–) with respect to each other.


It took me more time to fully appreciate the holistic mathematical significance of the second set, which relates directly to the dynamic interaction as between whole and part.

Now remarkably, the whole notion (in the authentic appreciation of qualitative interdependence) is imaginary (i) with respect to the corresponding real (i.e. quantitative) interpretation of a unit (1).

Basically the imaginary notion represents the attempt to express the holistic notion of interdependence in an indirect analytic type manner.

Now, as we have here the unconscious appreciation of qualitative interdependence, this entails the negative direction of understanding i.e. whereby positive recognition of the exclusive independence of 1  (i.e. as pole or direction) is thereby to a degree successfully eroded.

Thus, this unconscious appreciation of negation is 2-dimensional (as it dynamically also necessarily includes the positive direction). And to express this in the standard linear (1-dimensional) manner we take the holistic equivalent of a square root (i.e. in expressing what is 2-dimensional in a reduced 1-dimensional manner).

So the relationship of part to whole (quantitative as to qualitative) is as "real" to "imaginary".

And the imaginary - like the real - likewise has two directions that are positive (+) and negative (–) with respect to each other.

Because holistic qualitative notions are inherently of an unconscious intuitive nature,  to indirectly recognise their nature, they must necessarily be projected into conscious experience.

So the positive direction arises when we are aware of how an object indirectly conveys a holistic meaning.  For example, an athlete might have a dream - say - to one day win Olympic gold. So an Olympic final would not just be understood in a conscious manner, but would likewise serve a powerful holistic unconscious purpose. 

Now in similar manner, all mathematical symbols, not only serve a real quantitative, but likewise an imaginary qualitative purpose.

So not alone do "real" and "imaginary" have an important quantitative interpretation (as in Conventional Mathematics) but equally an important holistic interpretation!

So from this qualitative perspective, the great limitation of present Mathematics is that it is conceived solely in "real" terms (i.e. with respect solely to its quantitative aspect).

Therefore from this perspective, I am clearly maintaining that a comprehensive mathematical approach must be complex (i.e. with both real and imaginary components).  


So Conventional i.e. Analytic (Type 1) Mathematics, in this qualitative sense represents the "real" component of mathematical understanding.
Holistic (Type 2) Mathematics  represents the "imaginary" component and Comprehensive i.e. Radial (Type 3) Mathematics represents "complex" - both "real" and "imaginary" - mathematical understanding.

Wholes and Parts (2)

We have seen how our knowledge of number keeps switching - depending on context - as between part and whole aspects.

Indeed if we were to use a close analogy with quantum physics, number keeps switching as between particle and wave aspects.

So in this sense the wave-particle duality that applies to matter (especially at the sub-atomic scale) equally applies to number. Indeed I would maintain that this observed physical phenomenon itself ultimately reflects the whole-part duality of number!

We have also seen that conventional mathematical interpretation is inherently unsuited to dealing with this issue. Because of its unambiguous (1-dimensional) nature, it inevitably reduces qualitative meaning in absolute quantitative terms (whereby in effect the whole is reduced in terms of its constituent parts).

Thus again, if we were to use a close physical analogy, the present position in Mathematics is akin to the attempt to understand quantum mechanical behaviour in terms of standard Newtonian concepts.
Indeed in truth the problem is even more fundamental than this!

Nothing less therefore than a radical reformulation of the nature of the number system - and indeed by extension all mathematical notions - is now required.

For the simple fact exists that at present one cannot give a properly coherent interpretation of the simplest example of multiplication - indeed the simplest example of addition - in terms of the accepted mathematical paradigm.

Thus with respect to the number system, the present static absolute approach urgently needs to be replaced with a new inherently dynamic interactive interpretation, whereby the distinctive nature of the part and whole aspects of number can be properly preserved.


Therefore, for many decades now, I have proposed that rather than one natural number system - interpreted in an absolute rigid manner - that we need to recognise that there are two complementary aspects to this system, which interact with each other in dynamic fashion.

I refer to these aspects as Type 1 and Type 2 respectively.

Initially the Type 1 aspect would appear to bear the closest resemblance to to conventional interpretation.

So, again in conventional interpretation the natural numbers are listed as:

1, 2, 3, 4, ..........

Now in Type 1 terms, these are listed in more refined manner as:

1¹, 2¹, 3¹, 4¹, ……..

Again to simply illustrate in conventional terms, 

1 + 1 = 2.

Therefore both units are treated in an independent fashion as quantitative parts, with the resultant total representing the sum of these quantitative parts. 

Thus "2" - though referred to as a "whole" number - in this context, is given a merely reduced quantitative meaning (i.e. as the sum of constituent unit parts). 

However in Type 1 terms,

1¹ + 1¹  = 2¹.

The dimensional number (i.e. power or exponent) here refers implicitly to the corresponding whole status of the number.

Therefore to explicitly recognise that 1 + 1 = 2 (in a quantitative manner), one is implicitly recognising that 2 is equally associated with a new unitary whole status.

In yesterday's blog entry, I illustrated this with respect to the two slices (of the cake).

So the ability to recognise that that the combination (through addition) of each (individual) part slice resulted (collectively) in two part slices, implicitly requires recognition of the total cake as a whole unit. Thus the very ability to recognise the two individual slices as being related to the overall cake would be impossible in the absence of this implicit recognition of the cake possessing both a part and whole status. So again its part status is represented by its 2 individual slices. However its whole status is then represented by its distinctive status as 1 cake.    

And by including the dimensional number of 1, we are here recognising the corresponding whole identity of  the number "2".

The deeper implication of this is that this whole identity (in 1-dimensional terms) implicitly enables an interdependent relationship as between the two individual units (of 2) to be maintained.

So in ordinal terms, we would look on the two slices of our cake as the 1st and 2nd slices respectively. However we have now moved from the notion of independence (with respect to the two individual slices in cardinal terms) to the complementary notion of interdependence (with respect to the "same" two slices in an ordinal manner).  

And this equally applies to number. Thus in cardinal terms we can refer to 2 as 1 + 1 in quantitative part terms (where both units are independent in a homogeneous fashion).

However in corresponding ordinal terms, we can refer to 2 as 1st + 2nd in a qualitative whole manner (where both units are interdependent in a uniquely distinctive fashion).

And this whole nature of 2 comes from switching from its part status (as comprised of 2 independent units) to the new identity (as a unique whole in its own right).

Therefore, it is impossible to properly recognise the distinctive nature of the cardinal and ordinal interpretations of number, without also properly recognising the dual nature of number in terms of its part (analytic) and whole (holistic) aspects. 

Thus there is an underlying paradox here:

In explicit quantitative terms, we attempt to define each number as the part combination of individual units.

So again for example, 2 = 1 + 1.

However this part total of 2 itself represent a single unit (in qualitative whole terms). 

Therefore though we are indeed entitled to explicitly make clear quantitative distinctions with respect to the part nature of number (i.e. in analytic manner), implicitly we need to bear in mind the holistic qualitative nature of number, which makes these distinctions possible.

However, we equally have a Type 2 aspect to the number system. 

Now in the Type 1 aspect we have separate number (quantitative) objects (defined within a 1-dimensional framework).

However with the Type 2 aspect we have the same quantitative object (defined within multiple dimensional frameworks).

The easiest way to appreciate this is in terms of a unit line (in 1-dimensional terms) which is now used to define a unit square (in 2-dimensional terms).

Therefore through the quantitative nature remains unchanged as 1, clearly the dimensional nature of the number object has changed (from 1 to 2).

Now if one reflects for a moment on the 2 dimensions of a square object, clearly they cannot be independent of each other but must be related in a very ordered manner.

Thus the crucial point about the Type 2 approach is that we now are adding related (i.e. interdependent) units. Thus these units now represent wholes rather than parts (as was the case with the Type 1 aspect).

Thus when we add for example 1 + 1 (now representing wholes) the whole status i.e. the dimensional nature of the object is directly changed. 

Now the startling fact is that what represents addition with respect to this Type 2 aspect, represents multiplication from the Type 1 perspective.

So  1¹ * 1¹  = 1².    

And 1² = 1^{1 + 1}. 

However, when adding numbers as wholes (representing dimensions) the new qualitative change (i.e. the dimensional status of the object) can only explicitly be understood, through implicitly recognising the quantitative nature of the base unit (as measured in 1-dimensional terms).

Thus we can only combine numbers (as parts) through implicit recognition of their corresponding whole status. Likewise we can only combine numbers as wholes (representing dimensions) through implicit recognition of the quantitative nature of each dimension (in isolation).

Thus the Type 2 aspect of the number system is listed as:

1¹, 1², 1³, 1⁴, ……..

Note how it is the inverse of the Type 1 aspect, reflecting the change is explicit focus from the part (quantitative) to the whole (qualitative) nature of number.

However just as explicit part recognition implicitly requires corresponding whole, equally explicit whole recognition implicitly requires corresponding part recognition respectively.

In fact both forms of recognition are dynamically complementary with each other in a two-way manner.

Wholes and Parts (1)

I keep coming back to the most fundamental issue possible with respect to the true nature of number, which unfortunately due to the reduced nature of present mathematical interpretation is completely ignored.


It might help initially to explore this key issue in a concrete manner with a simple practical illustration.

Imagine that we have a cake that is cut into 2 (equal) slices.

Now as each slice comprises one distinct unit we could represent the cake as,

1 + 1 = 2.

In other words the two slices (comprising the cake) entail the addition of the individual (separate) units.

However this represents but a reduced interpretation of the relationship as between whole and parts whereby the (whole) cake is viewed in a merely fragmented manner as the quantitative addition of the individual unit parts.

So therefore from this reduced - merely quantitative - perspective the (whole) cake is represented as 2 (part) units.

However the cake has also its own unique whole identity, which would be represented as 1 (i.e. one whole cake).

So we have the paradox that the cake can be represented as 2 parts or alternatively as 1 whole.

So in the very dynamics of recognition, in order to relate parts and wholes we must implicitly switch as between both part and whole recognition (with respect to objects) or alternatively as between quantitative and qualitative recognition, which are dynamically related to each other in a complementary manner..

Thus again with respect to this example the quantitative recognition of the cake represents its 2 - relatively independent - part slices.

The corresponding qualitative recognition (in this context) then relates to the recognition of the cake as a whole unit (i.e. as interdependent with itself).

Of course the cake could now in turn attain a (part) quantitative status - say - as one of a collection of  cakes!

In our example, we initially treated the slices of the cake as quantitative parts (in relation to the whole cake).

However, each slice in turn has a qualitative identity whereby it is recognised as a whole in its own right. So if for example each slice contained individual components, these would thereby now constitute distinctive parts in relation to the whole slice!

In more general terms, phenomenal reality is necessarily composed of holons (i.e.whole/parts) whereby, in any context, what is whole (from one valid perspective) is equally part from an equally valid related perspective.

And in reverse terms, phenomenal reality is composed of onhols (part/wholes) whereby what is part (from one valid perspective) is equally whole from an equally valid related perspective.


So, in the example above, we illustrated how the whole cake (in relation to its 2 slices) could equally be part (as an individual item in a collection of cakes).

Equally, we saw how the part slices (in relation to the whole cake) could equally serve as unique wholes (in relation to constituent parts of each slice).

I cannot stress enough how important this distinction as between the part and whole status of an item (which is relatively quantitative as to qualitative and qualitative as to quantitative respectively) truly is, for when grasped, it leads to the need for a fundamental new interpretation of the very nature of the number system.

Basically in conventional mathematical terms, a merely reduced quantitative interpretation of number is given, which is of an absolute static nature.

So, for example, though we do indeed refer to a natural number such as 2 as a (whole) integer, in effect it is defined in a merely reduced part manner as quantitative.

Thus 1 + 1 = 2. In other words the whole number (i.e. 2) is treated simply as the quantitative sum of its constituent parts. So again, a fundamental reduction of qualitative in terms of quantitative meaning is thereby directly involved.

However, when we properly allow for the truly distinctive nature of both part and whole meanings in relation to number (which again are - relatively - quantitative as to qualitative and qualitative as to quantitative respectively) we must necessarily move to a new dynamic interactive treatment of the number system.

I will suggest the appropriate manner for achieving this in the next entry.