Wednesday, October 28, 2009

Radial 2 Understanding

I have for some time maintained that my scientific approach is designed as a - preliminary - Radial 1 account of reality. However, before we get carried away with this let's remind ourselves that Radial 1 is but the first of 9 radial levels!

Using mathematical terminology, the Radial 1 level is based on the two fundamental digits 1 and 0 (when appreciated in a holistic sense). So the purpose of this approach is to clarify how all stages of development are basically configured in terms of the interaction of linear form (1) and circular emptiness (0) respectively.
Though of course some level of duality is required even at this level it is confined to consideration of the three fundamental polarities internal/external), whole/part and form/emptiness that underlie all phenomenal processes (and which can be given a coherent holistic mathematical rationale).

So the Radial 1 approach lies closest to the purely contemplative vision of reality translated rationally into its most universal scientific features.

For some time I had considered that the Radial 2 account would then enlarge this holistic number mapping to the successful application of prime numbers to the basic phenomena of physical reality.
However I gradually came to realise that prime numbers are in fact interdependent with the natural so that the Radial 2 state is really concerned with establishing the basic relationship between both sets.

I also considered that this would be intimately connected with the equally important consideration of the the correct relationship as between primitive instincts and the natural experience of phenomena. This is of special relevance for a contemplative attempting to re-engage more fully with phenomenal reality!

Basically phenomenal attachment arises from an undue reductionism in the manner in which objects are experienced. Unfortunately conventional scientific understanding with its preoccupation on merely quantitative aspects, greatly fosters such reductionism.
However properly understood, associated with each primitive object phenomenon (as quantitatively understood), is a corresponding appropriate qualitative means of interpretation belonging to a unique dimension of understanding.

Now seeing as present scientific appreciation is so firmly rooted at the one-dimensional level, one might validly query our capacity to understand reality at higher dimensions (the range of which is potentially infinite). Fortunately, just as string theorists believe that the phenomena that normally arise in nature correspond to low-energy states of strings, likewise appropriate qualitative understanding is likely to be largely confined to the limited range of qualitative dimensions normally traversed through deepening of contemplative awareness!

The natural objects experienced then arise from highly composite interactions of these prime ingredients. So the key to avoiding attachment at the level of natural experience is correct understanding at the primitive level (which requires conscious and unconscious to operate in very close unison).

So for one approaching the Radial 2 level from a predominantly contemplative type experience of reality, considerable attention must be devoted to this process of constructing natural experience from "virus free" primitive understanding thereby avoiding rigid attachments setting in at the natural level.


It is tempting therefore to associate the most basic particles directly with prime numbers from which natural experience then emerges in a highly composite manner.

Not surprising therefore my latest insights on strings have emerged directly from a long period of immersion with the Riemann Hypothesis.

If one accepts that string reality relates to the most primitive levels of matter, then the same principles as underline the behaviour of primes should likewise underline string behaviour.

Finally looking forward, the Radial 3 level would be more directly related to substantial contemplative involvement with the natural world. Here because of greater mastery of primitive instinctive behaviour, the proficient would now feel freer to engage intensely with natural phenomena (without the danger of attachment) .

So from a properly integral perspective, there should be no real distinction as between a desire to understand the physical behaviour of "strings" and the corresponding desire to properly uproot primitive attachment in the spiritual life.

The Number 137

The number 137 has raised considerable interest. Its reciprocal (1/137) approx. is referred to as the fine structure constant in physics and is related to the probability of electrons (or other particles) emitting or absorbing particles.

Much has been written regarding the "mystical" properties of this number. Indeed some years ago my attention was drawn to its significance through correspondence relating to Jungian archetypes.

And just recently an interesting article by Giorgio Piacenza has been published on Frank Visser's Integral World web-site.

Without wanting to claim too much for the "mystical significance" of this number, I would like to initially broaden the topic to highlight some important general properties of prime numbers (of which 137 is a specific example).

From one perspective prime numbers can be viewed as the basic building blocks of the natural number system (which we literally view in a linear manner as stretched out on a straight line).

However when we take the reciprocal of prime number a fascinating transformation is involved whereby a highly circular number pattern is involved, with the same sequence of digits continually recurring in a predictable manner.

So for example when we obtain the reciprocal of 7 the six digit sequence 142857 continually recurs.

This represents a fine example of a prime number whose reciprocal has a full periodic cycle.

A full cycle is measured as one less than the prime number in question which in this case gives a cycle of 7 - 1 = 6.

Though many other prime numbers e.g. 17, 61 and 97 exhibit full cycles in the digit sequence of their reciprocals, most prime reciprocals operate with a cycle of digits that is an exact fraction of the original prime number less one.

So when we look at the reciprocal of 137 we find that the eight digits 00729927 continually recur. So the cycle here represents 1/17 of 136. (i.e. 137 -1).

Indeed we can suggest a formula which universally holds for the sum of digits of these uniquely recurring sequences. If n = number of digits in sequence then the sum of digits = (9*n)/2.

Therefore in this case as n = 8, the sum of digits = 36.


What is especially interesting about this sequence is that ignoring the two zero digits we have in the remaining digits an example of a number palindrome.

Now a palindrome has an especially interesting psycho-mathematical connotation as a symbol of integration.

Thus whether we read the number sequence from the left or right direction, the number is the same. So these two polar opposites are thereby reconciled in a similar identity.

Some years ago I made a study of palindromes from this psycho-mathematical perspective and made a number of fascinating discoveries.

If we take any number and then obtain the difference with respect to its reverse (or mirror number) the result will always be divisible by 9.

Furthermore the resultant number will exhibit marked palindromic tendencies. Frequently the result will be an actual palindrome. In other cases adjustment of one or two digits by 1 will yield a palindrome!

To illustrate we will take 137. Now the reverse of this number (obtained by reading from right to left direction) is 731.
The difference is then 731 - 137 = 594.

When we divide this by 9 we obtain 66 which is a palindrome.

So a kind of numerical alchemy is at work here greatly facilitated through division by 9 which serves as a special catalyst of such integration.

Not surprisingly 9 forms the basis for what perhaps is the best known of modern personality systems i.e. The Enneagram (which places special emphasis on the manner through which the various personality types can achieve integration).

What then is special about 9?

Well, when we obtain the reciprocal we get the digit 1 continually recurring. So we have here in its reciprocal the simplest possible example of palindromic behaviour (for 1 is 1 whether read in a left hand or right hand direction).

If we confine ourselves to primes the next simple example (after 3) where its unique digit sequence reveals a palindrome is given by 11. So the reciprocal 09 reveals the single digit 9 (where the zero is ignored).

Indeed 11 likewise plays a unique integral role in number behaviour (which again holds universally).

Once again if we obtain the difference of any number and its mirror the result will always be divisible by 11 (where the initial number has an odd number of digits).

However where the initial number has an even count of digits, when we add both numbers the result will always be divisible by 11.

So again taking 137 (which is odd) the difference from its reverse i.e. 594 is indeed divisible by 11 = 54.

However if we take a number with an even count e.g. 2009 when we add this to its reverse we get 2009 + 9002 = 11011 which again is divisible by 11 = 1001.


Now a special significance of 137 is that - apart from the special case of 101 - it is the next prime number whose reciprocal reveals this unique palindromic digit sequence.

So 729927 is clearly a palindrome (with digit sequence the same when taken from left hand and right hand directions).

So when we divide this number equally into its left hand and right hand components, each component is the reverse of the other. So 729 (left hand) is the reverse of 927 (right hand) and vice versa. Also the sum of the first two digits (72) and last two (27) = 99 (the middle two digits).
It is also divisible by 99 with the result 7373.


Once again without wanting to attribute a direct physical significance to 1/137 (which is really only an approximation of the truer number 1/137.03604), it still possesses somewhat unique attributes.

So 137 as a prime number, represents a linear (independent) extreme. 1/137 represents a circular (interdependent) extreme possessing a palindromic sequence of several digits where L.H. and R.H. components are the reverse of each other.

It thereby possesses certain unique properties in psycho-mathematical terms as an archetype of integration.

Monday, October 26, 2009

Prime Numbers and Strings

I have already mentioned my Riemann Hypothesis article. The basic finding of this article led me to an even more fundamental conclusion than I had imagined.

This conclusion can be summed up as follows.

(i) All dynamic processes entail two distinct logical processes that are linear (1) and circular (0) with respect to each other. Once again linear logic is of the standard either/or type based on the clear separation of polar opposites (such as objective and subjective in experience).
Circular (both/and) logic by contast is inherently of a dynamic nature based on the complementarity (and ultimate identity) of these same polar opposites.

(ii) The inherent nature of a prime number is based on the consistent interaction of these two processes

(iii) The Riemann Hypothesis is required to guarantee the consistent nature of this interaction.


Thus the true nature of prime numbers is of a radial mathematical nature (i.e. entailing the twin dynamic interaction of these two logical approaches). The Riemann Hypothesis thereby is the application of the basic axiom in this system ensuring the consistent interaction of the two approaches.

The Riemann Hypothesis therefore cannot be resolved within the conventional mathematical approach (based merely on the reduced application of linear logic).

From the (analytic) linear perspective prime numbers appear as the basic building blocks of the natural number system.

However from the equally important (holistic ) circular perspective these same prime numbers are intimately dependent on the natural numbers for their precise distribution.

As neither the linear nor circular aspect can be considered as prior to each other, to proceed with the understanding of the primes we must therefore establish the relationship between these two aspects (so as to ensure their consistency). Once again the Riemann Hypothesis can be viewed as the condition required for such consistency.

In my most recent work on the stages of development, I have carefully traced through all 7 bands (and their corresponding 21 levels) fundamentally outlining the characteristic relationship at each stage of both the linear and circular aspects.

Alternatively this could be described as the relationship between the structure and state aspects of each stage.

Prime numbers (as basic building blocks of the natural numbers) can be viewed as structures; however the overall holistic relationship of such numbers to the natural (that ultimately are infinite) defines their existence as state.
Thus in dynamic terms - as would be true of radial mathematical understanding - prime numbers have both structural and state aspects that are interdependent.

It is reasonable then to assume that the prime numbers have a special relevance for the interpretation of physical reality.

Indeed we have already a considerable overlap here with the nature of quantum processes which exhibit relatively independent structures (as manifest particles) and an underlying holistic interdependence as an overall quantum state.

However it is my contention that the true nature of prime numbers becomes especially manifest at the string level of reality.

In preparing to write that article on strings this Summer (mentioned in an earlier blog) I made an amazing discovery fully in keeping with a basic intuition on the relationship of prime numbers to string reality.

I was reading again Brian Greene's excellent book "The Elegant Universe" and his description of how the masses and force charges associated with each matter particle arise through the resonant dynamic vibrational pattern of strings.

Indeed he gave a diagram on p. 144 to describe such interactions which exactly corresponds to my holistic mathematical explanation of the nature of higher dimensional understanding corresponding to 2, 4 and 8 respectively).

Let's get to the nub of the issue here quickly.

The theory of strings in physics reflects merely the linear understanding corresponding to conventional (quantitative) understanding of reality.

However in a comprehensive understanding an equally important circular (qualitative) dimension - or rather dimensions - of understanding must also be included.
One of the great weaknesses of the understanding of strings as presently constituted is that the (quantitative) strings are considered against somewhat fixed background (qualitative) dimensions.
And this problem arises from the (merely) linear approach to interpretation.

However from an equally valid alternative pespective we can consider the strings as fixed with respect to dimensions which vibrate.

So in qualitative terms, each vibrational pattern of the string corresponds to a unique psychological interpretation of physical reality. The conventional scientific interpretation merely corresponds to the lowest energy state possible, based on one-dimensional linear understanding.

However higher energy psychological interpretations - reflecting more refined contemplative understanding of physical reality - correspond to more advanced dimensional structures and states (consistent with holistic mathematical appreciation).

And in then in the radial interpretation we strive to maintain full consistency as between both the quantitative and qualitative aspects.

So remarkably in the full radial interpretation we are led to the understanding that corresponding to each resonance pattern of a vibrating string corresponding uniquely to a physical particle, that we have a corresponding psychological dimension (relating to a unique interpretation of reality). And the most complete realisation of the nature of both physical and psychological reality arises when both aspects are in full harmony with each other.

Once again the idea that matter has some definite existence corresponds merely to the lowest energy interpretation (corresponding to conventional linear logic). With higher dimensional circular, i.e. contemplative understanding, such interpretation gradually dissolves (along with the corresponding dissolution of its material counterparts). Then with fully consistent understanding neither physical matter, nor its psychological interpretation, remains.

Thus all understanding of physical phenomena in material terms such as strings relates to appearances (rather than reality) corresponding to a specific psychological manner of interpretation. When qualitative and quantitative aspects are fully consistent with each other, such appearances eventually dissolve altogether leaving what is true reality (that is ineffable).


I reached the same conclusion at the end of the Riemann Hypothesis article regarding the ultimate ineffable nature of prime numbers.

Sunday, October 25, 2009

Science and the Stages of Development

Perhaps it will help to put my approach to integral science in better context by briefly discussing the range of possible stages that can unfold throughout human development.

In my latest version there are 7 major bands - a band is defined as a general group of stages with the same basic structure - with three main levels within each band. Thus we have 21 main levels in all. Though all of these bands are open in some degree to all, very little specialised development takes place in Western culture beyond the first two bands.

From a cognitive perspective, all of these bands are defined by the interaction of two distinct logical systems - the linear and circular - which form the basis in turn for the nature of science.

The linear system can be described as the logic of form and is the appropriate basis for differentiated (i.e. analytic) interpretation of reality of a quantitative nature. The line is - literally - one dimensional and so likewise analytic science (which is currently misleadingly solely identified as "science") is based on the number 1 (as qualitatively interpreted).

The circular system can be described as the logic of emptiness and is the corresponding appropriate basis for integral (i.e. holistic) interpretation of reality of a qualitative nature. The circle is - literally - non-dimensional (in the shape of O) and so likewise holistic i.e. integral science is based on the number 0 (as qualitatively interpreted).

So just as the the digits 1 and 0 - when used in a quantitative sense - serve as the binary system that can be potentially used for the encoding of all information, the same digits 1 and 0 - when used in a corresponding qualitative sense - serve as an equally important binary system that can potentially be used to encode all transformation processes.


I describe three fundamental scientific approaches:

Firstly we have analytic i.e. conventional science based on the specialisation of the linear (1) logical system.

Secondly we have holistic i.e. integral science based on the specialisation of the circular (0) logical system.

Thirdly we have radial i.e. comprehensive science based on the combined specialisation of both linear and circular logic.
Indeed properly seen the other two aspects (analytic and integral) are but partial aspects of the truly comprehensive (i.e. radial) view of science.

The various bands (and more detailed stages) can be viewed in terms of the interaction throughout development of the qualitative binary digits i.e. the linear and circular logical systems respectively.

When development commences both of these systems remain totally confused with each other with (conscious) differentiation completely entangled with (unconscious) integration.

The first band of development (sometimes referred to as prepersonal) is concerned with the gradual unfolding of conscious (differentiated) understanding.

The second band - which is especially important with respect to standard science - then concentrates on the specialised development of linear logic.
In practice in Western society development then significantly plateaus at this band.

However as is clearly evident from the various mystical religious traditions more advanced development can - and under suitable conditions - does take place.

The next band is then concerned with the gradual unfolding of spiritual intuitive (i.e. contemplative) type understanding of reality that is of a qualitative holistic nature based on the circular logical system. Whereas linear logic is based on the clear separation of polar opposites (such as external and internal) in experience, circular logic by contrast is based on their corresponding complementarity. Though ultimately such complementarity of opposites leads to ineffable understanding that is directly spiritual, between rational linear understanding and this pure state there are many dimensions of ever greater refinement where understanding of form becomes increasingly dynamic and highly transparent.

So in a sense integral science can be viewed as the application of these higher dimensions of understanding (which have a precise holistic mathematical rationale) to the interpretation of reality.

Just as the second band was devoted to the specialised development of linear, in like manner the fourth band is likewise devoted to the specialised development of circular understanding in the consolidation of a truly contemplative - as opposed to merely rational - perspective on reality.

The fifth band is then concerned with the first unfolding of radial development where
both rational (linear) and intuitive (circular) understanding interpenetrate in a mature manner.

Once again the next band (sixth) is devoted to the specialised development of radial interpretation. However even here a slight bias is likely to remain with some personalities inherently of a more active type (where linear logic predominates) and others of the more contemplative variety (where circular logic - as the indirect rational basis of intuitive understanding - likewise predominates).

So in my account the final seventh band is designed as the attempted correction of this bias.
And as it goes somewhat against the inherent grain of personality, it is likely to be associated with much suffering. So here the more passive type may finally accept the need to assume a leadership role in promoting some special cause. By contrast the more active type may now accept the need to step back somewhat from involvement in affairs so as to more deeply engage with the inner life.

Of course in practice very few are destined to traverse all seven bands in a manner related to their full development. Rather I have designed them in such a manner to serve as template especially for a fuller scientific appreciation of reality.

So beyond the first two bands, which are necessary for appreciation of conventional (analytic) science, we have five more bands of potentially rich scientific significance. The third and fourth bands are especially appropriate for the holistic qualitative appreciation that is integral science while the final three bands would then be required for radial science.

Though I would see radial science as ultimately both the most important and comprehensive of the three approaches, I have confined myself in my own work in the main to integral science (which is still greatly unrecognised). So without the development firstly of this important integral aspect, there is no way likewise of developing radial science (where this integral aspect serves as a vital component).

Thursday, October 8, 2009

Integral Physics

Just as standard i.e. conventional Mathematics provides the essential tool for analytic science (which we misleadingly define as "science") Holistic Mathematics provides the essential tool for a new form of integral science with a truly extraordinary potential for development.

When properly understood in a more comprehensive manner, every (analytic) branch of science has its integral equivalent based on a truly holistic appreciation of symbols.

So for example we can validly speak of Integral Physics and Integral Economics (just to give two examples with which I have been mainly concerned).

Though I had developed many general notions, with some important specific illustrations, concerning integral science, recently I returned to physics with a view to a more detailed investigation.

So during the Summer I completed three main articles, which still require some minor revisions, with the promise of a fourth to come (researched for, but not yet written).

The first article is more general in nature relating to the precise nature of an integral - as opposed to analytic - interpretation of physics.

One simple way of explaining the distinction is with respect to the comparison as between the rational and contemplative worldviews. So standard physics - in formal terms - is strictly based on rational criteria whereas integral physics is directly based on a contemplative worldview (indirectly conveyed through a distinctive paradoxical appreciation of rational symbols).

Now the most comprehensive form of physics - Radial Physics - then combines both rational and contemplative understanding in a manner that greatly enhances both single aspects. However - except in a very preliminary manner - I have not concentrated on Radial Physics as there is vast undiscovered territory yet to be explored at an integral level.

The second article explores Einstein's Relativity (Special and General) indicating the integral equivalent of key concepts. I found this very exciting as it opens up an entirely new appreciation of the nature of relativity.

Putting it briefly standard understanding is based on the quantitative interpretation of movement (within a fixed logical interpretative system).

Integral understanding by contrast is based on an alternative qualitative interpretation of movement (where logical interpretation itself changes becoming progressively more dynamic).
And this essentially is what happens through authentic contemplation. So in a way I am examining the precise implications of changes in contemplative understanding for the appreciation of physical reality.

The third article then applies the integral approach to quantum mechanical understanding. One key conclusion from my investigations here is that one cannot attempt to divorce physical understanding of reality at a quantum physical level from the corresponding manner of its psychological interpretation.
Once we properly realise this, the findings of quantum physics are fully in accordance with the enhanced intuition of the contemplative world view.

The final - and most ambitious - article proposes to deal with the world of strings.

We will perhaps return to give a flavour of this in future blogs.

Wednesday, October 7, 2009

Reflections on Riemann Hypothesis Article

It is now some months since I completed the article A Deeper Significance: Resolving The Riemann Hypothesis.

This is turn followed on more detailed work on the topic at my website Radial Development.
What I was attempting t0 do was indeed very ambitious i.e. resolving the Riemann Hypothesis.

Now from one perspective it might seem absolutely presumptuous that an amateur with no special ability in - what is conventionally called - mathematics should even attempt to tackle such a problem.
However through Holistic Mathematics - which I have been developing now for some 40 years - I was confident that I could approach the issue from an entirely new perspective.

So in the end I indeed resolved the issue to my own satisfaction, with the Riemann Hypothesis in the context of this new perspective having a remarkably simple resolution.

Just to briefly summarise my findings!

The Riemann Hypothesis is true! however this cannot be proven (or disproven) within the axioms of conventional mathematics .

Rather the Riemann Hypothesis arises as the direct consequence of the fundamental axiom of a more comprehensive mathematical approach - which I term Radial Mathematics - that combines twin aspects (relating to distinctive logical systems).

The first aspect is provided by Conventional Mathematics which is based on linear either/or logic (requiring the separation of opposites).
The second aspect is provided by Holistic Mathematics which is based on an alternative circular both/and logic (requiring the dynamic complementarity of opposites).

So a comprehensive mathematical understanding really requires both the analytic and holistic interpretation of symbols (represented by Conventional Mathematics and Holistic Mathematics respectively).

Just as in geometrical terms the line and circle are reconciled at the central point which is common to both, likewise linear and circular understanding are ultimately reconciled at a common point which is ineffable.

This provides the appropriate context for understanding the true nature of prime numbers. They inherently combine two logical systems the ultimate nature of which is ineffable.

The significance of the real part of 1/2 (to which all non-trivial solutions to the zeta function are intimately related) can best be seen as representing a golden mean as between opposite extremes. Reconciling linear with circular logic requires maintaining complete harmony as between dualistic opposites.

So inherent in the resolution of the Riemann Hypothesis - and also inherent in the fundamental nature of prime numbers - is the consistent reconciliation of the two distinctive types of logic that are themselves necessary for proper comprehension of the issue.

Thus the most fundamental axiom of Radial Mathematics (entailing the twin use of both linear and circular logic) entails the truth of the Riemann Hypothesis.

However as this truth relates to the relationship as between two distinctive types of logic it cannot be resolved with reference to just one!

Thus the Riemann Hypothesis - though necessarily true from a radial perspective - cannot be resolved (i.e. either proved or disproved) within the axioms of Conventional Mathematics.