Wholes and Parts (11)

Yesterday we saw how the Zeta 2 zeros (i.e. the non-trivial roots of 1) provide the all important interface as between the two extreme interpretations of number.

So from one extreme, each number approaches an absolute formal identity (in quantitative terms).

Once again with respect to the number 2, this quantitative notion of "2"  is composed of two independent units as constituent parts.

However at the other extreme, we have the qualitative notion of "2" (as twoness) composed of two interdependent units that ultimately approach total identity with each other as a pure energy state!

And whereas the quantitative recognition of "2" corresponds directly with rational recognition (of a conscious nature), the qualitative recognition of "2" (as twoness) corresponds by contrast directly with intuitive recognition (that is unconscious in origin).

Now strictly in experience, these two aspects of "2" are quantitative and qualitative with respect to each other in a complementary dynamic interactive manner.

So the two roots of 1, thereby provide the harmonious interface as between  the cardinal (quantitative) and ordinal (qualitative) notions  number.of

Thus with respect to "2" in cardinal (quantitative) terms, we can identify two - relatively - independent units (i.e. 1 + 1); likewise with respect to "2" in ordinal (qualitative) terms, an identify two - relatively - interdependent units (i.e. 1st + 2nd).

With one of these roots, ordinal is necessarily reduced to cardinal meaning. What this means in effect is that the recognition of interdependence must necessarily start from the initial recognition of independent units (that are thereby interdependent with themselves)!

So the non-trivial root (as the Zeta 2 zero) where n = 2, is – 1. In dynamic interactive terms, this is given both an analytic and holistic interpretation. 

Thus in analytic terms – 1 is considered as separate i.e. independent of + 1. However in holistic terms – 1 is now considered as fully related i.e. interdependent with + 1 (as an energy state).
Thus higher dimensions (where n > 2) reflect increasingly greater layers of interdependence (intuitively recognised) with respect to understanding.

Thus in dynamic interactive terms, understanding keeps switching in two-way fashion as between analytic and holistic aspects of interpretation.


Therefore dynamic interpretation is ultimately of an extremely subtle nature. We can start - in the Type 1 system - with the base natural numbers as quantitative and the dimensional exponent therefore of a -  relatively - qualitative nature.

The Zeta 2 zeros are then properly given their holistic interpretation.

However, when reference frames switch, we can start with the base natural numbers as qualitative and the dimensional exponent as - relatively - quantitative in this context.

The Zeta 2 zeros are then properly given their analytic interpretation.

Thus in the dynamics of experience, both base and dimensional numbers (with respect to the Type 1 and Type 2 aspects of the number system) keep switching as between quantitative (analytic) and qualitative (holistic) aspects.

Likewise the Zeta 2 zeros - which provide the crucial interface between both aspects - likewise keep switching as between their quantitative and qualitative aspects.

Thus the key overriding issue with respect to the individual natural numbers in this respect  is that consistency - expressed through the Zeta 2 zeros - can be maintained as between both quantitative and qualitative interpretations.

And remarkably, this fundamental issue cannot even be recognised in conventional mathematical terms, as it is already based, in every context, on reduction of the qualitative to the quantitative aspect!

Wholes and Parts (10)

As we have seen, the various roots of 1 provide the means of - indirectly - translating the true holistic nature of the natural numbers (that necessarily entail "higher" dimensions > 1) in the standard 1-dimensional manner.

In this way one is enabled to convert from  the (holistic) Type 2 to the (analytic) Type 1 nature of the number system.

So once again - to give the simplest example - the holistic nature of 2 consists in the intuitive recognition that the two opposite directions (+ 1 and  – 1 respectively) - representing polar reference frames - are identical, when simultaneously considered in a complementary fashion.  Such intuitive recognition thereby represents a pure energy state!  

We have already illustrated on many occasions this notion of 2 with respect to left and right turns at a crossroads.  Thus given the polar direction of approaching the crossroads (N or S) what is left (with respect to one polar reference frame is right (with respect to the other) and also what is right (with respect to the first frame) is left (with respect to the second).

Thus when considered in complementary fashion (with respect to both N and S directions simultaneously) left is right and right is left. In other words + 1 (representing the left direction) = + 1 (now representing the right) and + 1 (now representing the right) = + 1 (representing the left).

So from a rational dualiistic perspective, this interdependence of both directions represents pure paradox. However such paradox is directly appreciated in a nondual fashion (through holistic intuition).

So once again 2-dimensional appreciation is directly intuitive, entailing the simultaneous recognition of two (opposite) polar reference frames.

However when we reduce this analytically (i.e. in a 1-dimensional manner) we now recognise the two poles in a relatively separate dualistic fashion, whereby they are + 1 and  – 1 with respect to each other.

Thus in any actual situation, a turn can be unambiguously designated as + 1 or – 1 respectively. Thus if a left turn is denoted as + 1 , then in relative terms a right turn is – 1 (i.e. not a left turn).
However if the right turn is now designated as + 1, then the left turn is, in a  relative manner,  – 1 (i.e. not a right turn). 

Thus the relative independence of each direction is denoted by + 1 and  – 1 respectively through the circular number system (based on the n roots of 1 in the complex plane). And again in this case n = 2.

The corresponding interdependence is then denoted through the sum of the n roots = 0.

So when n = 2, the sum of the two roots = + 1 – 1 = 0. So the pure holistic notion of interdependence is thereby without quantitative significance! 

Therefore, we can now quickly generalise to state that the holistic notion of n (in Type 2  terms) can be indirectly reduced in a Type 1 manner through obtaining the n roots of 1.

Thus when  xⁿ = 1, xⁿ – 1 = 0 and the n solutions of his equation provide the n roots of 1.

<sup>However one of these roots will always = 1, where holistic meaning reduces to its analytic (1-dimensional) counterpart.

Thus to obtain the non-trivial roots (with true holistic meaning) we divide,</sup> 

xⁿ – 1 = 0 by x – 1 = 0 to obtain,

x¹ + x² + x³ + x⁴ + .........+ x^{n – 1} = 0.

This is what I refer to as the Zeta 2 function which complements Riemann's Zeta function (which in this context represents the Zeta 1 function).

The true importance of the Zeta 2 solutions is that they provide the means of switching as between qualitative notions (of relative interdependence) and quantitative notions (of relative independence) respectively.

Wholes and Parts (9)

In the last two blog entries, we have seen how 1) the holistic interpretation of number is directly related to its ordinal nature and likewise 2) that such holistic interpretation is intimately connected with the true inherent dynamic interactive nature of space and time.

This therefore implies that the ordinal nature of nature is directly concerned with the manner in which number is given a unique location with respect to both space and time.

Now, to spell this out more fully, we must once again remember that in dynamic interactive terms, the analytic and holistic interpretations of number are necessarily complementary with each other.

Therefore, when in analytic terms, we interpret the cardinal natural numbers in a 1-dimensional manner i.e. as lying on the number line, the ordinal nature of these numbers is then properly interpreted directly through intuition, in a complementary holistic manner (where they have a location in complex space and time).

However, as always we can switch reference frames.

Therefore, in an alternative manner we directly intuit in holistic manner, the cardinal natural numbers, again in 1-dimensional manner as lying on the number line, the ordinal nature of these numbers is appropriately interpreted in a complementary analytic manner (again with a location in complex space and time).

What this means from a psychological perspective, is that we keep switching as between conscious and unconscious type interpretation with respect the the nature of the natural numbers in cardinal and ordinal terms.

Now, with conventional mathematical interpretation, these two-way dynamic interactions are reduced in a static analytic fashion.

So the holistic aspect is reduced to the analytic in a merely linear (1-dimensional) fashion. The ordinal aspect of number is thereby reduced in a cardinal fashion. Likewise, the time dimension of number is ignored with space treated in linear fashion.


Properly understood, the dynamic nature of number is bound by two aspects that lie at opposite extremes to each other.

Therefore from the analytic perspective, number approaches the totally rigid state of absolute form.

Again in formal explicit terms, Conventional Mathematics interprets the natural numbers in an absolute manner. However properly understood, their nature always is merely relative, whereby a rigid absolute type interpretation can indeed be ever more closely approached, without however fully attained.

However at the other extreme from this fixed analytic aspect, we have the pure holistic interpretation of number (where complementary poles of recognition simultaneously interact with each other in a directly intuitive manner). In fact here, number approaches a pure energy state (without form) in both a physical and psychological manner.
Again, this pure energy state of number which in the limit is ineffable, can be ever more closely approached without being fully attained.

However, as stated before we can switch reference frames so that the holistic aspect is directly intuited from the relatively  fixed forms of natural number and then the complementary aspect - though of a highly dynamic interactive nature - now analytically interpreted with respect to separate frames.

Successful understanding therefore in this regard needs to be extremely refined with respect to both rational and intuitive modes of understanding. So rational understanding is required to cover the entire spectrum from fully linear to fully circular modes.
Likewise intuition is required to cover the entire spectrum from fully immanent to fully transcendent modes.

Once again to finish this entry, with the dynamic interpretation of number, we move away from the conventional - and utterly misleading - notion of number as in some sense existing in abstraction from experience.

Rather number is now seen as the inherent encoded basis of all phenomena (in both physical and psychological terms). And in this understanding both its physical and psychological aspects are understood as fully complementary. And the quantitative and qualitative aspects of number are likewise understood as fully complementary.

Wholes and Parts (8)

One consequence of the holistic appreciation of number, is that it opens up a remarkable new understanding of the true nature of space and time.

Again, because of the standard 1-dimensional approach, our notions of time (especially) have been strongly conditioned to view it in an extremely limited fashion.

So, for example, we are accustomed to think of time as moving forward in one positive direction.
Indeed it is essentially the same with respect to space. Clearly, though we view the rigid features of phenomenal reality as existing in 3-dimensional space, we view movement with respect to space (in each of these dimensions) as taking place in one positive direction!

However, I initially began to realise - directly with respect to my own experience - at a certain crucial stage in development, that the accepted approach made little sense.

I have frequently illustrated in the previous blog entries the holistic nature of "2" (and the 2-dimensional nature of experience to which it relates).


Therefore once again, phenomenal reality is conditioned, for example, by external and internal polarities which dynamically interact with each other.

Therefore if time moves forward (in a positive direction) with respect to the external world (in relation to self), then in relative terms, it must move backward in a negative direction with respect to the internal self (in relation to the world). Equally from the opposite perspective, if time moves forward in a positive direction with respect to the internal self (in relation to the world), then it must necessarily move backwards in negative fashion with respect to the external world (in relation to the self).

So depending on the polar reference frame, time moves in both positive and negative directions with respect to the two (real) polarities of experience. This paradoxical appreciation then leads to the pure intuitive realisation of a continual present moment with respect to all created reality.

Now, the reason why we - mistakenly - view time as moving in a solely positive direction (with respect to both the internal self and external world) is that we have attempted to abstract the internal observer in absolute fashion from what is externally observed.

So the perceived 1-dimensional nature of time simply reflects the (reduced) 1-dimensional manner of scientific interpretation on which it is based (i.e. where qualitative considerations are reduced in a mere quantitative manner). 

So 2-dimensional time  begins to reflect the inherent dynamic interactive nature of experience.
And the very nature of such time directly concurs with the holistic interpretation of "2".

And because we now view time in a relative manner, this implies that space also - at this level of holistic appreciation - also behaves in a relative manner.

Therefore in 2-dimensional terms, all movement in space likewise takes place (with respect to external and internal polarities) in complementary positive and negative directions.  

Then, when we move on to the vitally important 4-dimensional case (which includes both external/internal and part/whole polarity pairings), we now realise that all movement in time takes place - relatively - in positive and negative directions in both real and imaginary terms.

Likewise, all movement in space (in complementary fashion) likewise takes place - relatively - in both positive and negative directions in real and imaginary terms.


Before moving on, I will briefly elaborate here on these four directions relate to experience..

When we view the movement of time in a real manner, this directly concurs with the (localised) conscious appreciation of phenomena.

So once again, when the positive direction is identified with external appreciation (in relation to the self), the negative direction - relatively - is then identified with internal appreciation (in relation to the world).

And these dynamics likewise operate in complementary fashion with respect to our understanding of positive and negative directions in space!

When however we view the movement of time in an imaginary manner, this now concurs with the (holistic) unconscious appreciation of phenomena (whereby they thereby carry a deeper integral significance)

And once again when we identify the positive direction (of imaginary appreciation) with the external aspect (in relation to the self), the negative imaginary direction is then - relatively - associated with the internal aspect (in relation to the world).

And again these relationships operate in a complementary fashion with respect to the imaginary directions of space.

What is fascinating here is that we are now able to directly identify the true nature of space and time - or more correctly the key dynamics through which these dimensions interact - with the holistic appreciation of number.
 

And we have illustrated this briefly with respect to the holistic interpretation of "2" and "4" respectively.

What we now can clearly see is that the true nature of reality is necessarily complex (in a precise holistic mathematical sense) when we incorporate both (conscious) analytic and (unconscious) holistic type appreciation! 

However, in more general terms, associated with every number n, is a corresponding unique holistic interpretation (with complex co-ordinates that reflect the corresponding n roots of 1 of that number).

What this directly implies therefore is that associated with the holistic interpretation of each number is a unique configuration of space and time (in complex terms).

This then leads to the truly stunning realisation that phenomenal reality fundamentally reflects the dynamic interaction of number with respect to both its quantitative and qualitative features. In this sense phenomenal "reality" simply reflects the decoded nature of number.

Thus the qualitative features of phenomena, that thereby give them a distinctive unique identity, directly reflect the holistic nature of the number configurations (with which they are comprised).

What is also truly remarkable is that this whole new world of space and time - that embraces all the number types in holistic manner - directly concurs with actual experience (provided that understanding is sufficiently refined to properly uncover the dynamics involved).

For example when understanding is so developed in an unconscious intuitive manner to appreciate the two-way dynamic experiential interaction of external and internal polarities, one can then readily intuit that the interactions of space and time readily conform to the holistic mathematical interpretation of "2".

And then, at an even more advanced level, we have seen how the interactions with respect to both (pure) external and internal and whole and part coordinates conform to the holistic mathematical interpretation of "4".

With even more refined intuition, we will eventually be able to readily intuit more subtle interactions entailing the various possible configurations of both sets of polarities as conforming to all other possible numbers. And with further advancement, we will furthermore be able to give meaning likewise to the holistic interpretation of both real and imaginary numbers (in a rational and irrational manner).

And again what is truly remarkable, with sufficient intuitive and rational refinement we will be then able to see all these interpretations as directly relating to immediate experience of the world!


Thus again to sum up, associated with the holistic interpretation of each number is a unique dynamic configuration in both space and time, that is directly applicable to experience (in physical and psychological terms).

Just as the quantitative nature of reality is directly measured through the analytic interpretation of number, the qualitative nature will, in future, be directly measured through the holistic interpretation (reflecting these unique complex configurations of space and time). 

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.

Reflections on Number (5)

In my previous entries, I have stressed that every number can be given both an analytic and holistic interpretation respectively and that in the dynamics of experience, both aspects are inevitably intertwined, with one made explicit in conscious manner, with the other remaining - relatively - implicit in unconscious fashion.  .

And each number can likewise be given a base identity or a dimensional identity respectively.

So once more illustrating with respect to the dimensional aspect, the number 3 has an analytic interpretation with respect to 3 (referring to 3 dimensions in a quantitative manner). However 3 equally has a qualitative interpretation as the "threeness" or the quality of 3, which thereby enables the common identification of all members relating to a class of 3 dimensions (such as the length width and height measurements of different rooms).

However one might wish to probe further as to the precise difference as between the quantitative and qualitative interpretations.

So again, if I for example refer to the 3 dimensions with respect to the room of a house (length, width and height) this represents the accepted quantitative view.

However in conventional terms the distinct identity of a number (such as 3) used with respect to objects is not properly distinguished from what is used for dimensions.

But there are crucial differences. 3 as used for objects has a finite specific meaning i.e. as  3 unit objects). i.e. 3 = 1 + 1 + 1

However 3 as used for dimensions has by contrast a collective general meaning. Here each unit (i.e. separate dimension) applies potentially to every possible natural number in an infinite manner). So one more, length, width and height measurements could apply to 1, 2, 3, 4,.......rooms.

There is also another key difference:

When we use 3 in the restricted finite sense (where each unit applies to just one actual object) the units are treated as independent and homogeneous.

So when 3 = 1 + 1 + 1, the relationship between units is not considered.

However as far as dimensional "units" are concerned, this is not really the case. Here the units are not in fact independent but are related to each other in an ordered fashion as length, width and height respectively.

Thus treating the units as independent gives them a reduced meaning. Now it is true that from a quantitative perspective, that if we have 3 dimensions for a room, as length width and height respectively, the total volume will be the same (irrespective of the order in which they are taken).

So in fact when we multiply numbers a dimensional aspect is always involved. However in reduced quantitative terms this is ignored so that 2 * 3 * 5 for example = 30 (with no reference to the dimensional change involved).

In other words when we multiply 2 * 3 * 5 in this way, it is as if we accept that these measurements thereby belong to to the same dimension. So for example if we recognise the length as the only dimension, then 2 * 3 * 5 thereby represents the 3 numbers multiplied with respect to the same dimension. So the answer is thereby given in 1-dimensional terms.

So the very key to recognising higher dimensions (> 1) is that such dimensions by their very nature are not absolutely independent of each other, but must exist with respect to each other in an orderly manner.


So what we are faced with all the time is a constant dialectic as between notions of independence and interdependence respectively.

With independence, we view the units as quantitative in a cardinal manner.

So again 3 = 1 + 1 + 1.

However with interdependence, we view the units as qualitative in an ordinal fashion.

So here 3 = 1st + 2nd + 3rd.

And in experiential terms with respect to understanding, these two notions are necessarily of a relative nature in a dynamic complementary manner.

Thus we can only explicitly recognise cardinal units as independent (in an explicit quantitative manner), if we already implicitly recognise a corresponding ordinal relationship between units (in a qualitative manner).

Likewise we can only explicitly recognise ordinal units as interdependent (in a qualitative manner) if we already implicitly recognise a corresponding cardinal relationship between units (in a quantitative manner).  


So the key issue then relates to how we can successfully convert as between qualitative and quantitative notions respectively.