Thursday, June 30, 2016

Clarifying the Nature of Ordinal Numbers (2)

As we have seen, with the linear (analytic) interpretation of number, the ordinal notion is effectively reduced in cardinal terms.

Therefore we must move to a circular (holistic) interpretation, so that the ordinal aspect - while necessarily existing in relationship to the cardinal - yet preserves its own distinct identity.

When one reflects on it, the ordinal notion can only have meaning with respect to a group of numbers that are defined in a cardinal manner.

So for example one can validly seek to interpret the ordinal notions of 1st and 2nd (in the context of 2), the ordinal notions of 1st, 2nd and 3rd (in the context of 3), the ordinal notions of 1st, 2nd, 3rd and 4th (in the context of 4), the ordinal notions of 1st, 2nd, 3rd, 4th and 5th (in the context of 5) and so on.

So again to illustrate, if we have a class of 20 pupils (in cardinal terms), then we can provide ordinal rankings in an examination for 1st, 2nd, 3rd, .....20th positions.

What is fascinating here is that each ordinal position (e.g. 1st) acquires a uniquely distinct meaning (as the cardinal number of the group changes).

And in everyday terms, we can readily accept the relative meaning of such ordinal positions. For example, one can instinctively appreciate that 1st in a 2-horse race does not perhaps match the achievement of 1st in a 20-horse race!

So ordinal positions have a merely relative meaning (that keeps changing depending on context).

One could perhaps also inquire regarding the meaning of 1st (in the context of 1). In fact it is quite instructive! If we now consider for example a 1-horse race, the horse that comes in 1st equally comes in last! So this highlights the truly circular (paradoxical) nature of ordinal positions (that possess a merely relative validity).

Thus the question then arises as to how we can give an indirect quantitative interpretation for all these relative notions of ordinal numbers (which inherently are qualitative in nature).

And the answer is through simply taking successive roots of 1.

So for example, in the simple case where the cardinal group = 2, we can obtain an indirect quantitative expression for 1st and 2nd in this context.

Now to obtain the two roots of 1, strictly we must consider the two following equations

x2 = 11 and  x2 = 1respectively.

So what we are in effect seeking here is to reduce the Type 2 notion of number in a Type 1 manner.

Crucially therefore, we are attempting to "convert" the Type 2 notion indirectly in a Type 1 manner.

Therefore, though our original expression represents the "higher" holistic dimension of 2 (indicated by the power of x) the result from obtaining the two roots, represents the 1-dimensional "conversion" (that defines the Type 1 quantitative approach to numbers).

Indeed we can express the two results as fractions (in the Type 2 system).

So the first root is x = 11/2 and the second root is x = 12/2 = 11

In fact though not strictly required in this simple case, the general procedure for obtaining all the various roots of 1 is provided through the Euler identity (which we have already addressed at length in past blog entries).

eix  = cos x + i sin x = 1

Then when x is expressed in radians as 2π = 360 degrees.

e2iπ  = cos 2π + i sin 2π =  1 (i.e. 11).

So 11/2  = cos π + i sin π  = – 1.

Thus the two roots of 1 (i.e. of  11 and 12 respectively) are  – 1 and + 1.

So these results express in an indirect circular quantitative manner, the notions of 1st and 2nd respectively (in the context of 2).

Therefore, for example, to indirectly express in a quantitative manner the notions of 1st, 2nd and 3rd (in the context of 3) we would now obtain the 3 roots of 1 - strictly the 3 roots of 11 and 12 and 13 - respectively (i.e. 11/3, 12/3 and 13/3) which are – .5 + .866i,   – .5 .866i and + 1 respectively.

Note that the last root in each case, representing the 2nd in the context of 2 and the 3rd in the context of 3 respectively, = 1 (i.e. + 1). And in general terms, this will always be the case for the nth root of n.

So the reduction of ordinal to cardinal notions in conventional mathematical terms arises from sole consideration of this limiting case.

Thus in the case of 1, only the last unit ( which in this case is also the 1st) is considered. Then in the case of 2 units, only the last (i.e. 2nd) is considered; in the case of 3 only the last (i.e. 3rd) considered, and so on.

So we could say that the true holistic meaning of the ordinal relates to the other roots in each case
( 1). And in general, through obtaining the n roots of 1 (strictly 11 , 12, 13,..... 1n ), we can obtain fascinating indirect quantitative expressions for all possible ordinal numbers, 1st to (n – 1)th respectively, (within the range defined by the given cardinal number, n.

Wednesday, June 29, 2016

Clarifying the Nature of Ordinal Numbers (1)

To recap again briefly, during the 3 levels that characterise Band 3 (in my map of development), I had come to realise the extraordinary potential importance of the holistic dimensional notion of number (according to its Type 2 aspect).

Through this holistic mathematical interpretation of number, I could see how the dynamic interactive nature of experience is implicitly structured in mathematical terms, with each holistic dimension relating to a unique configuration with respect to the fundamental polarities of experience (internal/external, whole/part and form/emptiness).  Indeed as the 3rd polarity set here represents but a special requirement with respect to the first two sets, this entails that phenomenal reality (in physical and psychological terms) is fundamentally conditioned by the dynamic interaction of just two sets of polarities (i.e. internal/external and whole/part).

In psycho spiritual terms the "higher" dimensions (> 1) represent increasingly refined intuitive states, such as unfold through the process of authentic contemplative development. Then associated with these states are corresponding refined cognitive (and affective) structures of a circular paradoxical nature, which are precisely configured in a mathematical fashion as the holistic interpretation of number.

These holistic states and structures are however all initially grounded in the holistic "1" as the 1st dimension, which represents analytic understanding of a linear rational kind. So therefore, before nondual understanding of a true holistic nature can unfold, we must first initially understand phenomena in a dualistic (analytic) fashion.
Unfortunately in our present culture - especially in mathematical and scientific terms - an enormous degree of reductionism takes place, where the holistic nature of understanding is in effect completely disregarded!

I felt then that I had already made considerable progress with respect to this new holistic interpretation of number. I could readily appreciate how it was especially applicable to the "higher" levels of psychological development and how it would also form the basis for true integral scientific understanding.

However I had not yet reached the stage where I could properly relate this appreciation of mathematical symbols with the conventional accepted analytic interpretation. In other words, I had not yet properly attained to a radial interpretation of mathematical symbols (which always remained my true goal).

However as the descent through the various levels of Band 5 unfolded, this situation radically changed (especially with the onset of Level 2).

I had long puzzled over the precise nature of cardinal and ordinal numbers.

For example 2 (as a cardinal number) = 1 + 1.

So, in this definition of number, each unit is given a homogeneous impersonal identity (in a merely quantitative manner). Put another way, each unit here lacks any distinctive qualitative identity!

However the ordinal definition of number reverses the situation.

So, for example we could now refer to 2 = 1st + 2nd. However 1st and 2nd only have meaning in the context of interdependence. So ordinal notions necessarily express a relationship between numbers, which inherently is of a qualitative nature.

And it then became clear to me on reflection that the cardinal definition implicitly implied the ordinal and the ordinal in turn implicitly implied the cardinal.

Therefore when we use 2 in the customary quantitative sense (i.e. as a cardinal number) implicitly we realise that it is composed of a 1st and 2nd unit (in ordinal terms).

Likewise when we use 2 in the combined ordinal sense as comprising a 1st and 2nd unit, implicitly we must equally interpret these units as independent (in cardinal terms).

However this poses a huge problem, which is completely ignored in conventional mathematical interpretation.

Once again, the cardinal notion of number implies the assumption of "independent" units; the ordinal notion however implies the complementary notion of "interdependent" units. So properly considered, independence (in the numbers to be related) and interdependence (as the corresponding relationship between numbers) are complementary terms.  Therefore for cardinal and ordinal to be successfully combined, both the independence and interdependence of numbers must be properly interpreted in a dynamic relative - rather than static absolute - manner.

However in Conventional Mathematics,  the ordinal notion becomes effectively reduced to cardinal, with a static absolute interpretation of number (of a merely quantitative nature) resulting.

I then spent a considerable amount of time in properly clarifying how this reduction of ordinal to cardinal meaning occurs.

Let us take the simple example - which I have used before - of ranking two cars in size (with larger car ranked 1st). So say the choice is between  a BMW and a Panda. Then the BMW is ranked 1st and the Panda 2nd.

However say we now use a different ranking criterion e.g. age, with the newest car ranked 1st, and we are informed that the Panda was purchased in 2015 and the BMW in 2010. So the Panda is now ranked 1st and the BMW 2nd.

So the important point here is that implicit in the very notion of ordinal rankings is that the rankings can change (depending on relative context).

So what is 1st in one context, can be 2nd in another context and vice versa (as we saw with the ranking of the cars).

Therefore the very nature of ordinal positions from a holistic perspective - representing the potential or possibility of what can happen - is that in any grouping of numbers, ordinal positions can be fully interchanged with each other. We have already demonstrated this in the case of 2 members of a group where 1st and 2nd can be freely interchanged with each other. With 3 members of a group 1st, 2nd and 3rd can be interchanged with each other. Then more generally, with n members 1st, 2nd, 3rd,....nth position can be freely interchanged with each other.

Now in (analytic) cardinal terms, each number has a fixed identity (which is non-interchangeable). So 3 cannot for example be changed to 5 in some other context.

So we can perhaps see how the (analytic) cardinal and (holistic) ordinal notion of number are complementary with each other in an opposite manner. Whereas the cardinal can be represented in linear, the ordinal is best represented by contrast in a circular manner.

However in practise the ordinal is reduced to the cardinal, because in any actual context, no ambiguity will arise, due to the consideration of just one reference frame for ranking purposes. In other words, when we apply linear (1-dimensional) interpretation to ordinal numbers, they are effectively reduced in cardinal terms.

So in this reduced sense, each ordinal position is measured in static fashion as the last of the cardinal group in question.

So 1st equates with the last member of 1 (= 1). 2nd then equates with the last member of 2 (= 1).
3rd equates with the last member of 3 (with the other two positions already filled (= 1) and so on.

Therefore the ordinal identity, for example, of 3 = 1st + 2nd + 3rd = the cardinal identity of 1 + 1 + 1.

However implicit in the very ability to rank numbers (according to some set criterion) is the  appreciation that these rankings are unique for the criterion adopted. Thus we have no difficulty in accepting the "paradox" of the car rankings in the example above, because we implicitly recognise that 1st and 2nd can interchange (depending on context).

In other words, underlying our customary analytic interpretation of ordinal numbers, where they are given an absolute fixed identity in linear terms, is an unrecognised holistic appreciation, where all positions are interchangeable with each other.

And this is vital in the very ability to understand 1st, 2nd, 3rd .... in a qualitatively unique manner.
However, remarkably, the fundamental appreciation that these ordinal numbers represent qualitative - rather than strict quantitative - distinctions has been all but lost in Conventional Mathematics. Thus the true qualitative nature of ordinal numbers is carefully concealed through the use of the more neutral terminology of "ordinal rankings". 

Indeed we had another example of this ordinal dilemma before in the example of the crossroads where what is unambiguously left or right (when the crossroads is approached from just one direction) is paradoxically both left and right (when we admit an approach simultaneously from both directions).

Therefore I clearly realised now that the fundamental interpretation of ordinal numbers in Conventional Mathematics is strictly untenable in any meaningful sense. In other words a limited special case of a more general phenomenon is used in a way that reduces interpretation in a highly distorted manner.

Therefore though in truth number is of a dynamic relative nature, it is represented in a merely static absolute fashion; likewise though both quantitative and qualitative aspects define the interaction of all numbers (in cardinal and ordinal terms), conventional interpretation is subsequently reduced in a merely quantitative manner.

So the next stage of my investigation was to seek a means of "converting" the inherent qualitative nature of ordinal numbers in an indirect quantitative fashion.

Alternatively, this could be expressed as the attempt to express the Type 2 aspect of the number system in a conventional Type 1 manner.  

Tuesday, June 28, 2016

Dynamic Interacting Nature of Number

Before proceeding further, I will once again attempt to make clear the distinction that I make as between the two notions of number - in effect particle and wave aspects - that continually interact in experience as analytic to holistic (and holistic to analytic) respectively.

I define firstly the Type 1 aspect (where the number in question is referred to as the base).

So every base number here is defined with respect to a default dimensional number of 1.

Thus 2, for example in Type 1 terms is defined as 21.

I then define the Type 2 aspect (where the number in question is referred to as the dimensional number).

Every dimensional number here is now defined, in inverse fashion with respect to a default dimensional base of 1.

Thus 2 now in Type 2 terms is defined as 12.

Now it is vital to appreciate these two aspects of the number system operate in a dynamic relative fashion as complementary, where they keep switching (both in base and dimensional terms) as between analytic and holistic meanings respectively.

Thus when we start by defining the base in Type 1 terms as analytic, the corresponding dimensional number is then defined - in complementary Type 2 terms - as holistic.

However if we alternatively start by defining the base in Type 1 terms as holistic, the corresponding dimensional number is then defined - in complementary Type 2 terms - as analytic.

So with respect to our example of "2", it thereby keeps switching in experience as between both analytic (particle) and holistic (wave) aspects in both base and dimensional terms.

This then can be intimately linked to the psychological manner we experience numbers (and indeed  all phenomena).

The base number can be directly linked to the psychological perception of that number.
However the dimensional number is then linked - in complementary fashion - with the corresponding psychological concept of the number.

So in analytic terms, 2 as a base corresponds with the standard quantitative appreciation of the actual number (as comprising two independent units).

However in holistic terms, 2 as a dimension corresponds with the - largely unrecognised - qualitative appreciation of the number as potentially applying to all numbers (defined with respect to 2 as a dimension).  For example square or rectangular measurements with respect to a field would be 2-dimensional. However all measurements of such fields, which would be necessarily finite in nature would correspond with the quantitative analytic notion of 2 (as dimension). So the holistic notion is quite distinct in being understood as potentially applying to all such measurements (in infinite fashion).

So the key distinction as between analytic and holistic aspects respectively is that the former applies in a finite quantitative manner (in actual terms); the latter however applies in an infinite qualitative manner (in a potential manner).

By far the greatest and most profound reductionism therefore in Mathematics is the way in which holistic (i.e. potentially infinite notions) are continually reduced in an actual finite manner.

Therefore though once again the holistic notion of number properly is qualitative in nature (applying in an infinite potential manner), it is effectively reduced from a conventional mathematical perspective in merely quantitative terms (as applying directly to actual finite data).

Whereas psychologically, the analytic interpretation of number is directly rational (in a linear manner), the corresponding holistic interpretation is directly intuitive in nature, which then indirectly can be expressed in a circular (paradoxical) rational fashion.

So in Conventional Mathematics, the constant reduction of holistic notions in analytic terms is directly associated with the corresponding reduction of intuitive appreciation is a merely rational manner!

Therefore, using 2 as an example, here is a brief outline of how one experiences the true dynamic interactive nature of number!

When the perception of "2" is formed in a rational analytic manner, this links up directly in complementary fashion with the corresponding concept of "2" (implicitly in intuitive holistic terms) as potentially applying to all instances of "2".

Then in reverse fashion, we also have in experience the perception of "2" implicitly in an intuitive holistic fashion that links up directly with the corresponding rational analytic interpretation of the concept of "2" (as applying in reduced fashion to all actual data).

Whereas the rational analytic interpretation of "2" as a perception corresponds with its impersonal quantitative nature, the intuitive appreciation of "2" by contrast corresponds with  unique qualitative identity. So when we are aware of  a particular example of "2" as a unique qualitative expression of the notion of "twoness" then we are appreciating the number in a holistic - rather than analytic - manner.

So expressed in an equivalent manner, when we experience the number "2" we keep switching in dynamic fashion as between the analytic notion of 2 and the holistic qualitative notion of "twoness" both with respect to perception (as base) and concept (as dimension).

Without the potential of "2" to exist in holistic terms (as infinite), we could not recognise any actual example of "2" in an analytic manner (as finite). likewise in reverse without actual examples of "2" in analytic terms, we could not move to form potential notions of "2" (as applying in all circumstances).

In fact the true dynamics are even subtler than represented here as a continual interaction likewise necessarily takes place as between the external notion of number (as object) and its internal counterpart (as mental interpretation)!

However the crucial point to appreciate is that the true notion of number is of a dynamic relative nature, entailing both quantitative (analytic) and qualitative (holistic) notions in complementary fashion.

And of course by extension, this applies to all mathematical and scientific notions.

However, we have attempted to convince ourselves - quite wrongly - that the nature of number is inherently static and absolute and can be successfully abstracted from the rest of experience. 

When properly understood, this view is revealed as quite untenable. Worse still, its widespread acceptance has greatly distorted the very way we look at reality (in mathematical and scientific terms).  

Though, in a general manner, I had held these notions for several decades it was only now (around 2005) that I began to see them in a novel fashion  that gradually began to reveal for me the true nature of the great unsolved mathematical problem i.e. the Riemann Hypothesis.

And this quest for radial mathematical understanding (where quantitative and qualitative aspects are equal interacting partners) then became inseparable from my on-going psychological quest for the experience of spiritual union.  

Friday, June 24, 2016

Holistic Numbers

I have spoken many times before how Band 3 development, in my experience, was characterised by a keen realisation of the holistic mathematical nature of number (as relating directly to dimensions).

Now Band 2 - on which conventional scientific notions are firmly based is strictly 1-dimensional in this qualitative sense. This entails the absolute separation - in formal terms - of the key polarities of experience i.e. external/internal, whole/part and form/emptiness respectively, so that in any context involving these polarities, interpretation takes place with respect to just one fixed pole.

So conventional mathematical and scientific interpretation is based on the attempt to abstract objective (external) reality from mental (internal) interpretation. Likewise it is based on a mere quantitative relationship as between whole and parts (whereby the whole is viewed in reduced terms as the sum of its parts). Finally it is based on the consideration of phenomenal notions of material form in separation from empty (formless) spiritual notions.

So all in all, 1-dimensional interpretation is based on the reduction of qualitative to mere quantitative notions.

Holistic mathematical understanding of number (as dimension) can only be properly realised when the 3 key polarity sets (already mentioned) are viewed in a dynamic interactive experiential fashion.

In holistic mathematical terms, I had associated  the number "2" with Level 1 (Band 3). Here experience is mainly concentrated on the two-way interplay of  the horizontal polarities (external and internal).
One therefore now understands all relationships as comprising the complementary interaction of external and internal polarities that are - relatively - positive and negative with respect to each other. This holistic intuitive appreciation of complementary positive and negative poles (+ 1 and – 1 respectively) as interdependent thereby parallels the corresponding analytic interpretation of the reduced linear nature of "2" as representing dimensions, where the two roots of 1 (.+ 1 and – 1)  are considered in either/or terms as independent. 

I then later holistically associated the number "4" with Level 2 (Band 3) where experience is now additionally concentrated on the two-way interplay of the vertical polarities. Here, whole and part retain their uniqueness through two "imaginary" poles (that are indirectly expressive of the holistic intuitive nature of the unconscious).

Thus the proper mediation of whole and part aspects in experience i.e. without undue reductionism, entails the refined mediation of the intuitive unconscious - in an indirect conscious manner - as imaginary.

So the dynamic interaction of both conscious and unconscious in experience is now expressed through the holistic dimensional appreciation of  the number "4", where both real and imaginary polarities - that are relatively quantitative and qualitative with respect to each other - are considered as dynamically interdependent (in positive and negative terms). This in turn parallels the reduced analytic nature of 4 as a dimensional power (given by the 4 roots of 1). 

Finally the most refined interaction of polarities comes at Level 3 (Band 3) with 8-dimensional holistic understanding, which entails the closest approach in experience to the simultaneous identity of form and emptiness. 

This basically requires the ability to preserve both real and imaginary polarities in an extremely close balance (in the ultimate identity of both conscious and unconscious). Again this is paralleled in reduced analytic terms by the 8 roots of 1.

Though I have concentrated here on the holistic significance of 2, 4 and 8 respectively, the remarkable finding of Band 3 understanding is that every number has an important holistic dimensional significance, whereby its true qualitative significance is expressed.  

Basically this entails that associated with each number (in holistic terms) is a unique means of configuring the dynamic relationship as between the 3 central polarity pairings, which necessarily underlie all phenomenal relationships.  

Also in this regard, I came to understand that a significant distinction applies as between even and odd numbers, which has a parallel significance with respect to the Riemann zeta function (to which the Riemann Hypothesis applies).

Basically the even numbers are associated with "passive" (contemplative) states of experience where a dynamic equilibrium representing the balancing of complementary opposites applies. This again parallels the analytic situation, where one obtains an even number of roots with respect to a number.

However the odd numbers are associated with more "active" (asymmetrical) states, where one remains temporarily in a disequilibrium position from a psychological perspective. However the resolution of this problem then leads to the seeking of "higher" equilibrium states (that quickly give way in turn to further temporary states of disequilibrium).

However the problem with my Band 3 development was that experience was mainly geared to transcendent type development.

Though I was initially reluctant to accept this, it coincided with an unbalanced emphasis on specialisation of the even dimensions (especially 2, 4 and 8).

In other words I was placing too much attention in the ascent, with respect to "higher" contemplative states without sufficient emphasis on counterbalancing outward activity. So the odd numbered dimensions - including all the  primes other than 2 - were effectively becoming bypassed to a considerable extent in the process. And remarkably this implied equally in psychological terms that the deeper instinctive primitive levels of the psyche were being avoided.
This in turn coincided with the significant bypassing of the linear levels of Band 2 through the attempted integration of the "higher" levels of Band 3 with the "lower" levels of Band 1.

So as the descent through the levels of Band 5 gradually ensued, this led to a significant new perspective on the holistic mathematical notion of number, which equally provided a remarkable manner of uncovering the most primitive areas of psychic development.

Thursday, June 23, 2016

Riemann Hypothesis in Context

I would like to convey here the true significance of the Riemann Hypothesis.

However it is vital to appreciate from the outset that this significance cannot be properly understood within the accepted confines of Conventional Mathematics, which solely recognises, in formal terms, the quantitative (analytic) interpretation of its symbols.

Indeed from one important perspective, the Riemann Hypothesis - when adequately interpreted - points directly to a critical limitation in the overall accepted framework of Mathematics.  

The number system is fundamental to the reality we call Mathematics. And what in turn is central to the number system is the relationship between the primes and the natural numbers.

Now a prime has no factors (other than itself and 1). So 7 for example is an early example of a prime. All other natural numbers (i.e. composites) can then be uniquely expressed as the product of primes. So for example 30 ( = 2 * 3 * 5) is therefore composed of 3 prime factors and does not allow for any alternative factor combination.

Therefore the primes have been long accepted in conventional wisdom as supremely important, which - rather like atoms in physics - serve as the basic “building blocks” of the natural number system.

However it has also been keenly recognised that the individual behaviour of the primes is highly unpredictable with no regular pattern occurring.

Then at the beginning of the 19th century it was discovered that their overall frequency did indeed correspond to a simple log pattern.

Thus an overall dichotomy became apparent as between the - apparent - random behaviour of each individual prime and an increasingly regular pattern that characterised their overall frequency.

It was then Bernhard Riemann who on 1859, made immense strides in reconciling this discrepancy as between whole and part (i.e. the  regular collective order of the primes with their random individual behaviour).

However it is my strong contention that the overall philosophical significance of his seminal paper on the primes, has never been properly grasped.

For to put it simply, Riemann's findings point to the fact that underlying the conventionally accepted natural number system of a discrete “particle” (i.e. part) is a highly intricate continuous “wave” like system of equal importance, that in truth is dynamically inseparable from the manifest nature of natural numbers.

This discovery took place long before the quantum revolution that shook the physical world in the 1920’s. Here paradoxical features of behaviour were revealed as the inherent nature of sub-atomic matter. So for example at this level it was now understood that physical behaviour manifested itself in a complementary dual fashion with both particle and wave like features.

One might ask why parallels were not then drawn as between this behaviour at a physical level and the inherent nature of number revealed by Riemann more than half a century earlier!

However quite simply, the accepted  framework of Mathematics is so rigid that no proper accommodation of these findings could be made within its accepted confines.

Then starting from the 1970’s evidence began to emerge suggesting striking parallels as between the “Riemann zeros” (to which the wave like features number relate) and findings regarding energy levels in atomic physics.

However though mathematicians can no longer ignore these findings, they still lack the means of properly explaining their nature.

In fact the real position is so revolutionary that it undermines the very paradigm that Mathematics has been built on now stretching back over several millennia!

So what has wrongly emerged in our culture is the view that numbers - and indeed all mathematical relationships - can be interpreted abstractly in an objective manner with respect to their mere quantitative (analytic) characteristics.

However in truth numbers - and by extension all mathematical relationships – are inherently experiential in nature, comprising the dynamic interaction of both quantitative (analytic) and qualitative (holistic) characteristics.

And it is only in this relative interactive context (comprising both analytic and holistic aspects) that the true relationships as between the particle and wave features of number can be properly interpreted.

Putting it simply, one overriding confusion lies at the heart of all accepted mathematical interpretation. This relates to the attempted reduction - in every context - of qualitative notions of interdependence in a merely quantitative manner (i.e. as independent).

So for example we can indeed attempt to look at the individual primes in a quantitative analytic manner (as independent).

However the overall collective relationship of the primes (to the natural numbers) properly relates to their corresponding qualitative nature (as interdependent). However we cannot then also attempt - without resorting to gross reductionism - to express this qualitative nature in an analytic manner! Rather we must now switch - in complementary relative terms - to a corresponding holistic interpretation of symbols.  

In psychological terms this clearly entails incorporation of both conscious (rational) and unconscious (intuitive) modes of understanding, with the unconscious aspect indirectly subtly conveyed in a circular (paradoxical) rational manner.

However no recognition whatsoever exists - at a formal level - in present Mathematics of the qualitative holistic interpretation of mathematical symbols (which in truth is of equal importance to the analytic).
This is why I stress once again without a hint of hyperbole that the greatest revolution yet in our mathematical history - indeed in our intellectual history - is now required where both quantitative (analytic) and qualitative (holistic) aspects of interpretation are accepted as equal interacting partners, both in relation to the fundamental nature of the number system and by extension to all mathematical and scientific relationships.

So in this context the present accepted analytic approach represents but a limited special case of an altogether much more comprehensive understanding.

The great German mathematician Hilbert in responding to what he considered the greatest problem in Mathematics once replied,

“The problem of the zeta zeros, not only in Mathematics but absolutely (the most important)”.

Now once again the “zeta zeros” relate to the variety of solutions to a certain key equation - the Riemann zeta function - from which the wave pattern of number is constructed.

Over the past 10 years or so, I have come to agree with Hilbert, though for reasons that he would have been loath to consider!

The famed Riemann Hypothesis is also based on a certain assumption regarding these zeros i.e. that they all lie on an imaginary line drawn through .5 on the real axis.

In fact, when properly interpreted this assumption serves as the central condition for the ultimate identity of both the quantitative (analytic) and qualitative (holistic) aspects of number.

In other words the Riemann Hypothesis serves as the key requirement for the mutual consistency of mathematical symbols in both quantitative and qualitative terms (which is fundamental for all subsequent mathematical interpretation).
However once again this clearly cannot be properly appreciated through the present mathematical paradigm (which recognises solely the quantitative aspect).

Wednesday, June 22, 2016

New Transition

We return now to a personal account of the journey through the 3 main levels of Band 5.

Once again these bear a complementary relationship (on the spiritual descent) to the same three levels (on the spiritual ascent).

However whereas as the direction on the ascent is primarily of a transcendent nature, where nondual spiritual development is - to a considerable extent - at the expense of customary dualistic understanding, here a reverse immanent emphasis applies, where one now gradually attempts to incorporate the dual in a balanced manner with the nondual.

And once again, these 3 levels (of Band 5) are characterised by a special emphasis on each of the corresponding key polarity sets.

And just as with Band 3 the journey initially commences at Level 1 with intense exposure to the (horizontal) interaction of external and internal polarities, equally it is similar at Band 5.

However whereas at Band 3, this largely represents the gradual breakdown of rational analytic type appreciation in favour of a more intuitive - ultimately nondual - holistic worldview, here the reverse is in evidence as one attempts to indirectly communicate in a somewhat paradoxical fashion the nature of this worldview through the normal medium of rational interpretation.

Therefore in an important sense, this represents the first real attempt to properly harmonise both the analytic and holistic aspects of experience.

In my own case this initially coincided with the time of about 4 - 5 years (1997 - 2000), when I was intensely involved in day-to-day dialogue on the various integral forums.

For me the critical challenge that arose during this time was the requirement to effectively communicate my vision in a scientific objective manner, without becoming distracted internally by - at times - considerable personal opposition to my stance.

And though overall, I felt that I managed quite well in this regard, inevitably over time secondary attachment arose manifesting itself through a growing irritation in dealing with "difficult" participants and through suffering increasing levels of psychological stress.

In particular, I gradually had to face a crucial imbalance in that preoccupation with communicating an overall global vision of reality was interfering with my ability to properly carry out everyday responsibilities at a local level. Thus as my intuitive energy continued to be strongly focused on the former, more mundane activities gradually became drained of all inspiration.

This then led to gradual withdrawal from involvement on the forums punctuated by occasional returns on a less sustained basis.

However just as on the outward journey, this phase gradually ran its course with the realisation that once more deeper exploration of the unconscious was required in order to properly address the limitations that I had faced during the previous stage.

So a familiar pattern again reasserted itself. I had been working on a detailed account of the binary approach to development. Here every stage represents a unique configuration of both linear (1) and circular (0) aspects - representing both differentiation and integration - respectively, with a precise holistic mathematical rationale.

However having carefully mapped out the 21 levels (of the seven major bands) with respect to both (external) physical and (internal) psychological structures and then later both their characteristic states and structures, I found that my attention was becoming increasingly absorbed  by - what is perhaps - the key unsolved problem in Mathematics i.e. the Riemann Hypothesis.

I strongly sensed that issue was not only of fundamental importance in mathematical terms but was now - when properly understood - inseparable from my on-going psychological development.