Integral Science

Synthetic Biology

2012-01-21T13:12:00.000-08:00

It was another Horizon programme on BBC that recently set me thinking.

It was on the new field of "Synthetic Biology" where very dramatic changes are likely to take place in a short period of time.

We are already well used to the notion of computer programming. However this is going a big step further in that direct intervention with life forms can now take place through genetic engineering.

So already one can through the Internet purchase bio bricks and then seek to assemble the in a manner that never existed before in the natural world. So genetic advances that took many millions of years to evolve can now - apparently - be reprogrammed in an instant with potentially devastating consequences.

Now the initial phase of this engineering seems admirable enough!

For example we are introduced to three revolutionary new goats which are genetically part spider. Apparently a particular type of spider naturally produces a very strong type of silk thread which has all sorts of possible commercial applications. However with reliance on natural production the amount of thread produced would be minimal. However by now genetically combining the spider with the goats, such silk can now be produced in much greater quantities (when extracted as an additional protein from the goats' milk).

We also saw another example whereby brewers' yeast, instead of producing alcohol can be now genetically modified so as to produce diesel oil. So with the likely prospect of growing shortages in natural oil in the future one can readily appreciate the commercial possibilities of this new development.

Other potential uses again seemed in principle of admirable intent. For example the prospect of using genetic engineering to modify pathological body cells such as cancer would seem very welcome.

However what would worry me greatly here is the issue of control. When people are free to buy DNA parts from the Internet and experiment in creating life forms (that may never have existed before) it is potentially very worrying. Also when amateur clubs have already been set up in parts of the US with the intention of doing likewise one wonders where this will go.

Also while at present the main emphasis is on "lower" life forms, can this situation last? Given the pace of these new developments it may not be long till science fiction truly becomes a reality and all sorts of new creatures can be created in the lab. And then what happens if these then escape or are let loose with the capacity to multiply (perhaps at dramatically fast rates).

I even raised myself in "Brave New World" that it may be eventually possible to create intelligent life forms with the capacity to evolve much more rapidly than humans. This would indeed raise startling new practical and ethical problems.

In a best case scenario the possibilities opened up by this new life technology could be used greatly to benefit rather than hinder mankind.

However evil will always remain so it is only probable that the same technology could be carelessly used without adequate knowledge of the consequences or even deliberately designed as a of form of terror against one's enemies.

At present we seem to be still largely in control of this new life technology. However sooner than we realise the fruits of such technology could threaten to control mankind in an altogether unexpected manner.

A New Big Bang

2012-01-13T03:38:00.000-08:00

I will briefly attempt here to provide an alternative Big Bang scenario that properly caters for both the quantitative and qualitative aspects of phenomena.

In this scenario we always start from a present moment (which continually exists). Phenomenal notions in space time then obtain a merely relative meaning with respect to this absolute ineffable source (and end) of existence.

So before created phenomena come into being we have mere potential for existence. This can be represented in holistic mathematical terms as the total confusion of union (1) relating to quantitative notions of form with nothingness
(0) relating to corresponding qualitative notions of emptiness.

Earliest physical creations begins with duality (2) in both quantitative and qualitative terms. This quickly through a dynamic iteration process generates almost immediately all prime (and natural numbers).

If we could conceive of a world of merely of prime numbers then quantitative and qualitative aspects would remain identical. Put another way in a universe of merely prime numbers (as the most fundamental physical "objects") we would by definition have as many qualitative dimensions (as quantitative objects) in an ineffable manner. However the rapid combination of prime "objects" and "dimensions" quickly generates natural objects (which implicitly do have a phenomenal physical identity).

However this rapid generation of natural objects quickly leads to a collapse in corresponding dimensions (as matter achieves a more stable phenomenal form).

So the real issue in earliest creation is how the increasing identification of phenomenal objects necessarily leads to a dramatic loss in the unique dimensional qualities of these objects, thereby enabling them to attain an ever more common collective identity (i.e. with characteristics shared in common).

In the holistic mathematical sense in which I use dimension, remarkably what we term "science" as a rational means to investigate such evolution is by definition 1-dimensional.

Thus in the extreme desire to understand the nature of phenomenal objects we have reduced the dimensional qualitative aspect - literally - to an absolute minimum of 1. Therefore though conventionally we speak of a 4-dimensional physical world, we actually create significant asymmetry by treating the 3 spatial dimensions in a reduced quantitative manner.

What is remarkable is that if we now wish therefore to properly understand the nature of earliest creation (which existed in an unreduced multiple-dimensional framework) then we must psychologically experience reality in such higher dimensions.

Again using my holistic mathematical approach, I have demonstrated how nature through an advanced contemplative type intuitive vision actually corresponds to this higher dimensional perspective.

The clear implication is that to experience the beginning of creation we must experience from the end of creation i.e. through the psycho-spiritual attainment of pure spiritual union (which equally of course is an emptiness).

Thus in properly understanding the refined psychological dynamics prior to total union we would recreate in reverse complementary fashion the phenomenal structures of earliest creation.

What this demonstrates is that what we can know about physical reality ultimately is entirely mirrored by the psychological means by which we interpret this reality. Truly this reality and its appropriate interpretation are as mirrors to each other. So when they completely mirror each other there is no longer any separation but a union (that is also nothingness).

Finally, though I have been very critical of string theory there are marked similarities evident in the two approaches.

For example in both cases ultimately reality is seen in a mathematical fashion (though I would lay much greater emphasis in holistic mathematical interpretation in this regard).

Also both approaches see early creation as entailing a much higher number of dimensions (which become subsequently reduced with the existence of stable phenomena).

However again - unlike conventional string theory, I would draw a strong distinction as between both quantitative and qualitative interpretation. Indeed as I have written elsewhere the key notions of string theory can be given an alternative "imaginary" interpretation (in qualitative terms).

More on Higgs Boson

2012-01-12T02:48:00.000-08:00

I am returning to the topic of the last entry (i.e. the Horizon Programme on the hunting of the Higgs Boson).

There are many reasons why I have strong reservations regarding the interpretation of recent findings.

The finding of the Higgs Boson is required to bring a greater degree of completion to what is called the Standard Model which deals with the interactions between known particles and forces. Now admittedly this has proved remarkably successful in predicting an enormous range of physical events with astonishing accuracy!

However a major limitation is that it excludes gravity.

So to give an analogy; the search for the Higgs Boson represents the equivalent of finding a crucial weapon in a crime investigation. However in this investigation, the key potential suspect has been eliminated from investigation (due to the difficulty of searching for evidence). So in the absence of this key suspect, police are now attempting to pin the crime on someone else. Then to build a case against another suspect some key incriminating evidence is required. So if a weapon - deemed to belong to the suspect - can be found at the scene of the crime, police will be ready to press charges!

However one could validly question this whole exercise as a somewhat artificial attempt to solve a crime (in the absence of the chief suspect).
So even if an incriminating weapon is found at the scene, a variety of possible explanations could be given. Indeed the chief suspect way even have planted this evidence so as to falsely incriminate another person.

In the same way, even if the Higgs Boson is eventually deemed to exist it still begs a lot of questions in the absence of the inclusion of the gravitational force in the Standard Model. Indeed one obvious suggestion is that the very existence of this boson intimately depends on the gravitational force!

So again even if the Higgs Boson is confirmed to exist, it represents important information, which however in all probability requires a much deeper explanation (than can be provided within the Standard Model).

Also it must be stressed again that the very rationale of the Standard Model is of an extremely reduced - and ultimately untenable - nature.

The idea here is that the Universe - as we know it - is ultimately comprised of basic "building blocks" of matter (which hopefully can be experimentally discovered through ever more powerful particle accelerators).

However in order to give fundamental particles a meaning, they must be placed within a preconceived environment of space and time.

This in turn reflects the reduced orientation of Conventional Science (where qualitative considerations are reduced to the quantitative).

So we cannot conceive of the specific (quantitative) aspects of matter in the absence of the holistic (qualitative) dimensions they inhabit.

Therefore the true task of science is not to just to explain the basic "building blocks" of existence but rather the manner in which both particles and dimensions - which have no strict meaning in the absence of each other - come into existence.

So it is strictly futile attempting to construct reality from its basic "building blocks" when these phenomena cannot be understood in the absence of corresponding physical dimensions!

The real task therefore is to explain how matter and dimensions mutually arise through dynamic interaction with each other. And this requires recognition of both the standard and - as yet unrecognised - holistic aspects of science.

And the ultimate implication of this approach is that what we call "reality" in fact strictly represents appearances of a merely relative nature arising from the interaction os twin aspects that are quantitative and qualitative with respect to each other. So true "reality" would be then understood as that absolute ever present source (and goal) of all that phenomenally exists in relative spacetime.

Indeed there is already a clue in existing thinking to this new way of looking at reality.

Moderns physics is strongly based on the notion of symmetry. However the paradox then remains that the perfect physical theory (based on total symmetry) would mean that reality as we know it - which requires a degree of asymmetry - could not exist!.

However we can now perhaps see that the perfect versions of physical theories are pointing directly at true "reality" as the ever present source (and goal) of what phenomenally exists.

There was also an interesting indication of this in the programme when Michio Kaku referred to the beautiful objects all around us in nature e.g. snowflakes as fragments of an original perfect world (before phenomenal existence) where perfect physical symmetry reigns.

What is fascinating about this that we can give a direct psycho spiritual counterpart to this observation in that all beautiful objects in nature represent reflections (or archetypes) of an ultimate perfect reality (that is spiritual in nature).

This would then immediately suggest that the physical and religious quests to know reality are themselves truly complementary. So perfection (in physical or spiritual terms) is in the realisation that both the quantitative and qualitative aspects of reality are ultimately fully symmetrical!

The truly great limitation of Conventional Science is the manner in which it attempts to abstract physical reality in a reduced quantitative manner from what is spiritual.

In the end one cannot attempt to properly understand the physical world without equal recognition of its spiritual aspect. So for every quantitative type relationship a qualitative counterpart exists.

However whereas (linear) reason is the appropriate vehicle of the quantitative aspect, intuition recognition (which indirectly can be given be a precise circular logical interpretation) is the appropriate scientific vehicle of the qualitative.

Higgs Boson or Higgs Illusion

2012-01-10T07:35:00.000-08:00

I was looking at the BBC Horizon programme last night on the Higgs Boson which proved quite interesting.

As was widely reported in the media late last year, a determined attempt has been made to find convincing experimental evidence for the existence of the Higgs Boson which if verified would help to complete the standard particle model of physics.

One outstanding problem with this model is that it had yet to provide a convincing explanation as to how particles acquire mass. And as this requirement is crucial for explaining the existence of all phenomena, the issue was of great importance.

It had been proposed in the late 60's by Peter Higgs that what gives mass to particles is related to a seemingly invisible field viz. the Higgs Field. And as all fields are associated with corresponding particles, it was postulated that if this supposition of the existence of a new field was correct that it should in principle be possible to detect its associated particle.

However the technology to conduct the kind of experiments capable of uncovering such a high energy particle remained insufficient for the task until recently.

So now using the 17 ml. long Large Hadron Collider built underground at Geneva, researchers now feel confident that they can settle this issue once and for all.

Great excitement was raised by earlier tests carried out before Christmas which seemed to suggest that perhaps the elusive Higgs Boson had been at last detected (though further testing will be required to reach a conclusive verdict).

However as I listened to so much scientific hype regarding this as the most significant development since Einstein's discovery of Relativity and that scientists could now concentrate on closing in further on the initial instant of creation, I experienced a great let-down and an inability to share the excitement in news that I do not consider as truly fundamental.

The great weakness - which seems largely unrecognised by scientists - is that we cannot hope to come to knowledge of the fundamental nature of the Universe while attempting to remain as passive observers of its assumed objective existence.

When one reflects on it we are obviously dealing with a self-referential system (where what is known about reality inevitably reflects the nature of our psychological manner of interpreting such reality).

So we cannot possibly know the nature of objective particles or discover what transpired during the initial moment of creation when our very means of interpretation is itself dependent on the considerable evolution of creation that has since unfolded.

Thus in a very real sense, the initial beginning of creation has no meaning in the absence of the subsequent evolution required to then enable intelligent minds to reflect on the nature of this existence.
The Universe therefore has so evolved, that we can now, as an inseparable part of this creation, attempt to look back and discover its meaning in both objective and subjective terms.

And once we accept the inevitable fact that human nature entails both physical and psychological aspects that are interrelated, then we cannot meaningfully attempt to portray the unfolding of one aspect of evolution i.e physical while ignoring the other aspect i.e. psychological. The very irony in this situation is that any attempted physical explanation of reality - even through conventional scientific means - is in fact impossible in the absence of corresponding psychological constructs (that must necessarily be used to interpret this reality). So we can never describe (objective) reality as such but rather a more subtle relationship entailing the dynamic interaction of both physical and psychological aspects.

This is not to suggest that there is no valid role for the present model of science (based on mere objective interpretation). However such science is best at the everyday level of macro reality where stable objects and relationships exist.

However as we approach the ultimate nature of reality, it becomes increasingly unsuited to discovery of its moment of origin in spacetime (where both the actual and potential capacities of existence remain considerably confused with each other).

The strange paradox is that in the most fundamental sense, the earliest moment of creation is in fact inseparable from the present most advanced moment (with all secondary measurements of space and time of merely a relative nature).

So the only meaningful manner to realise ultimate physical symmetry (before finite creation occurs) is in experiencing that ultimate psychological symmetry (representing pure spiritual union).

Ultimately the only way therefore to truly discover the full nature of physical creation is through equally experiencing the full realisation of this reality (where physical and psychological aspects are inseparable).

And in this moment of pure spiritual union continually existing in the present moment, creation comes to realisation of its eternal mystery.

What we are in fact approaching now are the limits of one particular model of science. In precise holistic mathematical terms, I refer to this as 1-dimensional science. The very nature of this model is that it inevitably reduces in any context qualitative to quantitative meaning. However, associated every other number as dimension are ever more subtle scientific systems (where quantitative and qualitative aspects while remaining to a degree distinct can yet dynamically interact).

What we need to clearly realise is the fact that what we recognise as fundamental particles are in fact just appearances of reality (with no ultimate substance). They only appear to have substance when we attempt to give them an independent (objective) existence.

So the Higgs Boson - even if verified satisfactorily according to present scientific understanding - properly represents a physical apparition (corresponding to a particular means of - arbitrary - mental interpretation).

In fact the "finding" of the Higgs Boson will not depend on locating its direct physical existence but rather in achieving a consensus with respect to an interpretation that the indirect effects of the physical experiments conducted are consistent with its postulated existence!

New Perspective on Harmonic Series

2011-08-05T06:39:00.000-07:00

The very important and well-known harmonic series i.e. 1 + 1/2 + 1/3 + 1/4 + 1/5 +... is especially associated with Pythagoras who reputedly found in these simple fractions definite links with the manner in which musical notes sound. This pattern of the simple natural fractions, in particular, seemed to correspond perfectly with - what we would recognise as - a harmonious sequence of notes and thereafter it has become known as the harmonic series.

So we have here a very clear link as between simple mathematics and musical harmonics. Subsequently it was shown that this series (and its associated series where the dimensional power of each fraction itself can vary from 1 and ultimately alter over the entire range of complex numbers) has intimate connections with the prime numbers!

So in a certain valid sense there is music in the primes. So just as we are accustomed to give a wave form to musical sounds likewise there is a wave pattern associated with each prime number.

Indeed we could go further and suggest that there are intimate links as between the prime numbers and quantum mechanics with each possessing particle (discrete) and wave (continuous) aspects. The real implication is that just as music itself has quantitative and qualitative aspects which interact to produce the experience that we recognise, likewise - when appropriately understood - prime numbers and quantum mechanics likewise arise from the interaction of quantitative and qualitative elements.

As we have seen the harmonic series is made up of the reciprocals of the natural numbers.

Now each of these numbers has a dimensional aspect as the corresponding fractional powers, which results in a circular - rather than linear - quantitative structure. The corresponding qualitative numbers are then simply the natural numbers, 1, 2, 3, 4, 5.. etc now interpreted in a holistic manner.

Recently I have come to the conclusion that quality in all its aspects ultimately relates to this dimensional appreciation of number (interpreted in a holistic manner). In this context each dimensional number represents a unique way of configuring the fundamental polarities of experience (internal/external and whole/part).

So quality in the most basic sense always arises from a certain manner of configuring these basic polarities (which is given by the qualitative interpretation of dimensional numbers).

Seen in this light the qualitative correspondent of the harmonic series is simply the natural number dimensions. So it is not surprising that our fundamental appreciation of harmony (relating to the manner of configuring polarities) corresponds directly with the simplest dimensional numbers. These are the very ones with which we are implicitly programmed in out attempts to discern qualitative meaning!

A Double Code

2011-08-01T08:30:00.000-07:00

In a post on my blog on the Riemann Hypothesis, I referred to a new Mathematical series on the BBC hosted by Marcus du Sautoy called "The Code". I have already seen several such programmes with du Sautoy who I enjoy watching for his enthusiasm and obvious love of the subject. I also enjoyed greatly reading his books "The Music of the Primes" which played a large role in pushing me on to develop my own own insights on the primes. More recently I read his later book on Symmetry - an area in which he specialises - "Finding Moonshine".

Now the code that du Sautoy is referring to to relates to the quantitative use of numbers that is so wonderfully successful in helping to clarify so many of nature's secrets.

However Mathematics equally contains another marvellous code in the qualitative interpretation of these same numbers. However, seemingly there is as yet little or no recognition of the potential significance of this latter code.

I have frequently explained how the qualitative meaning of numbers is intimately tied up with mathematical dimensions. Unfortunately from a conventional perspective, only a reduced linear rational interpretation - that is literally 1-dimensional in qualitative terms - can be given.

In the most fundamental sense this true qualitative nature of dimensions relates to the dynamic manner in which the polarities of experience interact. In the most basic terms these polarities come in two pairings. The first relates to external and internal whereas the second relates to whole and part distinctions.

Now the conventional scientific (and mathematical) perspective is thereby linear as it seeks to define one unambiguous direction with respect to these polarities. In other words reality is here identified in a non-interactive fashion with what is understood as external to the observer. It then deals with wholes and parts in a reduced quantitative manner (whereby parts are simply viewed as fragmented constituents of the whole).

However though explicitly such scientific interpretation is indeed linear, implicitly in experience dynamic interaction in varying ways still takes place allowing for a much more varied dimensional appreciation (where many dimensions can interact to a degree with each other).

What I have recently come to realise is that our qualitative experience of reality in fact relates to the manner in which these various dimensions (representing the interaction of the fundamental polarities) takes place. And as each unique dimensional configuration is represented by a number, this thereby entails that all experience can be represented as the dynamic interplay of numbers (with respect to both their quantitative and qualitative aspects).

In a direct sense this qualitative dimensional aspect relates to the affective function in experience (which can then indirectly be translated mathematically in a cognitive rational manner).

So understood from this perspective our emotional experience with respect to sense and feelings represents the manner in which we configure various number dimensions (from a qualitative perspective).

One especially fruitful area of investigation would relate to musical appreciation. It has been demonstrated - at least since the time of Pythagoras - that there is a marked quantitative aspect to the manner in which we experience sound. So sounds that appear to us especially harmonious relate to the simplest ratios involving the natural numbers.

However an even deeper explanation of why this appears to be the case would relate to appreciation of the qualitative - as opposed to the quantitative - interpretation of number.

What I am suggesting therefore is that because our qualitative experience of dimensions remains somewhat undeveloped in present human evolution, we find it difficult to appreciate all but the relationship as between the simplest natural number dimensions. So the harmony in sound that we thus experience relates to the greater familiarity we have with the easiest dimensional configurations.

Put another way, for someone who in this affective context has implicitly achieved a more developed dimensional understanding, then a much more refined appreciation of sound would then be possible. Then what might appear dissonant at a customary level of experience, might now appear quite harmonious!

This also provides an interesting explanation as to the difference as between artistic and scientific type truth.

The very basis of current scientific truth is the universal acceptance (explicitly in formal terms) of its qualitative 1-dimensional manner of interpretation. However the basis of artistic appreciation is that people implicitly, as we have seen, use varying dimensional number configurations in attaining their qualitative understanding. And as such configurations can vary widely from person to person this implies that universal agreement as to what constitutes artistic merit is not therefore possible.

Of course in some respects a conventional consensus (at least in certain sections of society) may exist as to what constitutes artistic value but this simply reflects a shared qualitative perspective among those with influence, who thereby configure the qualitative dimensions in a largely similar manner.

Also it must be said that even where the same qualitative dimensions are involved, marked variations in subsequent experience (relating in turn to considerable differences as to the degree of spiritual refinement with which they are experienced) can exist.

Brave New World

2011-06-30T04:41:00.000-07:00

I was reading Paul Allen’s “The Idea Man” recently which I found fascinating on several levels (especially with respect to his relationship with Bill Gates and the founding of Microsoft).

Clearly both Allen and Gates were extremely talented individuals with a special gift for programming. What I did not realise however was the extraordinary amount of work that they put in eventually leading to the founding of Microsoft.

In Jungian personality terms, whereas Gates would have been an S, Allen clearly was more of an N type. Though the relationship between them was never easy, the combination of Gates’ pragmatism and ruthless realism allied to Allen’s creative vision and shrewd reading of technical developments effectively led to the founding of Microsoft.

However as regards the survival of the fittest with respect to Microsoft ultimately only one person would assume control which inevitably meant Bill Gates.

In the end whereas Gates was intent on making Microsoft his life, Allen wanted the freedom to devote his energies to a hotch potch of differing interests (which his immense wealth has facilitated).

Reading this book got me thinking again with respect to my own “visionary perspective” for the future of computers.

Once again the present IT revolution depends heavily on the quantitative use of the binary system i.e. in the ability of using a series of 1’s and 0’s to encode information.

However I have long advocated a second qualitative binary system that would make use of the holistic interpretation of 1 and 0 with the potential power to encode all transformation processes. 1 in this context relates to linear logic as the actual basic for encoding (unitary) form whereas 0 relates to circular logic as the corresponding indirect basis for encoding emptiness i.e. nothingness (as the potential source of all form).
And it is in the interaction of form and emptiness at all levels of reality that we have continual transformation.

Now up to this I have always tended to associate this second type in its most developed state with the human species especially with respect to the potential for considerable spiritual transformation.

And as evolution of such spiritual potential in general is of a very slow nature, this would place considerable limits on the possibilities of radical “computer” developments in the near future with respect to the use of this qualitative binary system. Indeed - as I have so often stated - it is not even formally recognised in current scientific thinking where the basis paradigm is almost exclusively built on the recognition of - merely - one qualitative digit (i.e. 1 corresponding to the linear logic of form).

Now the human species represents an organic biological life process (which is distinct from the mechanistic type of systems associated with quantitative IT type developments).

However it struck me recently that perhaps it is mistaken to confine possible developments with respect to the alternative qualitative system to advances in human spiritual evolutionary development (which could take a long time).

In other words other biological life processes could perhaps be created through laboratory means with far greater potential for rapid spiritual development. If this is true then this would entail that an alternative qualitative IT revolution could then be directly associated with the use of these biological life forms that would be capable of acting in a highly creative manner.

Indeed from one perspective, just as current IT devices can be seen as an extension of the human personality with respect to facilitating information of all kinds, in time these alternative qualitative IT devices - relating directly to “creative” intelligent biological life forms - would then be seen as an extension of the human personality with respect to facilitating transformation, leading to the potential for very rapid spiritual evolution in humanity.

However this brave new world would raise very grave issues regarding democracy and control with perhaps the human species having to recognise that it no longer can dominate in this regard but rather would have to concede considerable rights to new intelligent life forms that have been initially biologically created through artificial means.

Of course the ultimate IT revolution would then combine both the quantitative possibilities for encoding information with the qualitative possibilities for encoding transformation.

In fact I am slowly coming to the realisation that - when appropriately appreciated - what we know as reality is in fact but the operation of this ultimate IT system (most of whose secrets have yet to be revealed).

One obvious realisation that will flow from the mastery of this final comprehensive IT system is that phenomenal reality itself - on all levels - is capable of direct transformation.
Put another way phenomenal reality - when properly understood - can be seen to represent but the dynamic configuration of the binary digits (with respect to both their quantitative and qualitative aspects).

Therefore the changing of phenomenal reality in any context ultimately relates to the ability to appropriately reconfigure this digital system (with respect to both its quantitative and qualitative aspects).

We are very far from proper realisation of this at the moment. However the very fact that I am even writing about such possibilities would suggest that the starting recognition of such an ultimate IT system has already commenced.

And in the deepest sense this equally implies recognition of the true nature of our essential personalities as God incarnated in phenomenal form.

Prime Mystery

2011-06-15T01:41:00.000-07:00

As always we can provide a complementary psychospiritual account of the phenomenal activity (corresponding to what in physical terms lies below the Planck length).

As we have discussed before with increasing contemplative activity, phenomena of form take on a more transparent elusive quality. This ultimately relates to the dynamic manner in which the basic polarities of experience increasingly interact.

So - quite literally - experience becomes of a higher dimensional nature. Ultimately therefore the relationship between polarities becomes so rapid that it is no longer possible - in explicit terms - to follow their movement. So at this stage it is thereby not possible to provide a coherent refined rational interpretation of the subtle inter-relationships implied by these dimensions. In other words as the very attempt to provide such an interpretation would itself interfere with the dynamic level of interaction already attained, one must abandon formal interpretation in any explicit manner.

Put another way as relationships are now of an increasingly (circular) paradoxical nature, any attempt to objectively explain their nature would introduce an undue linear influence that would thereby distort their intrinsic nature.

This corresponds well with the nature of physical relationships below the Planck length where relationships become so circular and paradoxical that they strongly conflict with the standard notions, such as independent existence and unambiguous movement in space and time that so characterise the linear approach.

So whereas present M-Theory is defined for a 11 dimensional framework for physical activity down to the Planck length (setting thereby a limit to explicit physical investigations of matter), we can safely say therefore that the implied physical reality lying below the Planck length would exist with respect to an increasingly large number of dimensions.

Once again interpretation here is shrouded in deep paradox. Ultimately we have maintained that what defines the prime constituents of reality is that each unique "object" is associated likewise with a unique "dimension". However as we approach closer to recognition of the unique dimensional (qualitative) nature of each "object", the inter-relationships with other "objects" becomes so dynamic that we are unable to preserve any independent identity to these objects.

It may be instructive to explain a little more what is meant by multi-dimensional in this context.

As stated before the dimensions relate to directions with respect to the fundamental dimensions polarities of experience (internal/external and whole/part).

With linear interpretation (which is 1-dimensional in this perspective) external is separated from internal and then identified with whole/parts as independently existing objects.

2-dimensional interpretation is more refined where one accepts that a continual interaction as between internal and external polarities takes place leading to a consequent continual transformation in what is observed.

Then with 4-dimensional interpretation increasing interaction as between holistic and analytic notions of whole and part also take place.

Basically with progressively higher dimensions, one obtains an increasing ability to precisely control in experience the degree of emphasis in any context that is given to (internal) psychological interpretation or external physical observance from one perspective and from the other the precise emphasis on analytic (part) or (holistic) whole notions.

Now with higher-dimensional appreciation one realises that in phenomenal terms that we are always relating to mere appearances with respect to reality (which constantly change). Therefore one thereby surrenders any belief in precisely fixing the nature of such appearances. And with this one likewise surrenders the use of any explicit linear aspect with respect to interpretation.

So in similar terms at the higher dimensions below the Planck length (in physical terms) manifestations with respect to phenomena likewise change so quickly that it becomes impossible to preserve any fixed element. So relationships become increasingly paradoxical and circular in nature confounding all accepted scientific notions (with respect to independent existence and conventional notions of space and time).

So independent objective reality as we know it is associated with the need to continually reduce the qualitative nature of objects by likewise reducing the number of dimensions used to interpret reality. Indeed one could argue that the very need to still use four collective dimensions for identifying all macro objects sets a limit on the independent nature of these objects (as they still all require this same collective spacetime background).

In corresponding psychospiritual terms as one grows in awareness of the truly unique nature of objects (in what spiritually is referred to as the immanent aspect), ultimately again such objects become so transparent that they ultimately lose any independent characteristics. And at this stage interpretation of experience becomes so highly multi-dimensional that one can no longer preserve any explicit rational interpretation of such experience.

So properly understood we can never divorce the ultimate nature of physical reality from the corresponding means by which it is psychologically interpreted for both aspects are in truth complementary (and ultimately identical).

There is another fascinating point that can be made in this context.

Just as the original nature of matter is prime (where quantitative objective and dimensional qualitative aspects start from a state of total confusion), ultimate psychospiritual realisation in experience entails the same prime relationship (where however quantitative and qualitative are now maturely integrated).

So - quite literally - the very process of achieving such ultimate psychospiritual realisation requires the rooting out of all unreformed primitive instincts at an unconscious level.

Thus in the truest possible sense the resolution of the mystery of the prime numbers cannot be divorced from the realisation of ultimate spiritual union (where one approximates to removal of any residual conflict as between the quantitative and qualitative aspect of experience).

I Can Hear Music!

2011-06-13T04:30:00.000-07:00

I have often marvelled at the enormous significance of the numbers 1 and 0. Not alone are these sufficent in quantitative terms to provide a means for potentially encoding all information, but likewise in qualitative forms they provide the basis for holistically encoding all transformation processes.

I have even suggested in previous blogs that there is a strong case for suggesting that all reality is number through the dynamic interaction of both the quantitative and qualitative aspects of these digits (that ultimately represent both form and emptiness).

It is therefore tempting to extend this perspective to both the prime and natural numbers to see clearly how they are involved at a more detailed level of investigation in creating the reality that we share.

This is also intimately related to the problem of explaining how original emptiness (which is likewise the potential for all phenomenal form) gives rise to the continually changing universe that we inhabit.

So emptiness (that holistically is represented by 0) is also the potential for all subsequent form. And as the basis of identifying such form is based on recognition of a separate unit of existence, then in holistic terms this is represented by 1.

Thus the recognition of these fundamental numbers 0 and 1 are already inherent in experience as eternal archetypes.

The question then arises as to how emptiness gives rise to form. Though ultimately this process remains utterly mysterious, we can however suggest the means by which it phenomenally occurs.

This is where the prime numbers come into play. The very essence of a prime number is its uniqueness. In conventional quantitative terms this would be expressed by saying that prime numbers are linear in nature (with no factors) and therefore represent the basic building blocks of the natural number system.

Likewise from a physical perspective prime "objects" would represent the basic building blocks of all physical matter.

However what is generally missed regarding the nature of prime numbers is that they are also unique in representing dimensional numbers. Now the dimensions relate directly to the qualitative nature of matter. So the natural i.e. in which conventional spacetime is experienced represent the varied multi-interactions of prime dimensions.

So if we are to enquire regarding the most fundamental "objects" of nature we would have to say that they are prime. Now this in fairness is what the concept of "strings" attempts to get at by defining them in linear terms. However the problem again with the conventional approach is that it gives these "objects" an actual existence in quantitative terms, whereas properly they have a merely potential identity. Secondly such prime "objects" are directly paradoxical in that they have a complementary dimensional (qualitative) aspect that is - relatively - of a circular nature.

So remarkably at the most fundamental level each prime "object", which still has a merely potential existence, is equally through its very nature a prime "dimension".
One way of understanding this is through recognition that inherent in such "objects" are both quantitative and qualitative characteristics that are utterly unique.

Now once again in a limited sense, string theory attempts to address this issue by admitting that in some way the dimensions of space and time are embodied in the strings. However because of the merely quantitative emphasis of Conventional Science, such dimensions are reduced to the "objects" concerned, leading in effect to the unsatisfactory requirement of giving these "objects" a pre-existing dimensional location in space and time.

So - properly understood - the very essence of a prime number is that it equally combines both quantitative and qualitative characteristics that are paradoxical in terms of each other. Indeed this is the lesson of the Riemann Hypothesis which is really a statement regarding the requirement for reconciling these two aspects.

So remarkably, potentially preceding phenomenal creation, unique prime "objects" serve equally as unique prime "dimensions". Put another way this implies that each unique "object" potentially possesses both a quantitative and qualitative aspect (as dimension).

However as soon as prime "objects" attain a fleeting phenomenal identity, the uniqueness with respect to qualitative characteristics gradually gives way to a shared dimensional context. In other words through interaction with other prime "objects", gradually a more stable natural existence is achieved. So by the time we reach the Planck length, sufficient stability will have been reached to enable starting identification of particles to take place.

What is remarkable is that by this stage the number of dimensions, which is potentially unlimited for prime "objects", will have been reduced to a small finite number. So String Theory - or rather M-theory - is now conceived in 11 dimensions. And indeed my own investigations from the psychological perspective, places considerable emphasis on 8-dimensional reality.

So one way of looking at physical reality below the Planck length is with respect to activity that takes place in a much higher number of dimensions. However because such activity still largely relates to prime notions (with little composite organisation of a natural kind yet achieved) phenomenal reactions are too short-lived with respect to space and time to enable detection. Therefore though some actual activity does necessarily take place, it remains too elusive to be phenomenally detected in explicit terms. So reality still remains far closer here to the emptiness (from which it emerged) than actual phenomena.

It is also tempting to conclude that these original prime "objects" with the mere potential to exist are in fact the prime numbers (with respect to both their quantitative and qualitative characteristics).

Once again these prime numbers therefore represent eternal archetypes which then become inherent in fundamental processes (with respect to both their quantitative and qualitative characteristics).

So therefore, seen in this light, in the deepest sense all natural phenomena of form actually represent the interaction of prime numbers with respect to both their quantitative (object) and qualitative (dimensional) characteristics.

In this sense the music of the spheres is but the phenomenal resonance of the prime numbers whose ultimate secrets remain hidden in eternal emptiness.

Good Vibrations!

2011-06-12T11:34:00.000-07:00

In physics the Planck length is the order of magnitude of the vibrating strings that form the physical particles.

It is then postulated that it is not meaningful to probe below the Planck length. However it seems to me that this masks a considerable degree of philosophical confusion with respect to the very notion of strings.

For if it is not in principle possible to give physical meaning to distances less than the Planck length which is incredibly small at about 10^(- 20) the size of a proton, then how can strings themselves be given a physical meaning?

As I have stated before there are inherent philosophical difficulties with the manner in which physicists attempt to define strings.

Indeed from one valid perspective, what they are attempting to do is to achieve the reductionist fallacy of defining the ultimate constituents of the universe in merely quantitative terms (devoid of any qualitative meaning).

So from this perspective physical particles (which necessarily have qualitative characteristics) come into existence through the vibration of strings.
However such vibration could only have meaning in the context of a pre-existing spacetime environment for the strings. However these same spacetime dimensions are supposed to be in some way derived from the strings!

So really this attempted explanation of the nature of strings lacks any proper coherence.

I have long maintained that the notion of a string has a merely potential - rather than actual - significance. Secondly this notion has two complementary aspects which through interaction are linear and circular with respect to each other.

For convenience we can identify the quantitative aspect of the potential string with the linear aspect. Now this is what the conventional approach likewise defines. However it then makes the untenable reductionist assumption whereby the qualitative is reduced to quantitative interpretation so that the string is misleadingly defined in actual terms.

We can then identify the qualitative aspect of the potential string with the circular aspect. In mathematics is we raise 1 to a fractional power (dimension) its result will lie on the circle of unit radius in the complex plane.

In corresponding fashion to properly relate the linear quantitative aspect of the potential string with its - relatively - corresponding qualitative dimensional aspect, we must employ circular logic with respect to this latter aspect.

It is only through the dynamic interaction of both aspects of the string that physical particles can achieve a manifest actual physical existence (that exhibit both quantitative and qualitative characteristics).

Physicists maintain that the existence of such particles can only occur at or above the Planck length.

Now the very measurement of physical attributes in standard scientific terms assumes that linear type notions can be applied to the physical world e.g. where particles assume a certain independence and distances in space and time have an unambiguous positive meaning. We can then detect - literally - natural particles.

Natural numbers are made up of the primes and other composite numbers (derived from the primes).

In like manner natural phenomena are made up of prime and other composite varieties. It is therefore highly likely that what seem the basic natural particles already involve a high level of composition of more fundamental elements (below the Planck length).

Because the prime and natural are in dynamic terms interrelated they necessarily always co-exist together. However as we approach closer to the emptiness from which all matter emerges, the prime aspect becomes more pronounced, with any natural element arising, of an extremely fleeting short-lived nature.

So it is certainly possible that below the Planck length that a dynamic reality does exist that however in practical terms remains unobservable.
Such a reality would then exist overwhelmingly - not in linear spacetime - but rather in a paradoxical prime spacetime.

I have stated before that the very essence of a prime number is that it combines unique quantitative and qualitative aspects (that are linear and circular with respect to each other). So in extremes, this leads to the direct confusion of linear with circular notions. For example in linear terms a particle exists at some definite location. In circular terms it is however in an undefined location existing holistically everywhere. So prime spacetime involves the attempt to reconcile both notions (which are paradoxical in terms of each other).

We can perhaps portray what is involved here best with reference to psychological notions.

What is most prime i.e. primitive at this level entails the direct confusion of conscious (linear) with unconscious (circular) meaning. Thus the more primitive an instinctive impulse, as in earliest infancy, the more short-lived and fleeting it necessarily is in experience. Thus the attempt to embody what is holistic and universal directly in a limited phenomenal manner leads to the collapse of any dimensional context for the impulse (with the corresponding disappearance of the associated phenomenon).

In like manner at the most prime level of physical reality, the attempt to directly identify a dimensional context of space and time (circular) with a physical phenomenon (linear) leads to the immediate erosion of both aspects (i.e. spacetime and phenomenon).

So at a very low level of composition, prime reality would remain undetectable in phenomenal terms. However as levels of organisation increase, phenomena would attain sufficient stability to become detectable. So they would then emerge into a natural physical environment dictated by increasingly linear notions of space and time.

In this context the inability as yet to detect the graviton (the supposed transmitter of the gravitational force) is revealing.

I have likened before (in physical terms) the gravitational force to the role of the unconscious (from a psychological perspective).

This would entail therefore that any transmission of this force would relate to an extremely early stage of prime spacetime. This insight is also replicated in string theory by the prediction that the graviton corresponds to the lowest possible energy level of the string!

This could therefore imply that the phenomenal element associated with gravity is so minuscule that it will remain impossible to detect (in any physically discernible manner).

Put another way, this is a phenomenon related largely to circular (holistic) rather than linear (analytic) notions.

Moreover a considerable amount of activity is likely to take place below the Planck length governed mainly by circular rather than linear notions. So what we term the fundamental particles may largely represent - simply - what can be phenomenally detected (in linear terms) while concealing many layers of further physical organisation that take place in prime spacetime (below the Planck length).

As to the nature of the emptiness from which prime and natural physical actvity ultimately emerge "God Only Knows".

Strings Again!

2011-05-18T03:09:00.000-07:00

I have already written quite a lot regarding the conceptual difficulties surrounding String Theory.

As it stands I would say that from a qualitative perspective, it presently does not constitute a meaningful physical interpretation of the Universe.

However once some of its key ideas are properly decoded we can begin to appreciate what it really is all about.

Conventional Science is heavily based on the linear rational approach which is thereby 1-dimensional in qualitative terms. This attempts to give one unambiguous direction to understanding e.g. interpretation that is (merely) objective, space distances that are (merely) positive, propositions that are (merely) true etc.

Not surprisingly - in terms of this approach - the most fundamental constituent of matter i.e. the string is viewed likewise as 1-dimensional in quantitative terms.

Common sense notions of the nature of space and time likewise reflect the 1-dimensional qualitative approach. Here we view 3 dimensions as spatial (with direct quantitative characteristics) and the remaining dimension in qualitative terms separated as time.

Now from a quantitative perspective, String Theory has to admit the existence of extra dimensions. So with its later incarnation in M-theory, we now have the inclusion of 7 extra dimensions (giving 11 in all). However a direct correspondence with the 1-dimensional qualitative approach is maintained through treating all of these extra dimensions in spatial terms (maintaining just 1-dimension in time).

So the very interpretation of extra dimensions - which has no intuitive resonance with common-sense understanding of space and time - is itself a product of a reduced level of qualitative understanding (that remains 1-dimensional).

The clear corollary here is that once we concede the need for extra dimensions to physical reality (from a quantitative perspective) equally we need to concede the need for extra dimensions also in terms of qualitative interpretation of this reality.

So the real message of String Theory is that the present (1-dimensional) scientific paradigm is no longer adequate for interpretation of the fundamenntal nature of physical reality.

Bearing this in mind I have already proposed that the very notion of a string must be given two complementary interpretations (in quantitative and qualitative terms).

So from the quantitative perspective, the fundamental building block is the 1-dimensional string which then through various dynamic vibration patterns, produces the physical particles.

Equally from the qualitative perspective, the fundamental building block is the 1-dimensional (holistic) string, which likewise through dynamic vibration patterns produces various psychological interpretations of these particles. Properly understood as phenomena and dimensions are quantitative and qualitative with respect to each other, these implies that we cannot properly hope to interpret the dimensional aspect of strings without incorporating the qualitative element.

And for proper intuitive recognition of what in physical terms String Theory entails, higher dimensional qualitative interpretations of reality will be clearly required.

In my own work such higher dimensional interpretations are provided through Integral Science.

For some 20 years now I have maintained the key importance of 8-dimensional qualitative interpretation for a comprehensive holistic TOE. Now it is hardly a coincidence that this thereby implies 7 extra dimensions of interpretation (in terms of the conventional approach) while M-theory itself likewise postulates 7 extra space dimensions in quantitative terms.

However once again because - from a qualitative perspective - String Theory remains rigidly rooted in a 1-dimensional interpretation, its quantitative findings can establish little or no resonance with common sense intuitions regarding the world.

There is another extremely important (hidden) dimension to this new understanding that can ultimately make the (quantitative) findings of String Theory intelligible. This is, relative to the quantitative aspect (which is interpreted analytically in dualistic terms), the qualitative aspect is appreciated holistically in nondual terms (which indirectly can be given an imaginary rational basis).

Thus seen in this light, String Theory is actually pointing to the fact that at its most fundamental, physical - which is now inseparable from psychological - reality corresponds to the operation of a dynamic interactive binary system (combining both analytic and holistic elements). So incorporated in this system are the means of both encoding (and decoding) all information and transformation processes.

So coming back to the string it now is given a complex interpretation in qualitative terms (as both real and imaginary)

Thus in terms of this definition, if the real aspect is identified with the quantitative aspect, the imaginary - by contrast - is then identified with the qualitative. And of course both of these aspects keep switching through dynamic interaction with each other.

Apart from this interaction, physical reality in its most fundamental manifestations has no strict meaning.

Put another way at this level we cannot hope to separate physical data (in quantitative terms) from the qualitative means through which they are interpreted.

Thus the ultimate nature of reality, in the resolution of this interaction of both quantitative and qualitative, is utterly ineffable (both beyond and preceding phenomenal investigation).

String Theory then stands as an opening physical bridge in terms of connecting what is empty and ineffable with the world of phenomenal form.

Is Reality Just Number? (3)

2011-03-31T01:48:00.000-07:00

One issue that needs to be clarified here is the relationship as between number and other mathematical symbols. On the one hand it would be valid to maintain that the fundamental nature of reality (as phenomenally revealed) is mathematical. However I would also contend that it is more precisely related to number.

Basically I would maintain that number in fact implicitly requires other key mathematical symbols and relationships.

For example I have frequently identified 1 with the (straight) line and 0 with the circle. So there are intimate links here as between key numerical and geometrical notions.

Also the important operations of addition and subtraction are again implied by the most basic of numbers (1 and 0). The very recognition of form in any context implies 1. So for example if one distinctly recognises an object this implies the corresponding inherent recognition of 1. So without such recognition one would not be even able to differentiate objects in experience.

However equally implied by such recognition is the (conscious) positing of the object (which is the holistic meaning of +).

The awareness of emptiness - which in holistic terms is 0 - by contrast requires the ability to (unconsciously) negate recognition of distinct objects. So quite literally in both analytical and holistic terms 1 - 1 = 0.

Without this ability to dynamically negate - to a degree - distinct phenomena in experience, integration would not be possible.

So what we have here established is that from a holistic mathematical perspective, 1 and 0 are intimately related to both differentiation and integration respectively. And these two processes are fundamental with respect to all phenomena (in both physical and psychological terms).

The derivation of further numbers then arises from these basics.

If instead of the complementary relationship 1 - 1 we consider 1 + 1, we then generate 2 (as a new number) which in holistic terms defines all duality in experience. This also serves as the first prime number. Now we can generate all the natural numbers through considering 2 + 1, 3 + 1, etc.

However equally these natural numbers will serve as unique expressions of the product of prime numbers. So in this relationship as between prime numbers and natural numbers (and natural numbers and prime) the operations of multiplication and division are born.

Ultimately I would maintain that all the key mathematical operations, symbols relationships etc. are implied through the recognition of numbers (when considered in both analytic and holistic terms).

More correctly perhaps it should be said that they imply each other. So it is not strictly the case that the numbers come first in recognition but rather that they both simultanously arise (in implicit fashion).

So in this sense to say that all reality - at the phenomenal level - is number, is really a more emphatic way of maintaining that such reality is mathematical.

In a previous contribution I mentioned that the qualitative nature of phenomena arises through differing configurations of space and time (to which these are related).

Now once again these configurations are related directly to number (when interpreted in a holistic sense).

In this context 1 can be identified with linear type reason that serves as the basic for scientific understanding as conventionally understood (and by extension standard conceptions of space and time).

0 by contrast would be identified with pure intuition at the other extreme from reason. This would be especially relevant to a contemplative type vision of reality. Unfortunately it is not incorporated in current notions of science which therefore possess a strong - merely - quantitative bias.

However all other numbers (apart from 0 and 1) are associated with intermediate type interpretation where both reason and intuition are formally included. So when such numbers are holistically used as dimensions we generate an unending range of new configurations (real and imaginary) through which space and time are configured.

Once again, as I have repeatedly explained in these contributions, the appropriate holistic structure associated with a number (as dimension) is intimately related to its corresponding root form.

The two roots of 1 are + 1 and - 1. In corresponding complementary fashion 2 as holistic dimension combines both positive and negative polarities of form in a dynamically interactive manner (where linear and circular notions of understanding are involved). Applied to space and time this likewise implies that too must be conceived at this level of understanding in terms of complementary opposite polarities.

Ever more intricate configurations with both real and imaginary aspects are then associated with dimensions > 2.

The qualitative nature of phenomena then arises from the varied mixing of such dimensions giving unlimited complexity with respect to possible space and time configurations.

So we can see in this manner that the true nature of space and time has a profound holistic mathematical rationale (as befits its inherent qualitative nature).

Thus object phenomena that are initially conceived in a quantitative manner continually interact with space and time dimensions (that are qualitative in nature). So the uniqueness of the various phenomena we experience, coincides with the corresponding (holistic) uniqueness of the dimensions with which they interact.

Is Reality Just Number? (2)

2011-03-30T04:04:00.000-07:00

In the last contribution I suggested that - what we know as - reality at its most fundamental level represents a simple binary digital number system operating interactively in both an analytical (linear) and holistic (circular) manner.

The importance of this digital system in analytical terms is well recognised in IT technology where it has the capacity to successfully encode all information.

The corresponding importance of this same system in holistic terms - though not as yet recognised - is that it has the capacity to successfully decode all transformation systems i.e. as the dynamic interaction of both linear and circular logic.

However the most startling revelation comes from the incorporation of both systems whereby reality itself in all its diverse and complex intricacies (both physical and psychological) thereby represents both the encoding and decoding of the combined two systems.

So one way of looking at the nature of reality is as a dynamic computer programme that operates in a creative intelligent fashion.

However the implication of what is being expressed here is that at its most fundamental level reality has no discernible phenomenal basis.

This clearly has important implications for physics for the quest to unravel the secrets of nature (with reference to its phenomenal material characteristics) must ultimately be in vain.

As the great mystics of all ages clearly recognise, reality is utterly ineffable (in phenomenal terms). In other words it is of a pure spiritual nature (which is equally the ultimate nature of physical matter).

However at the next level which essentially acts as the bridge as between this underlying infinite (non-phenomenal) and the finite phenomenal realms, mathematical reality is born.

So ultimately mathematical meaning - when truly appreciated - is rooted in this great need to provide a connecting bridge - as it were - as between the finite and infinite realms.

Of course the value of mathematical symbols in understanding the finite side of this bridge has long been recognised.

Indeed in the quest to understand the ultimate nature of physical reality, string theory has now become so mathematical in nature that it has become increasingly difficult to give its findings a coherent physical explanation.

However the equal value of the same mathematical symbols - when understood in a qualitative holistic sense - for approaching close to infinite spiritual meaning has been all but lost in our culture.

So what we have with present science is a merely quantitative type approach (largely devoid of qualitative meaning).
One of the implications of this approach is that it has led to fundamentally distorted notions of space and time.

Properly understood the initial starting base for physical objects comes from the quantitative aspect of mathematical understanding.

However the correct starting base for the dimensions of space and time that they inhabit, should come from the qualitative (holistic) aspect of mathematical appreciation.

So our current notions of "default" common dimensions that are shared by all physical phenomena is the result of unbalanced thinking (i.e. where qualitative aspects are reduced to mere quantitative interpretation).

However correctly understood the great qualitative diversity of phenomena derives from the fact that they are defined with respect to relatively unique configurations of space and time (which yet also have shared common characteristics).

Without (qualitative) dimensions, phenomenal objects have no meaning. Likewise without (quantitative) phenomena, dimensions likewise have no meaning. So reality at all levels - physical and psychological - necessarily represents the interpenetration of merely potential quantitative and qualitative aspects (as considered separately) in the dynamic display of what represents actual phenomena.

One startling conclusion of all this is that in approaching ever closer to the quantitative origin of nature, we equally possess the power to give it differing qualitative configurations in space and time.

In other words as ultimately phenomena have no rigid tangible basis, we have potentially the power to literally create them as we wish.

At some future more advanced stage of evolution, far greater levels of spiritual realisation will provide the conditions for radical transformation of the qualitative dimensional nature of matter (thereby changing its phenomenal characteristics).

Our true identity is indeed God. So as we approach closer in realisation of this identity, the free creation of phenomena will thereby become possible.

And these new powers will have a corresponding scientific explanation in an enhanced paradigm that incorporates both quantitative and qualitative understanding (which at its deepest level is purely mathematical in nature).

Is Reality Just Number?

2011-03-28T01:49:00.000-07:00

It should be apparent in the first instance that number is essential in quantitative terms.

The very notion of quantity implies some form of numerical ordering! What is not however so obvious is the realisation that number is equally essential from a qualitative perspective where it plays a holistic role complementary to what is accepted in conventional (quantitative) terms.

To appreciate this important point we can initially confine ourselves to the two most fundamental numbers 1 and 0. As we know these two digits can be used - as in modern computers - to successfully encode all information (in quantitative terms). However what is not commonly appreciated is that the same two digits can be potentially used to encode all transformation processes when used with respect to their holistic qualitative meaning.

So, as I have repeatedly stated, 1 in this context relates to the linear use of logic in the analysis of form and 0 to its corresponding circular use (as an indirect expression of the holistic awareness of emptiness).

Some appreciation of this latter holistic use of number can be obtained with reference to the mystical traditions.

In Western tradition the linear notion of form tends to dominate understanding. Not surprisingly, ultimate spiritual meaning is generally expressed as union with reality, which relates directly to the holistic appreciation of 1. In the more intuitively based Eastern spiritual traditions, by contrast ultimate reality is often referred to as "a void" or "emptiness" (i.e. nothingness) which represents the corresponding holistic appreciation of 0.

So all information and transformation processes can ultimately be encoded in terms of number. The next giant leap is then to appreciate that the phenomenal characteristics of reality that we observe are of a secondary nature representing dynamic number configurations (that combine both quantitative and qualitative characteristics).

Therefore at the most fundamental level of appreciation we cannot hope to interpret reality with respect to these mere phenomenal characteristics. This has important implications for physics as it implies that ultimate explanations cannot be obtained from the behaviour of phenomena but rather from the mathematical structure inherent in such phenomena. And this behaviour must combine both conventional analytic and the - as yet - (unrecognised) holistic aspects of interpretation.

Stating it again directly, reality as phenomenally observed represents the dynamic configuration of numbers with respect to both their quantitative and qualitative characteristics. And at the most general level this can be understood merely in terms of 1 and 0 (which is sufficient to provide a basic encoding for all information and transformation processes).

Parallel Wonders

2011-03-25T05:27:00.000-07:00

Following on my last contribution it would be instructive to demonstrate futher important points regarding the "true" nature of physical reality (through corresponding reference with the complementary psychological connection).

In physical terms we have (partial) physical objects that dynamically exist through interaction with a (holistic) dimensional framework of space and time. In corresponding psychological terms we have (specific) perceptions that dynamically interact with (universal) conceptual classes. So for example a specific electron perception necessarily interacts with the universal conceptual class of electron. In other words an electron as quantitatively perceived thereby must relate to the qualitative concept of "electron". The psychological dimensional framework of each electron in space and time is provided through its conceptual appreciation. However as we can have many different types of object perceptions, likewise we can have many corresponding dimensional concepts.

For example the concept of a proton or neutron is qualitatively distinct from that of an electron. The clear implication therefore is that in a very important sense every distinct object phenomenon relates to a unique dimensional form of space and time.

So with respect to physical reality, rather than one space and time reality (to which all objects collectively exist) we have in truth an unending series of uniquely distinctive dimensional realities in continual interaction with each other. The reason why we do not recognise this important fact is again directly due to the limited nature of the scientific paradigm we employ. The rational paradigm is solely geared to quantitative type appreciation.

Therefore it can only deal with the qualitative dimensional aspect through a process of reductionism where all distinct objects are assumed to belong to the same standard collective framework of space and time.

However once we explicitly allow for qualitative type consideration through indirect incorporation of circular (intuitive) type appreciation, then all objects have both quantitative and qualitative aspects.

In complementary fashion all dimensions have both quantitative and qualitative aspects. So correctly understood in qualitative terms corresponding to each unique object class is a unique dimensional configuration of space and time which continually interacts in quantitative fashion with one collective quantitative interpretation. As we have seen, correctly understood gravity has both physical and psychological aspects that are complementary.

In physical terms, gravity is increased through the erosion of the "empty" physical dimensions that inhabit the very structure of matter. In complementary psychological terms, gravity is increased through erosion of the material physical phenomena relating to dimensions. So for example a deepening conceptual understanding requires the consequent erosion of attachment to specific phenomena of a perceptual nature. Not surprisingly a person who is deeply rooted in holistic concepts is likely to develop a more "grave" or serious personality (as the surface light associated with superficial phenomena is eroded).

As I have commented on before, in physical terms the degree of gravity is associated with the corresponding extent of physical mass entailed. In complementary psychological terms the degree of gravity is associated with the corresponding loss of matter entailed. So for example with the death of a loved one a profound loss - of what in literal terms greatly matters - is involved leading to the customary experience of grief (directly representing psychologically gravity).

And such grief can lead to a curving of psychological space and time. Due to the unconscious development that such an experience entails, a marked transformation in one's customary (conscious) perceptions and concepts can take place thereby altering experience of space-time phenomena. Finally it is quite clear from the psychological perspective that matter and energy on the one hand and dimensions and gravity on the other bear a direct complementary relationship to each other.

Thus deepening psychological gravity at an unseen unconscious level is later the very source of expansive spiritual light (in visible conscious terms). So in complentary physical fashion this also necessarily applies. Therefore deepening gravity (as for example in black holes) is later the very source of the expansive physical light than extends the phenomenal universe.

In fact in holistic mathematical terms, we have the beautiful relationship that gravity and (electromagnetic) energy are real and imaginary with respect to each other. So "real" gravity is "imaginary" energy; likewise "real" energy is "imaginary" gravity. And this relationship applies equally in both physical and psychological terms. Indeed its very recognition requires that both physical and psychological aspects of reality be viewed in a complementary manner.

The Wonders of the Universe

2011-03-23T04:15:00.000-07:00

It seems that the BBC has now found their answer to Stephen Hawking through their own series “The Wonders of the Universe” hosted by Professor Brian Cox. Now Brian Cox, a former member of the 90’s pop band D-ream is something of a wonder himself as he looks far too young to be a professor. However there is no doubting his communication skills and his undoubted knowledge and love for his subject.

Indeed I found his programme demonstrating how elements here on Earth have emerged from the high temperature processes within stars compelling viewing as it clearly showed how we are all truly children of the Universe (and indeed of the original Big Bang which is believed to have started it all)!

The following programme "Falling" on gravity proved of equal interest as he showed how this mysterious force is responsible for the structure of all phenomenal forms in the Universe.
Initially he confined himself - which accepting that it was not fully satisfactory - to Newton’s conception of gravity. However towards the end of the programme he showed how it failed to precisely predict the orbit of Mercury around the Sun and how Einstein’s new theory of Gravity in “The General Theory” perfectly corrected this problem.

However though Einstein’s Theory has been brilliantly successful in its own right he admitted at the end of the programme that even this was not a complete theory.

I would agree with this statement. However the true implications are much more profound than what most physicists would be willing to accept.

To properly understand the role of gravity we need to look on physical in a manner complementary to psychological reality.

In psychological terms it would be readily accepted that our experience is based on the inevitable interaction of both conscious and unconscious aspects. In direct terms the conscious aspect enables us to differentiate phenomena analytically as parts. In corresponding fashion the unconscious enables the integration of such phenomena in a holistic unseen fashion.
Now from a scientific perspective, rational understanding corresponds to this conscious aspect of experience whereas – in direct terms – the holistic aspect is provided through intuition.

However the big problem with present scientific understanding is that is formally based on mere rational modes of enquiry. Therefore it inevitably reduces - in any context - the holistic intuitive aspect to rational interpretation.

As I have stated, properly understood the physical view of the universe corresponds in complementary fashion to its psychological means of interpretation. Therefore when we allow for both conscious and unconscious modes of experience this entails that in physical terms we allow for a physical aspect (that is phenomenally visible) and a holistic unseen aspect (that is empty of form).

So then correctly understood in dynamic interactive terms what we term the physical universe represents (visible) phenomena of form in ceaseless interaction with an invisible holistic ground.
This then directly corresponds with a rational (conscious) mode of interpretation with respect to visible phenomena and an intuitive (unconscious) mode with respect to the underlying invisible ground of reality.

Now of course when we adopt a reduced mode of understanding as with Conventional Science we attempt to view physical reality as merely visible phenomena corresponding to a rational mode of interpretation.

The key reason I would suggest as to why the gravitational force has proven so elusive and mysterious is precisely because of the inadequate mode of present scientific enquiry.
In fact correctly understood, gravity plays a complementary role in the physical universe to the unconscious in psychological understanding. Therefore we cannot properly appreciate its nature through the standard paradigm of scientific understanding that altogether ignores in formal terms the role of the unconscious.

In physical terms we have material phenomena which are in some sense visible always exist in a dimensional context of space and time (that serves as an invisible background). Now of course space and time indirectly (though this dynamic relationship with matter) can have phenomenal aspects (such as distance).

However correctly understood, phenomenal objects in space time represent the continual interaction of what are specifically visible in the context of a holistic environment (that is invisible). In other words pure space and time (if we could conceive of their nature) would be fully empty of phenomena. And without matter, such dimensions could not therefore exist. So such emptiness relates to an underlying present moment that continually exists.

Now gravity arises from this interaction of (specific) phenomena of form with its (holistic) dimensional background of space and time.

In normal circumstances gravity is an extremely weak force. For example the gravitational pull of the entire Earth would not be enough to stop one easily lifting an object such as a stone from the ground!

This weakness of gravity reflects the fact that in practice what we consider as dense matter is almost entirely made up of “empty” space.

This weakness of gravity led Newton for example to consider the phenomena of the Universe as existing in flat space and time (as non-interacting containers of all objects that exist). This in turn is consistent with linear notions of space and time.

Now there is a complementary parallel to this view as given by the standard scientific paradigm (based on mere rational interpretation). Here unconscious interference is so weak that the physical universe can seemingly be understood in a merely conscious manner (based on the linear use of logic).

So Newton’s worldview in physical terms is properly consistent with linear rational interpretation (which despite many recent paradoxical findings remains in effect the undisputed paradigm for science).

I have always found Einstein’s contributions fascinating as their true implications have only been partially recognised. The great paradox with Einstein is that his contributions overthrew the Newtonian world view of absolute space and time. However Einstein still remained a steadfast believer in the classical scientific paradigm (that is properly consistent with Newton’s views).
In his General Relativity Einstein showed that gravity can exert a considerable force where extremely large mass is concerned. Then it would act to “bend” in some measure space-time. In particular with black holes where “empty” space would get squeezed out entirely from matter, space-time would then become completely curved.

However what is not at all properly realised is that this inevitable curvature of apace and time (in the context of strong gravity) equally applies to the qualitative manner in which we seek to interpret physical reality.

Indeed a fascinating correspondent exists on the psychological side to the notion of a “black hole” as exemplified most starkly in the writings of the great Spanish mystic St. John of the Cross.
One can view authentic contemplative life as starting from the standard linear conscious worldview. However, as the influence of a developing spiritual intuition (based on the unconscious) becomes ever stronger, a dramatic change takes place in one’s customary understanding. So St. John dramatically portrays the most intense dark night where understanding becomes completely curved (i.e. purely intuitive). This happens through a process of detachment from all conscious understanding of phenomena (which literally happens through a squeezing out of space and time). Not surprisingly one feels at such time as if in a dark and narrow dungeon where one is scarcely able to breath (due to the suffocating congestion). And the reason why this experience is a “dark night” is precisely because conscious light - based on linear notions of understanding - is not able to escape during such a transformation.

So with pure contemplative spiritual awareness, space and time become completely curved representing an extreme in terms of unconscious type intuitive development. Likewise as we have seen linear assumptions with respect to "flat" space and time represent a corresponding extreme with respect to conscious rational understanding.

The clear implication of all this is that a comprehensive paradigm for science necessarily must combine both conscious and unconscious through the explicit integration of both (rational) linear and (intuitive) circular modes of interpretation.

Though the nature of gravity necessarily remains mysterious, its true nature in physical terms relates to an inevitable dynamic interaction as between (visible) phenomena of form and a holistic (invisible) dimensional background of space and time.

A true understanding of gravity therefore equally requires from a psychological perspective a complementary dynamic interaction as between rational (conscious) and intuitive (unconscious) aspects of understanding.

So gravity is a force which has complementary physical and psychological interpretations.
This is even evident in conventional understanding. Gravity of personality for example reflects an experience of the world that is rooted in a deeper holistic structure of meaning. By contrast a superficial or “light” personality operates at a very low level of psychological gravity through relationship with mere conscious appearances.

New Vision of Physics

2011-01-12T03:08:00.000-08:00

I have just been completing four articles that seek to outline a new vision of physics (which I believe is greatly needed at the present time).

For very good reasons, I do not share the present optimism regarding physics, which seemingly assumes that at last we are on the threshold of unveiling a Theory of Everything .

Rather than finding this illusionary TOE, we are about to painfully discover how limited in truth is the present scientific framework!

So as I would see it comprehensive scientific understanding will require at least three great phases.
What we are witnessing therefore at present is but the peaking of the first great phase that is geared merely to the specialised quantitative interpretation of physical reality.

However properly understood science possesses an equally important holistic qualitative aspect (which I refer to as Integral Science) . However this aspect has remained for all practical purposes entirely undeveloped. In fact most scientists would have great difficulty in accepting its potential relevance!

So it will be a very long time before this hidden qualitative aspect gradually gains acceptance among the scientific community. And this will only arise as scientists are gradually forced to face up to the limitations of the present quantitative approach.

When this qualitative aspect (i.e. Integral Science) achieves appropriate specialisation (which itself will require a considerable transformation in the manner we experience reality) can we then hope to emerge on the third comprehensive phase of scientific understanding (in what I term Radial Science).

So Radial Science entails the dynamic interaction of both specialised quantitative and qualitative type appreciation.

However, we are still a long way from attaining such comprehensive understanding.

In the first article I outline the key general features of this qualitative integral aspect of science (contrasting them with the corresponding features of the accepted quantitative approach).

This integral appreciation in no way represents merely an extension of present type conventional understanding. Rather it requires a fundamental transformation in the very nature of consciousness so that it becomes inspired with an authentic contemplative vision. This does not exclude the need for precise rational interpretation - indeed it increases such a need - but without this underlying holistic vision (based on genuine spiritual insight), the qualitative approach cannot prosper.

In the second article I apply this new integral type appreciation to some of the Einstein's key findings with respect to Relativity (both Special and General).

One key finding is that the present understanding of the relative nature of space and time (as promoted by Einstein) is somewhat limited and thereby incomplete. In other ways relativity - in this context - applies not merely to the quantitative measurement of space and time but equally to the qualitative manner by which such notions are interpreted.

So in qualitative terms there is not just one acceptable manner of interpreting reality (which for Einstein was the classical paradigm) but potentially an infinite number (of ever more refined interpretations).
Thus when we properly allow for this qualitative aspect we can see that an inevitable uncertainty principle likewise applies at the macro universal (as well as the quantum micro) level of reality.

Putting it more generally this thereby establishes that we cannot hope to distinguish the "objective" (quantitative) nature of physical reality from the psychological (qualitative) means through which it is interpreted.

And ultimately these two aspects can only be reconciled in the pure mystical experience of oneness whereby physical creation momentarily realises its own inherent empty nature (that both precedes and transcends all phenomenal attributes).

The third article then deals with with the qualitative interpretation of quantum reality indicating that such behaviour - when understood in the appropriate manner - is fully in correspondence with its corresponding manner of interpretation. This once again establishes that we cannot hope to divorce the manifest "objective" (quantitative) nature of reality from the "subjective" (qualitative) means by which it is interpreted.

Also by showing in parallel fashion how the Uncertainty Principle equally applies at both the quantum (micro) and relativistic (macro) levels of reality through incorporation of both quantitative and qualitative aspects of science, it indicates clearly the means through which both of these areas can be reconciled.

The final article - which is the most ambitious - indicates how present string theory concepts are hopelessly inadequate (from any acceptable qualitative philosophical perspective).

Unravelling this problem then requires a corresponding holistic qualitative counterpart for all existing string concepts.

So a properly formulated String Theory (where concepts are given an acceptable intuitive explanation conforming with actual experience of reality) requires that concepts be given be both parallel physical and psychological interpretations.

This clearly indicates demonstrates that present hopes in a mere quantitative type TOE are unfounded.

Indeed the message of this reformulated String Theory once again points clearly to the fact that we cannot hope to ultimately divorce quantitative understanding of physical reality from the qualitative means through which it is interpreted.

Therefore from this more refined standpoint, the ultimate issue in understanding does not relate to the nature of what is known but more subtly to the relationship as between the (psychological) knower and what can be (physically) known.

And this relationship cannot be resolved in a phenomenal manner!

So the starting basis for Radial Physics is that all phenomenal explanations of reality have a merely provisional validity. However a high level of importance still attaches to the precise nature of such provisional explanations (that are deemed appropriate in any context).

Thus such scientific interpretation - viewed in a flexible interactive manner - can serve as a very important catalyst for an underlying contemplative vision (seen as both the initial source and ultimate goal of all such interpretation).

Fermat's Last Theorem Revisited

2010-10-30T12:16:00.000-07:00

Looking again at the Horizon TV programme on Fermat’s Last Theorem proved a very rewarding experience. Unlike the first time I was able to appreciate much more of the fine detail (e.g. with respect to elliptical curves and modular functions). Also it got me thinking again on a number of levels regarding my own mathematical journey.

Like Wiles as a child of about 10, I too had heard of Fermat’s Last Theorem. The problem seemed so beguilingly simple that in my naïveté I thought I would be able to solve it. However after many hours of futile endeavour I abandoned this quest in failure. Nevertheless as Mathematics remained my favourite pursuit I hoped to major in the subject at College. However after a troublesome first year when I became greatly disillusioned with the mathematical treatment of the infinite, I dropped out of the class.

Many years later I became interested again in Fermat’s Last Theorem from a very different context. I had been giving a great deal of attention to my new pursuit of Holistic Mathematics (where every mathematical symbol can be given a well-defined qualitative as well as quantitative meaning). and was looking for a problem to demonstrate its potential value. So I hit upon “The Pythagorean Dilemma” relating to the irrational nature of the square root of 2. Then to my considerable satisfaction, I felt that I was able to provide a coherent qualitative solution to this problem.

Now the Pythagoreans would not have been satisfied with a merely quantitative explanations as to why the square root of 2 is irrational. They really wanted a deeper philosophical explanation as to why in qualitative terms such a number can arise!

The implication is that there are really two as aspects to proof (i) (quantitative) analytic and (ii) (qualitative) holistic.
Comprehensive mathematical interpretation then requires both types of proof.

The problem with respect to the square root of 2 arises in the context of the famous Pythagorean triangle so 1^2 + 1^2 does not result in another rational number that is squared. Now as Fermat’s Last Theorem is closely connected with this problem, I decided to look at again (strictly from this new holistic perspective) and came up what I would see as a partial qualitative explanation for its truth.

Central to holistic mathematical appreciation is the notion that mathematical dimensions have a coherent qualitative interpretation whose structure is inversely related to the quantitative root form of those dimensions. And to see this structure we obtain the successive roots of 1.

For example 2-dimensional interpretation is linked to the two roots of 1 which are + 1 and -1 respectively. Thus qualitative 2-dimensional interpretation is based on the complementarity of opposite poles in understanding (such as external and internal).

Now the significance of all dimensions higher than 2 is that the corresponding root structures will contain both real and imaginary parts.

This means in effect that mathematical interpretation at these dimensions entails both real (conscious) and imaginary (unconscious) aspects with the imaginary expressing the unconscious aspect in an indirect rational manner!

Real understanding in qualitative terms relates to strictly rational type understanding (corresponding to rational numbers in quantitative terms).

Thus in Conventional Mathematics, when irrational numbers are used they are given a strictly reduced interpretation.
However imaginary understanding corresponds to the rational means of conveying true holistic meaning (of a circular kind).

So the qualitative explanation for Fermat’s Last Theorem relates to the basic fact that 3-dimensional interpretation and higher can never be conducted in a merely real rational manner. Likewise in quantitative terms, when we attempt to add two quantities - which represents the simplest form of linear transformation - raised to such dimensional powers, the result likewise cannot be expressed in merely real rational terms.

We can even give a simple geometrical rationale for this.

Clearly from a linear (1-dimensional) perspective, when we add two rational numbers the result will also be rational. The very essence of the linear interpretation is that qualitative considerations are ignored. Therefore no qualitative transformation in the numbers can take place.

Now 2-dimensional interpretation is a half-way house as between linear and circular which can be easily illustrated. Here the two roots of 1 lie on both a straight line diameter and also on its circular circumference. Therefore in quantitative terms when we add two numbers (raised to the power of 2) the answer can be either linear yielding another rational number (raised to the power of 2) or circular (i.e. an irrational number raised to the power of 2).

However for all dimensions greater than 2 the corresponding roots (of 1) cannot lie on a straight line. So both real and imaginary aspects are necessarily involved. The corresponding corollary in quantitative terms is that when we add two rational numbers (raised to such dimensions) then an inevitable qualitative transformation in the nature of the number is involved. So the resulting number (raised to the power of n > 2) must be irrational.

I think it is even possible that Fermat’s “truly marvellous demonstration” of this fact might have related to this simple insight (i.e. that all root structures greater than 2 necessarily entail imaginary as well as real components). Now a demonstration does not amount to a proof and perhaps Fermat subsequently realised how difficult it was to build on such an insight to establish a proof!

Returning to the programme on Wiles, one striking paradox that hit me was how much the very process of his discovery runs counter to the established mathematical notion of rational proof.

So for example Wiles through his early discovery of Fermat’s Last Theorem was inspired by a powerful childhood dream i.e. one day to find the proof to this great puzzle. So his initial motivation relates more to the holistic unconscious (than rational thought). Also his subsequent voyage of discovery in many ways paralleled that of a spiritual contemplative seeking union with God.

Indeed with his thin frame and quite demeanour he very much fitted the part of the religious ascetic. He then pursued a very uncertain journey in considerable isolation for seven years gaining total immersion in his problem. For much of the time he wandered in darkness, hanging on in faith and hope of an eventual resolution.
Then finally after long and painful endeavour he received a special moment of illumination in that Eureka moment when he finally resolved his problem. So great was his emotion at this final revelation that he could not even attempt to describe the feeling but only to say that he would never experience a moment like it again!

Wiles proof is rightly hailed as a truly remarkable mathematical achievement. However the point that I am making is that the actual experience of discovering such a proof entails much more than what is formally recognised.

So Mathematics - especially with truly creative discoveries - entails both intuitive (unconscious) as well as rational (conscious) processes. In particular Wiles’ decisive final insight was of a (holistic) intuitive nature. However this important holistic aspect is screened out totally from formal mathematical interpretation!

Though properly speaking there are two aspects to Mathematics that are quantitative and qualitative, only one is recognised. Thus the interpretations of Conventional Mathematics are of a highly reduced nature (and are only strictly valid within the 1-dimensional mode of qualitative interpretation adopted).

This intimately applies to the nature of mathematical proof.

It is only within a linear (1-dimensional) interpretation that mathematical proof can be given an absolute meaning. So from this perspective Fermat’s Last Theorem is either true or not true. And the popular belief is that Wiles has now finally resolved the matter for once and for all by proving that it is indeed true!

However the actual process by which the validity of his proof was decided shows that such absolute interpretation is not strictly valid.

Indeed when Wiles first presented his “proof” in 1993, it was widely accepted by the mathematical community. It was only later that a referee found a flaw in reasoning at an important juncture. Even when this was pointed out to Wiles, it took him some time for him to recognise the true importance of the difficulty. So more than a year of further investigation was required (with the help of a talented student) before Wiles was finally able to amend his proof.

So in truth “mathematical proof” in such cases – indeed in all cases - represents but a special form of social consensus among the mathematical community and is strictly of a probable nature. In other words as time goes by with no other questions being raised regarding its validity it can be accepted with an ever greater degree of confidence as true. But this truth will still always remain of a merely probable nature.

There is an even bigger challenge with the accepted notion of proof that I am here raising. This relates to the fact that with the passage of time, our very understanding of the nature of “proof” is likely to become considerably more refined.

So when properly understood, each dimensional number in qualitative terms corresponds with a unique mode of qualitative interpretation with respect to mathematical symbols.

So there is not just one valid mode - as presently believed - of acceptable mathematical interpretation (but rather potentially an unlimited set).

This understanding can then be applied directly to the status of the proof of Fermat’s Last Theorem. Just as it is not possible to add two rational numbers (raised to the dimensional power of 3 or higher) together to obtain another rational number (raised to the same power), equally it is not possible to maintain a strictly linear interpretation with respect to Fermat’s Last Theorem when understanding takes place in 3 (or higher) dimensional terms.

In other words - in terms of the qualitative understanding of such dimensions - interpretation is subject to the Uncertainty Principle. What this means in effect is that Fermat’s Last Theorem would now be given both a conventional (quantitative) and holistic (qualitative) interpretation. Thus inevitably there would be a trade-off necessary with respect to both types of appreciation. Therefore the more definite the merely quantitative, the more fuzzy would remain the corresponding holistic dimension. Equally the more definite the holistic, the more fuzzy would remain the quantitative aspect.

Now it might be maintained that my own appreciation with respect to the Wiles’ quantitative proof of Fermat’s Last Theorem is still fuzzy; however while accepting this observation, it only helps to confirm the general point I am making with respect to the true nature of mathematical proof in the wider context of interpretation (where both quantitative and qualitative aspects are formally incorporated).

Happily, Fermat’s Last Theorem is at least capable of proof (in the 1-dimensional sense of current mathematical interpretation). However I believe there are other outstanding problems (such as Riemann’s Hypothesis) where even this type of proof (or disproof) will not be possible.

The Big Bang

2010-10-28T07:51:00.000-07:00

It is indeed a pleasure to watch so many beautifully produced programmes highlighting the wonders of the universe. Recently I was viewing "Stephen Hawking's Universe" on Channel 4 and found it fascinating (especially the last episode on the origins of creation).

The Big Bang about 13.7 million years ago has now become so commonly accepted as if it is an established scientific reality not to be questioned.

However I always like to take a wider perspective than mere conventional acceptance of present views. Just look at how our worldview has changed so much from even 100 years ago! Is it not reasonable to assume that perhaps even greater changes will take place in the next 100 years making much of what is presently gospel truth seem naive and even foolish!
So even if the Big Bang remains the accepted orthodoxy, I am sure that the manner in which it is understood will have changed considerably.

Indeed it seems to me somewhat ridiculous to attempt to describe in detail what happened during the first nanoseconds of the Big Bang for the simple reason that the linear notions of space and time on which this is all premised clearly could not have existed within the context of the Big Bang itself.

Our very notions of space and time (including the more relativistic notions of Einstein) are always premised on the clear separation of observer from what is observed. Therefore the very registering of space and time requires that the (internal) observer in some way be detached from the outside system (that is observed).

Now clearly in the context of the Big Bang such conditions would not exist, for the evolutionary potential which it contained for eventual emergence of psychic observers (of a physical universe) would have remained indistinguishable from the physical.

Put another way in attempting to travel back in space and time to the supposed beginning of the physical universe we are likewise - though not always realising this fact - attempting to travel back to the beginning of the psychological universe (for clearly our wonderful capacity for conscious investigation ultimately emerged from this initial state).

However once again we can only give a clear meaning to space and time (through detachment of the psychological from the physical aspect) and in the context of what we call the Big Bang this would be clearly impossible. So in attempting to give a linear history in space and time to our universe we are attempting to act as outside observers of an initial event (which clearly is not tenable in the context of that event itself).

Now I would accept that the birth of space and time coincides with the birth of the phenomenal universe. However in this emergent state circular - rather than linear - notions would be more appropriate. Thus any notions of forward movement for the universe would have to be countered by corresponding notions of backward movement. So the best we could say therefore is that the universe emerged out of the present moment which is continually renewed. Furthermore properly understood the very notions of space and time that we use are ultimately purely relative with respect to the present moment. So once again though space and time measurements have a certain validity when we abstract one part of the system from its environment, in the context of the overall universal system they are rendered ultimately paradoxical and meaningless!

Indeed this raises an even more fundamental problem in that the very way we scientifically attempt to interpret the universe is deeply flawed.

This is due to the failure to properly distinguish wholes and parts. Because science is based on mere quantitative type analysis of reality it thereby reduces the qualitative dimension (which is of distinct holistic nature) to the quantitative.

This thereby leads to the notion of one universe composed of many constituent parts.

However when we properly allow for the qualitative dimension, interpretation is much more subtle. Here we view the universe as the intersection of the one and the many.

In other words each part contains the whole (as an individual universe); equally the parts are contained in the whole (as the collective universe).

So what we refer to statically as the universe is in fact a dynamic interaction of many (micro) universes with the one (macro) and equally the reverse interaction of the one (macro) with the many (micro) universes!

And as I have stated on many times to talk about this in correct scientific terms we must incorporate the true holistic meaning of what is real and imaginary (in mathematical terms). Thus from one perspective we have the interaction of many real (individual) universes with the one (collective) universe that is - relatively - imaginary; equally from the other perspective we have the interaction of the one (collective) universe that is now viewed as real with the many (individual) universes that are - relatively - imaginary. Thus the one and the many have both a real and imaginary identity that keep switching through dynamic interaction.

I would also see the very notion of the Big Bang as deeply problematic for the concept of strings.

From the accepted linear viewpoint of the Big Bang, strings could not have existed before this event. Therefore once we accept this we are left with the problem of how they have emerged. And if science attempts to give a phenomenal explanation to this, then clearly it is pointing to something that is even more fundamental than strings.

One alternative would be to accept that strings are created - literally - out of nothing. However scientists would be loath to accept such an explanation which would be inseparable from saying "God created the world".

Of course the best solution - as I have suggested - is to abandon the very notion of strings as having any phenomenal identity. As I would explain it, strings possess merely the inherent potential for both the linear (independent) and circular (interdependent) aspects which necessarily underpin all phenomenal existence.

However properly incorporating such an explanation ultimately requires modifying the very nature of science.

In other words science does not entail mere rational interpretation of reality (at a conscious level); rather it entails the dynamic interaction of both (analytic) rational and (holistic) intuitive processes that operate in both conscious and unconscious terms.

So correctly understood all phenomena (as actually manifest) are in continual dynamic interaction with a fundamental holistic ground (as potential for existence).

So ultimately there is no scientific hope of appreciating the mystery of the origins of our universe without formally incorporating the role of the unconscious in interpretation.

More on Nature of Strings

2010-09-16T03:51:00.000-07:00

I have often stated that the current definition of "a string" is somewhat meaningless from any coherent physical perspective.

A string is viewed essentially as like a thin elastic band - extremely short in length - that is 1-dimensional (with no other spatial characteristic).

However this very definition requires the background existence of space. However it is then also admitted that the dimensions of space and time must in some way arise from strings (as the basic constituent ingredients of everything in the universe). So we clearly have an obvious problem.

Indeed there is a strong parallel here with a similar issue in mathematics relating to the prime numbers.

The prime numbers are conventionally viewed as the basic (independent) building blocks of the natural number system; however equally the general distribution of the primes intimately depends on the natural numbers.

Now I have explained that this issues of the primes in Mathematics ultimately relates to the fact that they can be given both a linear and circular interpretation which are relative.

So it is akin to the simple problem of defining left and right turns with respect to street directions. If for example one defines the direction of a road as either "up" or "down" (as independent) then left and right turns can be given an unambiguous meaning.
However when we treat them as interdependent with a relative meaning, then a turn on a road is both "left" and "right" (with the designation in any given circumstance purely arbitrary).

So with prime numbers when we try to treat the individual existence of prime numbers and the general distribution (overall) of prime numbers as independent, then both issues can be unambiguously studied through conventional mathematical methods.

However when we attempt to properly integrate these two areas, we require both a linear (quantitative) and circular (qualitative) treatment.

And indeed this is the very conclusion I reached with the Riemann Hypothesis which arises through attempting such integration in mere quantitative terms.

And of course it is exactly the same situation with respect to the world of strings.

So "a string" cannot be given an independent existence which already assumes a general background in space; likewise the dimensions of space and time cannot be given an independent existence as they only have meaning with respect to already existing phenomena.

So "a string" strictly has no phenomenal meaning as an independent physical constituent of matter; likewise "a string" has no phenomenal meaning with respect to the dimensions applicable to such reality.

It is only in relation to each other (where both the quantitative and qualitative aspects of matter dynamically interact) that phenomenal investigation of reality can begin.

And again the crucial point to realise is that this necessarily entails both quantitative and qualitative aspects of understanding (pertaining to both linear and circular interpretation of relationships).

Physical science as we know it is still based on a fundamentally false premise i.e. that phenomenal investigation of "specific objects" and corresponding investigation of the general dimensional background of such phenomena can both be interpreted in merely quantitative terms (using the standard rational linear approach).

However as we have seen object phenomena and the dimensions of space and time are in dynamic interactive terms quantitative and qualitative with respect to each other.

So therefore we must use a binary scientific approach combining both linear and circular type appreciation. Whereas the linear aspect corresponds to reason (as conventionally understood), the circular aspect corresponds directly to intuitive type appreciation.
However indirectly this intuitive aspect can be translated in a scientific rational fashion as "imaginary".

Thus a comprehensive scientific approach that can properly deal with the relationship of phenomena to space and time must be in qualitative terms of a complex rational nature (combining both real and imaginary aspects).

Properly speaking a string does not have an actual phenomenal existence (which would already imply space and time). Rather it represents the potential for manifest existence which has the two aspects of (differentiated) independence and (integral) interdependence.

So in a static sense, no distinction can be made as between the linear aspect of the string as (linear) differentiated form and the integral aspect as (circular) emptiness.
Thus in static binary terms therefore with respect to the string, the linear (1) cannot be properly distinguished from the circular aspect as nothingness (0).

However when these two aspects interact in dynamic manner (i.e. through the vibrations of the string) then we enter the world of actual manifest form. The most primitive excitations of the string would correspond in turn with the prime numbers (which in turn would represent the most fundamental particles).

And because at the lower energy levels very few particles become manifest which then serve as the basis for the natural phenomena we recognise, in like manner the majority of natural numbers arise as the composite expression of comparatively few early prime numbers!

So just as there are links in natural terms as between the original numbers (1 and 0), the primes and the natural numbers, there are similar links as between strings (defined in a binary fashion), the fundamental sub-atomic particles and natural phenomena.

However whereas both sub-atomic and natural phenomena enjoy a dynamic interactive actual existence in space and time, strings properly pertain to a mere potential for actual existence that underlies the manifest phenomena.

Thus irrespective of technological developments e.g. with respect to particle accelerators, it will never be possible to experimentally detect strings!

Once again from a more correct physical perspective, what are referred to as strings represent the inherent (linear) capacity of matter for differentiated form (as relatively distinct phenomena) and the equal inherent (circular) capacity for an overall qualitative integral relatedness (as holistic dimension).

Alternatively in more scientific language we can say that strings possess both a real and imaginary potential which in dynamic interaction generate both matter phenomena and the dimensions of space and time (which are inextricably linked).

Darwin and Riemann

2010-09-13T03:28:00.001-07:00

When doing some research for my articles on the Riemann Hypothesis, I made the interesting discovery that both Darwin's Origin of Species and Riemann's famous article on prime numbers were both published in 1859 (just over 150 years ago).

Indeed the historic connection can be shown to be even closer with the publication date of Darwin's book in November of that year while the full text of Riemann's article also appeared in November (in the monthly reports of the Berlin Academy) though Riemann actually had delivered his address on the contents of that article to the Academy in August, 1859.

However recently I have come to see an even greater significance to this interesting coincidence of publication dates (of what constituted truly ground breaking initiatives in two different fields).

In earlier blogs I addressed the issue that any attempted reconciliation of science and religion would require two key developments.

1) the recognition of an alternative qualitative aspect to science, utterly distinct though of equal importance to the present recognised quantitative aspect.

2) the demythologisation of the manner in which universal spiritual truths are conveyed in the major religious traditions.

It is with respect to the latter of these requirements that Darwin's work is of such enormous significance.
Whereas Newton especially had paved the way early for this change with respect to the natural sciences, Darwin above all has helped to extend it to the biological sciences.

For example in the Christian tradition the evolution of life on Earth, especially with respect to the development of the human species had been shrouded in myth for which no proper scientific basis existed. So in providing a truly coherent scientific explanation for evolution of all life forms, Darwin effectively unmasked the nature of literal Christian beliefs in this regard.
Of course a proper scientific appreciation of the nature of evolution does not affect the legitimacy of spiritual beliefs per se (but rather the manner in which they may be presented in the religious traditions)!

As a child I had already embraced evolution (having dismissed in my own mind any literal basis to the Genesis account of Adam and Eve). However I never saw this as having any direct bearing on spiritual truth (which for me still possessed a powerful significance).
Perhaps because of this early clash with religious orthodoxy I have remained open to the manner in which so many Christian doctrines are still expressed in the form of mythical explanations.
So I do see unquestioning acceptance of the literal meaning of these myths as a major barrier to genuine discourse with the scientific community.

It is with relation to the first requirement above i.e. the need for a qualitative aspect to science, that I now see Riemann's article as being of immense potential significance.

As is well known, Riemann's article was to give rise to the famous Riemann Hypothesis (which still remains unproven from a conventional mathematical perspective).

As I had for many years suspected a hidden qualitative aspect to the Hypothesis in recent years I have given it considerable attention with a view to unravelling the barrier to its resolution.

To my amazement, I eventually was able to conclude that - when properly appreciated - the Riemann Hypothesis is really a statement regarding the basic requirement for maintaining consistency as between both the quantitative and qualitative aspects of mathematical understanding.

One obvious implication of this new understanding is that the Hypothesis has no proof from a conventional perspective (where only the quantitative aspect is recognised). Rather it serves as a more general axiom on which those axioms already used in conventional interpretation depend.

So, the Riemann Hypothesis in fact serves as a powerful expression of the need to incorporate a complementary qualitative with the recognised quantitative aspect of present science.

Thus from my newly adopted perspective, the very basis of the two great revolutions that are required (before science can be be properly reconciled with religion) have already been sown in two major contemporaneous developments in thinking some 150 years ago.

The Uncertainty Principle

2010-08-16T07:52:00.000-07:00

Much is made of the Uncertainty Principle in Quantum Mechanics, whereby it is accepted that both the position and momentum of a sub-atomic particle cannot be precisely determined. So there is a trade-off involved with respect to both aspects with ever greater accuracy with respect to one aspect (e.g. position) inevitably being at the expense of the other (momentum). And this is an inherent problem with respect to the behaviour of such a particle (and not due to practical difficulties with measuring devices).

However there is a much wider context to this principle which is not properly recognised (due to the lack of any appropriate qualitative context to Conventional Science).

As I have stated before the very basis of Conventional Science is the use of linear rational logic (reflecting in turn the Middle Band of the psychological spectrum).

However just as electromagnetic energy has many bands (of varying wavelength and frequency) likewise it is true with the modes of possible rational understanding.

So besides the natural mode (which provides the common sense intuitive light informing normal macro understanding of reality) there are many "higher" modes of intuitive energy possible (that are consistent with a more refined circular type of reason).

Now the very essence of the linear mode is that it necessarily reduces (in any context) the qualitative aspect of understanding to mere quantitative interpretation.

Though such reductionism indeed enables an extremely good approximation with respect to the quantitative nature of reality at the macro-level of experience, it begins to break down badly at both "higher" qualitative and "lower" quantitative levels (which are complementary).

In other words to appropriately interpret the quantitative behaviour of a "lower" level of physical reality (e.g. sub-atomic particles) we must use a corresponding "higher" level in qualitative terms of refined rational understanding (with the degree of refinement depending on the quality of the corresponding spiritual intuition required).

So for all other dimensions other than the 1st (i.e. linear understanding) a necessary complementary relationship exists as between what is "objectively" known about reality (in quantitative terms) and a corresponding "subjective" manner of psychological interpretation (in qualitative terms).

So in the deepest sense this inevitable dynamic interaction as between quantitative and (complementary) qualitative aspects is what really underlines the Uncertainty Principle.

There are just two important aspects with respect to this wider Uncertainty Principle that I would like to highlight.

1) When one experiences "normal" reality from the authentic perspective of a "higher" spiritual contemplative stage, the Uncertainty Principle necessarily applies in qualitative terms to all scientific (and mathematical) understanding.

For example this finding intimately applies to the nature of mathematical proof.

At the linear (1-dimensional) level of understanding a mathematical proof (e.g. of the Pythagorean Theorem) is interpreted in a somewhat absolute fashion.
Though it may well be conceded that such a proof represents a "subjective" manner of interpretation (in the necessary psychological use of mathematical constructs) a merely static relationship is maintained with respect to the application of such constructs to "objective" reality. So there is an underlying belief here that the "subjective" mental interpretation absolutely represents the "objective" reality (with respect to the behaviour of right angles triangles).

However at the "higher" levels (where intuition and reason are explicitly involved in understanding) one realises that a dynamic interaction is necessarily involved as between the qualitative means of mathematical interpretation (i.e. the mental "proof") and what it relates to in quantitative terms.

So one's understanding of the proof (in this example of the Pythagorean Theorem) is now understood in a merely relative sense. Thus the truth is strictly approximate and thereby of a merely probable nature.

Thus a mathematical "proof" is now seen to represent but a special form of social consensus. However the degree of understanding which each person will form of the "proof" will vary (perhaps considerably). One could even in some respects form a faulty understanding of the logical connections while confirming a "proof" (without being aware of the fact). Indeed it is quite possible that for a time the mathematical community en masse could form such a faulty understanding and confirm a theorem as true (that is later shown to be false). This famously happened with the 1st "proof" by Andrew Wiles of "Fermat's Last Theorem". Now admittedly this problem was quickly addressed and as time proceeds with no further objections arising then we can assume that the theorem has indeed been proved with an ever greater degree of confidence. But all this merely confirms the probable nature of mathematical proof. So the best that we can hope for is that long accepted "proofs" are true to a high degree of probability!

This qualitative Uncertainty Principle likewise applies to all scientific findings such as physical laws.

I have discussed this previously with respect to Relativity and String Theory. So for example the relativistic nature of space and time does not apply solely to quantitative measurements but equally to the qualitative means by which we strive to interpret such findings - literally - at the "higher" dimensions of understanding.
And then the Uncertainty Principle applies to the dynamic interaction as between both quantitative and qualitative aspects. So once more precise measurement with respect to one aspect inevitably implies diminished measurement with respect to the other. So for example the zenith with respect to qualitative appreciation - which is inherently of an intuitive nature - comes through pure contemplative union with reality (where quantitative measurements lose any specific meaning!)

Indeed the very manner by which Einstein ignored the qualitative aspect of interpretation (in striving to understand Relativity) clearly demonstrates the linear rational nature of the classical paradigm from which he operated!

2) Just as the potential set of all possible numbers is infinite this applies likewise that the set of potential dimensional interpretations is infinite. Therefore the Uncertainty Principle - relating such dimensional interpretations to corresponding reflections of physical reality - has an infinite number of possible expressions.

However it is still true to say that a limited number of these expressions have special relevance. I will return to these in later posts.

Odd Numbered Dimensions

2010-07-29T05:12:00.000-07:00

I have found the odd numbered dimensions more difficult to properly understand than the even.
As we have seen the even are more properly geared for holistic integral interpretation of reality and as my main concern over the years has been to articulate such an approach in appropriate mathematical terms, it is not surprising therefore that the even dimensions held more resonance for me.

The clue to understanding however of the higher odd dimension (focusing initially on the positive) is the realisation that human development necessarily entails both differentiation and integration which need to be maintained in healthy balance.

So even for one committed to the process of growing in pure contemplative development a certain level of activity at each stage must be maintained.

Thus for example when one returns to active involvement (following the development of the more passive even numbered dimension), the next odd numbered dimension will then unfold.

Thus arriving at the first of the higher dimensions i.e. 2 leads to a new integral appreciation of phenomenal reality as being based on the dynamic complementarity of opposite real poles (e.g. internal and external respectively). This new "higher" rational appreciation of the structure of reality is in turn supported by an enhanced quality of spiritual intuition (not available at the linear level).

However such appreciation is of a holistic nature. So to maintain balance one must equally strive for appropriate appreciation at the more active level of dealing with specific phenomena. So this leads to the unfolding of 3-dimensional understanding.

If one looks at the three roots of unity, the first is + 1, which in a sense stands separate from the other roots (which are paired in a more complementary manner). And this feature is shared by all higher odd numbered roots.

The corresponding qualitative significance of this is that likewise associated with the higher odd dimensions is a certain return in qualitative terms to linear understanding (correponding with + 1). However unlike the earlier 1-dimensional level, this is circumscribed to a considerable extent by other forms of understanding.

It is easiest to illustrate here with respect to 3-dimensional interpretation and I will do so from my own particular experience.

Having gone through an extended "dark night" lasting about 5 years, which in holistic mathematical terms represented the negation of 2-dimensional understanding, I noticed a distinct lull with respect to customary purgation taking place. This gradually enabled me to turn the focus outwards again and resume many of the activities that I had given up for some considerable time. In one way it seemed like a welcome return to normality. As I came to share again the same interests as others, the preceding "dark night" appeared like an lengthy aberration that had now thankfully passed.

However I slowly realised that things were not quite the same as before and that I had become extremely sensitive to projections of all kinds. Now a projection represents an involuntary eruption of the (repressed) unconscious which then becomes embodied - and thereby confused - with conscious phenomena. So I found myself spending considerable time in the attempted negation of the many projections that had come to interfere with my activities.

Now if one looks at the two other roots (of the 3 roots of unity) one can see that there is a real part (which in both cases is -.5 and an imaginary part that is complementary (i.e. one with a plus sign and the other with a minus).

I began to understand that there is a fascinating holistic mathematical significance here (with intimate relevance for psychological development).

The real part represents the attempt to split dynamic negation of phenomena as between the external and internal aspects of phenomena. However because the unconscious is not yet sufficiently developed, this leads to projections becoming associated with both aspects.

So it is only when the unconscious has matured that such involuntary projections cease and the new integral 4-dimensional appreciation unfolds. Now in a suitably refined manner one clearly realises how the unconscious is necessarily involved in the appreciation of all phenomena so that involuntary projections (which represent a failure of such realisation) largely cease.

So in a valid sense the odd numbered dimensions can be seen as transition points between successively higher integral stages (represented by the even numbers).

In this way a new "higher" stage of integration requires a corresponding preceding stage of differentiation before it can successfully unfold.

Indeed this qualitative explanation would also help to explain why in the Riemann Zeta Function, pi values do not conveniently arise where odd numbered dimensions are concerned.

As we have seen integration is associated with pure circular understanding (based on the complementary of opposites). However for odd numbered roots we never obtain matching complementarity. And likewise with odd numbered dimensions, which represent a new form of differentiated experience, likewise we do not obtain full complementarity. Thus in a sense with each new stage of differentiation there is inevitably symmetry breaking with the circle therefore incomplete (until integration with the next even numbered dimension can be restored).

However it is the negative odd dimensions that are perhaps the most perplexing!

Once again if we go directly to the Riemann Zeta Function, we have the unusual finding that the value of the Function always results in a rational number (where the value of the dimensional power i.e. s, is odd and negative). Also this value keeps alternating as between positive and negative with successive "higher" (absolute) values of s.

So when for example s = - 1, the value of the Function = - 1/12 and then when s = - 3 the value = 1/120.

Now once again - though this is not yet recognised - such values (where s < 0) possess a holistic mathematical rather than conventional linear meaning.

Indeed there is a famous account of when Ramanujan first wrote to Hardy, he included the finding 1 + 2 + 3 + ......= - 1/12. Both Hardy and his collaborator Littlewood thought initially that perhaps Ramanujan was mad and then realised that this actually represented the Riemann Zeta Function for s = - 1. So quite independently of Riemann, Ramanujan without any formal training had been able to derive a key result requiring complex analysis. Not surprisingly their opinion as to Ramanujan's talent quickly changed.

Now from a holistic mathematical perspective, I grappled for years to find why this value is rational. And then eventually it dawned on me, providing in the process much greater clarity regarding the nature of spiritual development.

As we have seen the negative even dimensions are required through a necessary form of passive purgation (or cleansing) so that the spiritual light can shine as pure intuition without any phenomenal attachments remaining. In a scientific context, this would entail the removing of all rigid rational interpretation from such dimensions (whereby the spiritual light to a degree becomes reduced to the phenomenal structures used in representation).

However - though St. John does not specifically refer to it - there is an equal need for a new form of active purgation following each "higher" stage of differentiation. So in a more healthy - and balanced - view of the spiritual journey, passive does not replace active purgation at the more advanced contemplative stages, but rather such advanced stages require both active and passive purgation. Now of course the degree that is appropriate at each stage will largely depend on personality factors and special circumstances.

In this regard I remember being impressed by a lengthy account given by Eveylyn Underhill in her Chapter "Dark Night of the Soul" (from "Mysticism") on the Dominican monk Henry Suso. There is little doubt that he traversed an authentic spiritual journey (entailing a lengthy "dark night") that was very arduous. However many of his key trials - even at the more advanced stages - were decidedly of an active kind (i.e. relating to negative odd dimensions). So this served in my mind to counteract somewhat the account of St. John in his "Dark Night" where the emphasis is very heavily on passive purgation (corresponding in turn to the negative even dimensions).

We can now provide a fascinating qualitative reason as to why all the negative odd dimensional values of the Riemann Zeta function are associated with rational values.

Again in an intellectual context we explained how in the case of the even numbered dimensions, negation with respect to any (rigid) rational attachment would be required so as to enable purely intuitive understanding (which is literally 0 in phenomenal terms) to arise.

Now an exactly counter experience is required in order to sharpen reason. Reason is cleansed by in turn removing all unrefined intuitive elements.
Normally the use of reason is facilitated by a ready supply of intuition (like oil in an engine) which is taken for granted. However it becomes much more difficult to use the rational faculties when this intuitive light is taken away. What then happens is that one is forced to greatly economise on reason by concentrating on what is truly essential with respect to every task.

Now without such radical purgation, most people never use their minds efficiently constantly wasting intuitive energy on distractions. However when one is required to do tasks in faith (with all supporting consolation of the light taken away) reason is greatly sharpened.

So intuition is refinned and purified (through negating rational support); in counter manner, reason is likewise refined and purified (through negating intuitive support).

So in an intellectual context in this sense, pure (linear) reason is developed through the negation of the odd numbered dimensions (in the "dark night" of active involvement).

It has always been my position that reason and intuition are of equal importance (and essential for the proper working of either aspect). However what I did not realise until recently is that the holistic mathematical appreciation of the Riemann Zeta Function has an intimate and precise bearing on the manner through which "higher" spiritual psychological development unfolds.

We can in addition provide a holistic mathematical explanation of some of the other aspects of the negative odd dimensional values.

For example why do the zeta values successively alternate as between positive and negative?

Well the reason is this. Whereas the even numbered integral dimensions relate to the complementarity of both poles, the odd numbered relate to the temporary separation of the positive from the negative pole. Now it is in the very nature of differentiation therefore that either the external or the internal is given temporary prominence. So when negation takes place it tends to be one-sided. So typically at one stage the external aspect will be more dominant before being negated; then at the next stage the internal will now likewise undergo - relatively - more development and be negated etc. Therefore through the continual positing and negating of each pole separately as with an iteration one gradually moves to the position where both are equally refined.

Another fascinating aspect relates to the fact that the absolute value of the Functions keeps falling until s = - 5 where it reaches - 1/252, before then continuing to increase, gradually at first and then more and more rapidly.

The qualitative significance here relates to the fact that with contemplative development, rational type use of faculties tends to diminish to be used very sparingly (as the emphasis is primarily on intuitive awareness at this stage). Even before the 8th dimension is reached, a certain rebalancing is required so that the holistic transcendent drive to pure intuitive union with reality does not become too extreme. However by the time that the radial stages unfold, the value has reached 1. This would imply that only then has the emphasis of form been brought back into balance with (intuitive) emptiness. Put in spiritual terms, both the transcendent and immanent aspect would now be in substantial balance. And then the progressive increase in the value would signify an ever greater ability to use the rational faculties, as the radial stages proceed.

Now I can see why I laid so much emphasis on the dimensions up to 8 in my earlier approach (as these are the ones indeed most important in terms of holistic development). I had already worked out that a lengthy transition phase would be necessary to reincorporate the rational faculties (which I had identified with the dimensions from 8 to 16). So it is remarkable that the actual value for the Zeta Function only passes one, for the negative dimensional numbers of s between 15 and 17.

Interestingly where pure intuitive development reaches its zenith, one would expect that the positive and negative polarities of rational experience would be in substantial balance. And this indeed is the case for - 1/252 is followed by 1/240. So the absolute ratio of the two numbers is nearly 1 here (and much closer than the ratios of any other successive zeta values for negative odd dimensions).

All in all therefore, though it is more difficult to precisely "nail" the reasons for the precise magnitude of the vales of the Zeta Function (for negative odd values of s) we can once again show strong associations with spiritual psychological development amply demonstrating that these dimensional values (as with the even) have a precise holistic mathematical significance. In fact the meaning of all the values of the Function for negative dimensional values is - directly - of a holistic mathematical nature (as their values cannot be defined here in standard linear terms).

Negative Even Dimensions

2010-07-28T03:53:00.000-07:00

We have looked at the positive even numbered dimensions. These are the most suited for pure integral interpretation of reality and are always based on the matching complementarity of opposites. Within the even numbered dimensions 2, 4 and 8 would command a special importance (as suited for integral interpretation of reality).

However so far we have only considered the positive numbered even dimensions!
So our next task is to give a meaning to the corresponding negative numbers.

I have mentioned the philosopher Hegel as proving initially inspirational in terms of formulating the holistic mathematical meaning of 2-dimensional understanding.

However I gradually became disenchanted with Hegel's approach. It seemed to me that that he was led into a fundamental error in elevating the mere formal rational interpretation of his logic above the very spiritual intuition that it really illustrated. In other words though one can formulate in rational terms the basic principle that all phenomenal interactions are based on the dynamic complementarity of opposites, the true understanding of this principle can only be attained in pure intuitive awareness (where the separate identity of either pole is no longer seen to exist).

So I gradually realised that the paradoxical rational formulation itself is properly designed to lead one on to a purer spiritual intuitive realisation of the meaning it contains. And quite literally this requires the negation of any rigid attachment to the intellectual formulation.

I then discovered in St. John of the Cross a person who had addressed these very issues with a profundity that is perhaps unmatched in the Western tradition.

St. John principally in "The Ascent of Mount Carmel" and "The Dark Night" deals in detail with these problems.

Firstly he deals with the problem of the standard linear attachments that arise from normal day to day duties. So to remove such rigidities the "active nights" of sense and spirit are required. This entails the dynamic negation of linear i.e. 1-dimensional, understanding (or more correctly rigid attachment to such understanding). Sense in this context would relate to the more superficial empirical type of understanding while spirit would relate to more deep-rooted concepts and volitional capacity.

All going well in the contemplative life this leads on to the illuminative stages.
Now this would be characterised in our terms by 2-dimensional understanding with a much more spiritually intuitive worldview properly consistent with the both/and logic of the complementarity of opposites.

However St. John warns strongly of the dangers of secondary attachment to such illuminated understanding. In an intellectual context this would imply undue attachment to the rational forms by which 2-dimensional understanding is mediated (without corresponding development with respect to appropriate intuitive recognition).

So this leads on to need for the passive nights of sense and spirit. So once again whereas the active nights relate directly to linear type understanding, the passive nights relate to more refined circular appreciation (of which 2-dimensional understanding represents the earliest kind).

Thus in holistic mathematical terms, the passive nights refer to the negation of "higher" even dimensional understanding. And the goal of all this purgation is, as St. John emphatically stresses, "nada" (or nothing). In other words the purpose of passive purgation is to remove any undue attachment to circular type understanding (that mediates the spiritual light).

Now there is an extremely important - though completely unrecognised - connection here with the Riemann Zeta Function.

In the Riemann Zeta Function the values for all even valued dimensions is defined. And all cases the resulting values are expressed in terms of corresponding powers of pi.
As we have seen in the previous post, in holistic mathematical terms this rational interpretation expresses the relationship as between circular and linear understanding. So the paradoxical nature of such understanding arises from the attempt to convey circular type relationships through - the necessary use of - linear language.

So when undue importance is attached to this new rational logic, it leads to an inevitable reductionism of its true meaning (that is ultimately spiritual) in rational terms. So in the end Hegel tends to elevate philosophy (in the expression of such rational forms) over religion!

However when rigid attachment is negated with respect to these paradoxical rational forms, their true meaning is revealed in a purely intuitive manner (= 0 in phenomenal terms).

And the remarkable finding here is that in the Riemann Zeta Function, when the even dimension is negative, the value of the function is always = 0.

Now these are referred to misleadingly as the trivial zeros of the Function.
However what is entirely missing from present understanding is that these values actually relate to holistic rather than conventional mathematical interpretation.

Now if one for example tries to calculate the value of the Riemann Zeta Function when s (i.e. the value of the dimension) = - 2 in the standard conventional manner i.e. 1 + 4 + 9 + 16 + ..., the series will diverge (i.e. infinite in common expression).

However in the way the Riemann Zeta Function is defined, the value that arises from the Function = 0. So what is not explained - or indeed is even capable of explanation in standard terms - is to give such values an intelligible meaning.

However what is actually happening is that for negative values of dimensions that the Riemann Zeta Function switches from linear interpretation (where values are infinite) to holistic higher-dimensional interpretation (where values are finite).
Once again conventional mathematical interpretation is based on the default dimension of 1 (i.e. where all calculations are reduced in terms of their quantitative values in this dimension).
And for example in the Riemann Zeta Function whenever the value of s (representing the power or dimesnion involved) > 1, a finite value for the series emerges (using the standard default dimension of 1). However remarkably when the Function here is now calculated - not in terms of dimension 1 - but rather in terms of the actual qualitative dimension arising in each case, a finite result emerges.
Thus we have two methods of calculation a) the conventional linear manner (based on a default dimension of 1) and (b) the holistic mathematical calculation based directly on the qualitative dimension actually involved.

Now interestingly when s = 1, the Zeta Function diverges in conventional terms (with no finite value). And because we are here using the Function is already defined in terms of the default dimension of 1, the holistic mathematical calculation is exactly the same. And this is the one point on the Riemann Zeta Function where its value cannot be defined in finite terms (in either the standard or holistic mathematical fashion).

So when the value of s > 1) quantitative interpretation on the RHS (of the real line) can take place for values of the zeta function in the conventional manner. However qualitative interpretation of the holistic mathematical kind is required on the LHS for s < 0).
And indeed this relationship is even enshrined (again without its true significance being recognised) in the Riemann Transformation Formula!

So enhanced integral understanding actually requires two related aspects. In rational terms it requires appreciation of the higher-dimensional forms that I have been illustrating on these blogs. Though the 2-dimensional has indeed been well articulated in various spiritual and philosophical cultures, without holistic mathematical appreciation it is not really possible to formulate higher even dimensions in a satisfactory manner.
The second aspect relates to appropriate intuitive recognition of what these rational forms are supposed to represent.
Quite simply this cannot be attained without journeying through the various stages of contemplative spiritual development. So initially the rational structure of a certain dimension is posited through initial illumination. However sustained attainment of the appropriate intuition associated with this structure requires a very deep purgation or cleansing i.e. negation of the rigid phenomenal aspects of such understanding. And this is the process that spiritual writers such as St. John have already well documented.

Update on Dimensions

2010-07-27T04:50:00.000-07:00

As one may perhaps appreciate, the qualitative use of numbers (as dimensions) plays a key role in holistic mathematical understanding.

So I have spent at this stage more than 40 years in the slow - and often painful - task of unravelling the hidden meaning contained in all numbers (as dimensions).

In the attempt here to explain my present position, I will relate how this understanding actually unfolded in development.

The first key insight was in recognition of the linear nature of the rational paradigm which underpins interpretation of both mathematics and science.

So in qualitative terms, such understanding is defined in holistic mathematical terms by the number 1 (i.e. 1-dimensional interpretation). As we have seen this entails in Conventional Mathematics that all number quantities are ultimately defined in terms of a (default) power of 1. For example 2 ^ 2 = 4 (i.e. 4 ^ 1). In physics it implies for example that object phenomena are unambiguously posited in just one (external) direction.

The next key step was the recognition of the nature of 2-dimensional understanding. This arose through a deep interest for a time in Hegelian philosophy. Rather than a linear use of reason Hegel favoured a more circular form based on the complementarity of opposite poles (popularly represented as thesis and antithesis).

The key insight here was to see in holistic mathematical terms a direct connection between such poles and the square root of a number.

This can be generalised by saying that there is a direct connection between the n roots of the default number 1 in quantitative and the corresponding n dimensions in qualitative terms. However whereas linear either/or logic applies in the former case, circular either/or logic operates in the latter.

For example in quantitative terms the two roots of 1 are given as either + 1 or - 1.

However corresponding qualitative 2-dimensional interpretation is both + 1 and - 1.

What this means is that all understanding now entails a dynamic interaction as between the knower (as internal subject) and what is known (as external object) with a merely relative meaning so that these two poles keep switching in experience.

Though it may not be articulated in a holistic mathematical manner such understanding represents a normal stage with respect to authentic contemplative development. Here the rigid barriers as between the world (as objective) and the self (as subjective) gradually break down with neither having any absolute identity (in isolation).

The importance of this "higher" 2-dimensional understanding is that it is replicated at a corresponding "lower" level by the behaviour of sub-atomic particles. So the provision therefore of a coherent interpretation of quantum behaviour in qualitative holistic terms therefore requires (at a minimum) 2-dimensional understanding. What this means in effect is that in terms of such understanding quantum behaviour intuitively readily resonates with esperience (corresponding to such circular rational interpretation).

2-dimensional understanding represents the first of the truly integral approaches (based on the complementarity of opposites). I usually refer to this as the Integral 1 approach. Once again it is based on the complementarity of opposite real (conscious) poles

4-dimensional understanding is the next truly important integral approach (Integral 2).

Just as the earlier approach establishes the complementarity of internal and external aspects (the knower and what is known) this likewise establishes the complementarity of whole and part which is perhaps the most fundamental of all relationships in physics.

Standard linear interpretation cannot properly maintain the key qualitative distinction of whole and part. Instead it simply reduces (in any context) the whole to the parts in quantitative terms.

Not surprisingly, gross reductionism pervades the conventional approach to science and mathematics (which for the most part is not even recognised as such by its practitioners).

Precisely because the whole - when properly understood - is qualitatively distinct from the parts one cannot maintain such a distinction without incorporating the qualitative dimension of science on an equal basis with the quantitative.
Of course wholes and parts can be given a quantitative interpretation. However in the dynamics of recognition by which the mind switches from wholes to parts (and parts to wholes) an unconscious aspect is necessarily involved.

Properly understood therefore in the recognition of any object phenomenon an unconscious (as well as conscious) aspect is involved. Indeed even in popular language this is often recognised. For example one may speak of a "dream" house. So here the house has a local (conscious) identity; however equally it possesses a holistic (unconscious) identity as the embodiment of a deeper meaning.

However, strictly this necessarily applies to all phenomena. Indeed the very desire of scientists to study certain phenomena often speaks strongly of this holistic (unconscious) meaning (which cannot ultimately be separated from associated recognition of a conscious kind).

The key breakthrough here (which owed much to an interest in Jungian Psychology) was the recognition that the "embodied" unconscious aspect of understanding is "imaginary" in a precise holistic mathematical sense.
Now the negative direction (by which phenomena are literally negated in experience) represents the unconscious direction of understanding.Inherently this leads to a 2-dimensional form (combining positive and negative polarities). So to express such understanding in a (reduced) linear form we take the square root!

Thus "imaginary" interpretation is the every means through which the qualitative - as opposed to quantitative - aspect of understanding is expressed in a rational manner.

Mathematics recognises that numbers (as quantities) incorporate both real and imaginary members.
The holistic mathematical corollary of this is that science contains both "real" and "imaginary" aspects (i.e. as quantitative and qualitative aspects respectively).

So this whole blog on "Integral Science" is thereby designed to elaborate in various ways the nature of the unrecognised "imaginary" aspect of science.
And a fully comprehensive approach to science - which I term radial - would be "complex" combining both "real" and "imaginary" aspects (as equal partners).

As we have seen in the previous post, one important application of this relationship as between "real" and "imaginary" relates to object phenomena and dimensions (which are "real" and "imaginary" with respect to each other. Thus a clear implication of this is that we cannot hope to properly understand String Theory without incorporating a corresponding holistic qualitative aspect. And as I was mentioning all the important concepts in String Theory can be given corresponding holistic interpretations with a comprehensive understanding then relating to the relationship as between both aspects.

The next truly fundamental integral approach - which I term Integral 3 - relates to 8-dimensional approach. As well as the 4 dimensions (of Integral 2) this opens up 4 new dimensions (of a complex kind). However in geometric terms the diagonal lines representing the corresponding additional roots (on the circle of unit radius in the complex plane) can be represented as null lines.

The psychological implication of this is that pure spiritual attainment (that is nothing in phenomenal terms) is approximated when both conscious (real) and unconscious (imaginary) aspects are fully harmonised.

Remarkably the corresponding physical implication relates to the very nature of forces. For example light is often in Relativity Theory represented as a null line. So the deeper implication is that light (in itself) represents the complete harmonisation of phenomenal and dimensional characteristics.

Even in the Biblical account in Genesis, the World is created out of light. So the phenomenal and dimensional characteristics of the created Universe can be seen to emerge from a common origin!

For many years 8-dimensional interpretation represented the limit of what I felt could be achieved in integral terms.
Indeed I formulated A Holistic Theory of Everything based on such understanding!

Then shortly after that when studying the holistic mathematical nature of Jungian Personality Types I saw how to extend 4 to 24-dimensional interpretation through obtaining all possible permutations of the existing 4. This - as I have related elsewhere - then led to an extension of the Myers-Briggs system (to include 8 missing types). The realisation that each of these Personality Types represented a unique circular dimension (in the characteristic manner of configuring space and time) led me a similar notion within physical reality (which I now see as the means of interpreting dimensions in String Theory).

So just as with String Theory, I was left with 5 main integral models for holistic qualitative interpretation of reality.

It is only in more recent years - largely through association with the Riemann Hypothesis - that I have been able to significantly extend this understanding.

After some time I realised that 16-dimensional interpretation (and all subsequent powers of 2) could be explained like a compass. So as with the compass we have 4 main coordinates (applying to 4-dimensional interpretation. Then with higher powers of 2 we are able to define our directions ever more accurately. Likewise it is similar with higher dimensional understanding where 2 is raised to 0, 1, 2, 3, 4,...etc.

Then all other even numbered dimensions likewise have a direct integral significance.

The basis of integral understanding is the complementarity of opposites in understanding (with again 2-dimensional interpretation offering the simplest example).

However all other even based dimensions can likewise be represented by the complementarity of opposites (replicated in quantitative manner by the format of the roots of every even number).

Indeed there is an intimate connection here with the Riemann Zeta Function.
As is well known for all even numbered integer dimensions (powers) of the Function, an expression involving pi is involved.

Now pi represents in quantitative terms the relationship i.e. ratio, as between the circular circumference and its line diameter.

In like manner in qualitative terms all the even based dimensions (as interpretative models) represent the direct relationship as between circular and linear understanding!

String Theory Again!

2010-07-26T03:07:00.000-07:00

I have already contributed a number of posts to this personal blog on the subject of String Theory.

Though most of the unease within the conventional physics community relates to the difficulties in empirically testing String Theory (for some considerable time to come) I have been concerned with a deeper problem!

For in terms of offering a coherent philosophical view of the nature of physical reality the present position is utterly impoverished.

Now this failure to properly recognise a key weakness, relates to the traditional bias of science i.e. that is geared merely to quantitative analysis - rather than qualitative synthesis - of reality.
Though it is certainly possible that ingenious ways of an indirect nature may be found to remedy the empirical testing deficit, little hope however exists of offering any coherent explanation as to what the theory is supposed to represent.

Now scientists may immediately retort that they are not in the business of qualitative (i.e. philosophical) speculation! But I would reply by saying that the very failure to incorporate a qualitative dimension has led to a growing problem of overall (i.e. holistic) incomprehensibility with respect to modern theories of physical reality.

This has raised therefore a very interesting paradox. As physicists move ever closer to what they hope will be the "Theory of Everything", they find themselves ever less capable of offering any intelligible explanation of what the TOE is supposed to represent.

This problem when properly understood highlights a divide in science (and equally mathematics) that properly should never have arisen i.e. the pursuit of a merely quantitative approach to meaning (without emphasis on its equally important qualitative aspect).

At the normal level of macro investigation of reality this problem is not apparent. Here scientific rational explanations of reality (based on linear logic) correspond readily with everyday qualitative intuitions with respect to such reality.
Because this is so, science conveniently ignores the important role of supporting intuition in all interpretation, misleadingly identifying physical behaviour with mere rational type explanation.

However as with the electromagnetic spectrum there are many other types of spiritual light besides that corresponding to natural light!

So for example, where authentic contemplative development unfolds, various bands of intuitive energy (with their own unique characteristics) unfold that do not correspond to conventional (i.e. linear) rational interpretation. However these bands do indeed correspond with higher-dimensional rational structures!

Such "higher" contemplative states (with their supporting higher dimensional rational structures) have a vital relevance for interpretation of physical reality.

Just as horizontal conplementarity exists at each level as between physical and psychological reality (i.e. where physical reality acts as a mirror of a corresponding manner of psychological interpretation) likewise vertical complementarity likewise operates. What this entails is that corresponding to every "higher" contemplative level of dimensional understanding exists a corresponding "lower" level of physical reality.

In other words as we delve deeper into the "lower" levels of sub-atomic matter, we need the corresponding "higher" levels of dimensional understanding for appropriate qualitative understanding.

So this immediately points to the difficulties that physics faces in providing for example a philosophically satisfying interpretation of quantum mechanics (which appears counter intuitive in terms of everyday understanding of macro reality).

Quite simply, normal macro reality corresponds intuitively with linear (1-dimensional) understanding. However quantum reality belongs to a "lower" level of sub-atomic reality where at a minimum 2-dimensional interpretation is appropriate.

So what does this mean? Well in linear terms, "objective" reality is viewed as independent of the "subjective" observer.

However in 2-dimensional interpretation both objective (external) and subjective (internal) poles are considered as complementary. So strictly speaking where quantum events are concerned what is observed as objective cannot be considered as independent of the observer.
Though this fact is recognised by physicists - though in practice largely ignored - the appropriate intuition for ready comprehension (with corresponding 2-dimensional rational explanation) does not fit in with the accepted conventional paradigm which is decidedly linear. Therefore, even though physicists are able to deal with the merely probable quantitative aspects of quantum behaviour to a remarkable degree of accuracy they are unable to provide an appropriate qualitative explanation (which explains in a satisfactory manner why such behaviour arises).

Now again many physicists might try and retort that they are not concerned with philosophical aspects of understanding. However such a position is extremely short-sighted and ultimately untenable for without appropriate holistic understanding of a qualitative nature, overall interpretation of theory becomes incomprehensible.

Indeed this is major problem at present with String Theory. Though undoubtedly there are good reasons for excitement regarding the mathematical relationships that have been discovered, no coherent explanation has been given by physicists as regards the very nature of the reality that they are attempting to interpret.

For example one major problem attaches to the nature of dimensions in which the strings supposedly operate. In various approaches this has required more than the customary 4-dimensions with 26, 10, and now 11 being used.

However once again this is calling for the kind of multi-dimensional interpretation that the qualitative approach can provide.

We have already seen that in the 2-dimensional approach, external and internal aspects of reality dynamically interact in a relative manner (corresponding to positive and negative polarities that are understood in a real conscious manner).

However in the 4-dimensional approach, not alone do internal and external interact but also whole and parts in the form of object phenomena and dimensions of space and time. So properly understood phenomena and associated dimensions are now real and imaginary with respect to each other!

If we take these 4 poles (representing both real and imaginary coordinates with positive and negative directions) we can permutate them in various ways. So each permutation now represents a unique configuration with respect to space and time.

The need for higher dimensions in String Theory essentially points to this fact.

At this level of reality it is meaningless to try and consider space and time as separate from the object phenomena they contain. So a dimension now represents a unique configuration with respect to possible arrangements of what are 4 independent dimensions in linear understanding.

So the key to proper appreciation of what is going on is the recognition that one must now employ a circular (as opposed to linear) notion of dimension.

And as we approach closer to the origin of life - because of the greater interdependence both of internal and external aspects and also object phenomena and related dimensions - appropriate understanding of reality, in qualitative terms, becomes ever more circular. Put another way the standard linear approach of Conventional Science becomes ever more inappropriate in providing a coherent overall explanation of the dynamic relationships involved. So what we are left with is attempted quantitative interpretation of relationships completely devoid of any proper holistic context.

So just as the Riemann Hypothesis lies at the leading edge of Mathematics where both quantitative and qualitative aspects intersect, likewise this is also true of String Theory. Having attempted for some time to decode the philosophical meaning of the various concepts used, I made the amazing discovery that every notion that is currently used in the quantitative sense can be given an equally important qualitative interpretation. So the real truth that String Theory is pointing to is the inevitable conclusion that our quantitative explanations of physical reality have no meaning independent of the qualitative interpretations that we use as viewing lenses. So the origin of life cannot be quantitatively understood (nor qualitatively understood) in isolation. Rather its mystery is revealed when quantitative and qualitative aspects of understanding are fully reconciled (which represents pure spiritual awareness). And this experience is inseparable from the contemplative mystical desire to discover the ultimate goal or destiny of the Universe!
So realisation of the origin and goal of the universe are revealed in the same spiritual experience as the present moment continually renewed. Only here is any remaining gap as between the knower and what is known finally dissolved where both realise their common identity (in spirit).

And it is not really surprising that both the Riemann Hypothesis (relating to the nature of prime numbers) and String Theory are so closely related in this manner. For the most fundamental particles - derived from what we might call strings - are in fact the prime constituents of all natural physical reality!

Holistic Mathematics and Science

2010-07-24T08:43:00.000-07:00

In the last post I addressed the fact that corresponding to each number in Holistic Mathematics is a unique scientific interpretation of reality.

So the range of possible interpretations is infinite with Conventional Science based on just one of these numbers (i.e. 1).

This I believe is a truly remarkable finding - which if even remotely grasped - should end all notions of science having reached its zenith!

Indeed we can make further remarkable statements based on holistic mathematical interpretation. For not alone does every number possess a unique qualitative dimensional significance (as indicated) but equally every symbol and relationship - with an already established significance in Standard Mathematics - can likewise be given a unique holistic mathematical explanation with intimate relevance for psychological interpretation of reality.

And those theories and hypotheses that already have been shown to have a special importance in standard quantitative terms would possess an equally important significance in holistic qualitative terms.

Indeed to demonstrate this point to my own satisfaction I spent several years examining the Riemann Hypothesis in the attempt to establish its true qualitative psychological significance. Then as an altogether unexpected bonus this paved the way for - what I am confident - is a true resolution of the hypothesis.

So I established in the end that the famed hypothesis is actually expressing a fundamental condition for maintaining consistency as between both the quantitative and the qualitative interpretation of mathematical symbols.
Of course one clear implication of this is that it cannot be solved in standard mathematical terms (as it is based on recognition of the quantitative aspect only).

So correctly understood - as I would see it - the Riemann Hypothesis serves as one expression of a vital axiom in a more comprehensive radial mathematical approach (i.e. that explicitly entails both quantitative and qualitative aspects of interpretation).
So not alone is the standard approach to Mathematics inadequate for the task of solving the Riemann Hypothesis, it is even inadequate in terms of understanding its true nature!

However it would be mistaken to believe that the relevance of Holistic Mathematics (and related Integral Science) is confined to qualitative interpretation of a psychological kind.
For once one departs from the default linear (1-dimensional) interpretation of Conventional Mathematics (and Science), a two- way complementary relationship exists as between physical and psychological understanding. Thus at all other levels, what we "see" with respect to physical reality inevitably reflects in considerable measure the psychological manner by which it is interpreted.

Thus the physical and psychological aspects of (interpreted) reality are mirrors to each other so that the structure revealed in one is the same structure pertaining to the other.

The clear implication of this therefore is that not alone do we have an infinite set with respect to potential interpretations of reality, but equally we have an infinite set of (corresponding) physical realities.

So the great error that is committed by adherents merely of the standard linear approach is the belief that there is just one physical reality "out there" with just one valid manner of overall interpretation.

Though most scientists may not fully subscribe to the pure representation view (i.e. that our chosen mental constructs reveal reality as it actually is) in practice they behave as if this is exactly the case!

However once again when we recognise the legitimacy of the other dimensional numbers (as interpretative structures) we are faced with the inevitable conclusion that we cannot possibly view what is "out there" for in truth a potentially infinite set of interpretations (and corresponding physical realities) exist.

Putting this another way, the inevitable conclusion of adopting this more comprehensive approach is an acceptance that science can only attempt to grapple with secondary phenomenal appearances that serve in an infinite variety of ways to mask what is truly primary and essential (i.e. spirit).

Existing scientists are slowly coming to accept that physical reality in its deepest workings behaves very differently from what is seen at the normal macro level of observation.

So the 1-dimensional method of Conventional Science is best equipped to deal with normal macro reality.

Unfortunately while now recognising that sub-atomic reality conforms to different rules, essentially the same linear method is still being used for interpretation.

It is hardly surprising therefore that quantum reality for example appears so strange and paradoxical (as it is continually viewed through an inappropriate interpretative lens).

At an even deeper level string theory in the same attempt to pursue a merely quantitative type appreciation has rendered itself virtually meaningless as a coherent philosophical (qualitative) explanation of reality.

And this is where the new interpretative models based on higher dimensional numbers) come into play which lead among other things to a new appreciation of dimensions.

I will return to this briefly again in a further post!

More on Nature of Holistic Mathematics

2010-07-22T06:30:00.000-07:00

Just as the standard analytic approach to science is ably served by its corresponding mathematical tool i.e. Conventional Mathematics, likewise the (qualitative) integral approach to science is likewise potentially served by its own respective mathematical tool i.e. Holistic Mathematics.

A great barrier however that I have continually faced is the recognition that Holistic Mathematics - though using the same symbols - is radically different from what most people understand as Mathematics. So what certainly is not intended here is the standard use of Mathematics to deal with holistic type problems (which represents the old reductionist approach)! Rather it requires a new interpretation of mathematical symbols that inherently depends on corresponding appropriate level of intuitive insight for correct usage.

To put this in context, one must appreciate that just as there are many bands on the electromagnetic spectrum (with natural light comprising just one), likewise it is true with respect to the potential for human psychological development. Such development comprises many bands where the nature of the intuitive light - that necessarily informs all understanding - varies greatly.

So what informs rational interpretation - dictating conventional approaches to mathematics and science - is a natural light (in the normal intuition that underpins the use of such reason). However potentially there are many other bands on the spectrum with very different forms of intuitive energy so that here "normal" intuition is no longer appropriate.

For example higher bands of intuition unfold through the process of contemplative spiritual development and in the various mystical traditions their associated states have been documented to a considerable level of detail.

However what has been largely missing is the attempt to develop in systematic manner the implication of such states for understanding of science. For quite simply, a radically distinct approach is needed so as to maintain true compatibility with authentic contemplative understanding.

In my initial quest to deal with this issue I realised that a new type of holistic or qualitative science was needed so as to be consistent with the various stages of contemplative development.

I then further recognised that just as the most developed type of intellectual development would combine both reason and contemplative insight in a refined manner that likewise the most comprehensive kind of science - which I term radial - would likewise combine both analytic (quantitative) and holistic (qualitative) aspects in a balanced interactive fashion.

However I have since come to realise with respect to the qualitative holistic aspect that an infinite variety of interpretations potentially exist.
So what this all mean?

Using holistic mathematical language, I always term the basic approach of science (what one might refer to as its metaparadigm) as linear i.e. 1-dimensional.

Now customarily we think of numbers 1, 2, 3 as quantities. But as we know in Mathematics these same numbers can also be used to represent dimensions (as powers or exponents).

However, in re-opening a major misgiving felt even as a child, I realised that that when numbers are used to represent dimensions in Conventional Mathematics they are given but a reduced i.e. quantitative meaning.

For example when we raise the number 2 to the power of 2
i.e. 2 ^ 2, a qualitative as well as quantitative change takes place. So a table with each (straight) side = 2 metres would have an area of 4 square metres. Thus the 2-dimensional area here (in square units) is qualitatively distinct from the 1-dimensional measurement of each side (in linear terms).

However remarkably in Conventional Mathematics, the result of
2 ^ 2 is provided in a merely reduced quantitative manner as 4 i.e.
4 ^ 1. So this illustrates clearly the truly linear nature of Mathematics where the result of numerical calculations is ultimately expressed in a 1-dimensional manner (so that implicitly this result is raised to the power of 1!)

So - quite literally - Conventional Mathematics is confined purely to a 1-dimensional manner of interpretation (with the qualitative dimension thereby reduced in a merely quantitative fashion).

However associated with every number is a unique qualitative interpretation.
Therefore the default standard interpretation - in what we misleadingly call Mathematics - represents just one of a range of qualitative interpretations that is potentially infinite!

You might well ask what possible bearing this can have on scientific investigation!

In fact when appropriately perceived it has very important consequences.

We can illustrate the standard linear manner of interpretation in science very directly.

When one for example observes an object it is - literally - posited in experience as distinct. Now in holistic terms such positing is denoted by the mathematical symbol + (this time with a holistic meaning). Likewise implicit in the recognition of any phenomenon as distinct is its unitary nature (i.e. where it is literally identified as a unit). And here we have the holistic meaning of the symbol 1.

So standard scientific investigation is thereby one-directional which can be defined here in holistic terms as + 1.

Now if we take the 1st root of 1 (which is the same as raising to the power of 1) the answer remains 1.
More precisely the answer is + 1. So again in 1-dimensional terms, no distinction can be made as between quantitative and qualitative interpretation.

However when we obtain the two roots of 1 something strange happens in that the answer splits quantitatively in two possible directions i.e. + 1 and - 1.

The corresponding implication in qualitative terms is that when we raise the number 1 to the power of 2 that the answer now splits again in two directions + 1 and - 1.

As one would expect - because of its 1-dimensional bias - there is no recognition of this in Conventional Mathematics (which merely reduces the result in quantitative terms). So 1 ^ 2 is treated in precisely the same manner as 1 ^ 1 (= 1).
However in Holistic Mathematics when we raise 1 to the power of 2, it corresponds in qualitative terms to the taking of two roots.

So very importantly raising 1 qualitatively to D (as dimension) structurally corresponds to raising 1 to (1/D) in quantitative terms. However the qualitative inetrpretation is based on circular (both/and) rather than linear (either/or) logic!

So to make a long story short, 2-dimensional - as opposed to standard 1-dimensional interpretation - in a scientific observational context entails giving every phenomenon two directions that are + (positive) and - (negative) with respect to each other.

This in fact refers to the inescapable fact what all experience of reality (in any possible context) inevitably entails a dynamic interaction as between the knower (as subject) and what is known (as object). So from a 2-dimensional perspective an "object" phenomenon has no strict meaning outside this two-way dynamic interactive context.

So we now can only speak relatively of every phenomenon with two directions involved. Thus if we associate + with the external (objective) aspect, then we must associate - with the corresponding internal (subjective) aspect; however if we equally in any arbitrary polar context fix + with the internal, then in relative terms we must then associate - with the external. So in dynamic interactive terms positive and negative signs keep switching in a relative manner.

This type of circular understanding based on the complementarity of opposite poles, thereby enables one to deal more subtly and accurately with the actual dynamics of scientific experience.

Considerably more refined experiences are possible (and in the appropriate scientific contexts extremely important). For example with 4-dimensional understanding (which is especially significant) every object phenomenon would be given a 4-directional interpretation with both two real and two imaginary aspects that are each positive and negative with each other. This again would correspond, from a structural perspective, with the 4 roots of 1 (in quantitative terms).

What this means in effect is that as well as recognising the dynamic nature of the external/internal dialogue in experience, we also recognise the equally important interaction of whole and part (which inevitably defines all possible experience).

This leads on to the key insight that the imaginary symbol i corresponds directly with the unconscious recognition of an object (in holistic terms). Thus switching from whole to part (and part to whole) in experience always entails the interaction of conscious and unconscious. And in holistic mathematical terms whereas conscious understanding is "real" unconscious - by contrast - is "imaginary".

So the qualitative implication of the complex number system is the recognition that both rational (conscious) and intuitive (unconscious) aspects must be formally recognised and that correctly understood in 4-dimensional terms in any appropriate context the whole is always - relatively - "imaginary" with respect to the parts that are "real". So when we fail to recognise this fact (as in standard 1-dimensional interpretation which is solely "real") the whole is inevitably reduced to the parts.

So again we can characterise the limitation of the present accepted approach as one that formally recognises solely the "real" aspect of understanding.

Thus what I am attempting to address here is the need for recognition of the equally important "imaginary" aspect of mathematical and scientific understanding (i.e. Holistic Mathematics and Integral Science respectively).

Then the radial approach to both - combining them as both equal in importance - can thereby constitute in qualitative terms the complex rational approach. And when both aspects are equal they can be represented as null lines, likewise such an approach also represents the most simple (based on pure spiritual intuition).

We have not time here to deal with further dimensional interpretations (which are ever more demanding in terms of appropriate intuitive recognition).

However we can hopefully generalise what is involved in a meaningful manner.

Just as all the possible roots of unity can be represented by sets of complex numbers on the circle (in the complex plane), likewise all possible dimensions of understanding can be understood in circular logical terms as representing various configurations of both "real" (conscious rational) and "imaginary" (unconscious intuitive) understanding. (Once again what is "imaginary" in this context strictly relates to an appropiate indirect rational means of expressing holistic intuition!)

The End of Science?

2010-07-19T04:46:00.000-07:00

I have been reading again "The End of Science" by John Horgan which I find interesting on several levels.

Firstly, whether one agrees or not with its conclusion it puts forward a most provocative hypothesis i.e. that the great era of theoretical scientific discovery is at an end with diminishing returns with respect to further development now to be expected.

Secondly it attempts to cover a wide range of different scientific fields conveying in the process some flavour of the rich developments that have already taken place.

Finally - and most notably - Horgan managed to do an impressive amount of research in interviewing a significant number of the biggest names associated with these fields (at least at the time of writing in the mid 90's). What I like most about his approach is a certain innate scepticism which prevents him from ever appearing unduly awestruck with their strongly held beliefs.

My own approach would be somewhat different to Horgan's and more optimistic that even within the confines of - what is presently accepted as - science, further impressive developments in many fields (theoretical and empirical) can be expected for some time to come.

However in the quest to obtain final definitive answers to life's greatest mysteries e.g. the origin of life, science is in fact straying way beyond present boundaries without seemingly recognising the significant limitations of its present methods.

So in the opinions of many of those interviewed there is much reductionist thinking masquerading as scientific wisdom.

What is more likely is that we are now witnessing the peaking of the golden age with respect to a particular form of science i.e. analytic science geared to quantitative interpretation of reality. However by its very nature such science is not suited to deal with qualitative issues (except in a grossly reduced manner). Likewise it is leading to significant fragmentation with respect to knowledge already accumulated.

It has become increasingly obvious to me over the past 40 years or so that what we now need is a truly massive revolution in the accepted nature of science itself which will allow for the proper inclusion of an equally important - though utterly distinctive - holistic qualitative aspect. And then with gradual development of a more specialised appreciation of the nature of such qualitative science, we will then be in a position to unleash its true potential in a comprehensive approach that combines both quantitative and qualitative aspects as equal partners.

So I would say with considerable confidence that we are not witnessing the end of science. Rather we are perhaps witnessing the end of the total domination of just one aspect of science i.e. its analytic quantitative aspect.
However in terms of a truly comprehensive worldview, science is still very much in its early infancy.

More on Science, Art and Religion

2010-07-13T08:03:00.000-07:00

As we have seen science is based directly on the cognitive mode of reason.

More specifically - as conventionally understood - it is based almost exclusively on the use of linear reason. This requires standard either/or logic where opposite polarities can be clearly distinguished. So for example this assumes that "objective" can be clearly abstracted from its opposite subjective pole, that parts in any relevant context are distinct from wholes and - perhaps crucially - that phenomena of form can be distinguished from the emptiness (in material terms) of spiritual reality.

So the focus of science is to gain knowledge of reality in an impersonal detached manner. Though the affective and volitional faculties are likewise necessarily involved in experiential terms, their use is of a supporting indirect nature. For example even in the most abstract forms of mathematics, use must necessarily be made of symbols (that can only be verified at a sensible level). Also the very pursuit of science requires motivational desire (pertaining to the volitional faculty).

However having said this the nature of science is in practice most clearly defined in a limited rational manner (thus enabling it to be successfully separated from the other domains of art and religion).

However while accepting that ultimately science will always pertain directly to what can be cognitively understood (through reason) in an impersonal manner, my fundamental criticism relates to its present unduly limited context where it is through its analytic nature merely confined to deal with the quantitative aspects of reality.

However an equally important holistic type of science potentially exists based on the alternative circular both/and logic of logic (where the key polar opposites are understood as complementary) that is directly suited for an integral qualitative appreciation of reality.

Then the most developed form of science - which I term radial - would combine both of these approaches comprising what is in truth a binary digital system for encoding transformation processes using both linear (1) and circular (0) logic.

As we shall briefly see, though though there are at present insuperable barriers to the successful integration of science and religion (and to a considerable extent also science and art) these can be greatly removed though a more comprehensive understanding of the true nature of science.

As I would see it art (in all its varied forms) is based directly on meaning that can be conveyed through the affective function (of sense and feeling). However in practice it is much less limited in definition than science allowing potentially for considerable interaction with both the cognitive and volitional functions.

For example the underlying structure of much art e.g. painting and literature depends on a capacity for right order (that is of a scientific nature). Also considerable opportunities in most forms of art exist for combining the religious with the artistic impulses. Far for these being seen as necessarily incompatible, much of the most celebrated art in our culture is decidedly of a religious nature.

Also though several art forms are now distinguished in popular culture such as literature, music, painting, sculpture, the theatre, film and photography, the meaning of art in truth in a much wider sense is tied up with the everyday process of living. So perhaps the most pervasive "arts" relate to human relationships and the natural environment.

So the art of living essentially relates to the ability to continually derive meaning on a daily basis despite surface appearances that might seem incompatible with such meaning.

Religion however ultimately pertains to the primary volitional function (relating to will) and is the basis of that fundamental motivation or desire which is always a prerequisite for pursuing any purpose.

Though it is a often misunderstood term, the basic religious impulse relates to the exercise of faith in an unseen dimension to life (that cannot be directly identified with phenomena of form).

Whereas science relates directly to impersonal meaning (knowledge and order) and art directly to personal meaning (beauty and devotion), the religious quest is at once more fundamental and paradoxical. Though it combines in a sense both personal and impersonal meaning, yet it cannot be directly identified with either (in a phenomenal context). At its deepest level it relates to that spiritual meaning which both transcends all created phenomena (as goal) and yet is immanent in those very same phenomena (as their true source).

Again at an ultimate mystical level, the religious quest expresses that desire to be at one with all creation. However because the very nature of registering material phenomena requires that they be to a degree detached from us, such a union can only take place in spiritual terms (where rigid phenomena have melted away).

Of course in practice religion can become strongly identified with the affective dimension (e.g. in popular religious devotions) and the cognitive (with the structures and rules of the various churches). However the distinguishing irreducible element of the religious quest is that assent for spiritual meaning (that cannot be fully reduced to phenomena).

So properly understood science art and religion are inseparably linked, ultimately providing differing routes to an eternal reality which lies as the very essence of our being. Though I am of course uttering my own particular perspective, it would seem obvious to me at this stage, that ultimate fulfilment requires that we become reunited with our essential being (which in spiritual truth is inseparable from everything else in creation).

From this perspective science and art provide complementary perspectives for pursuing that ultimate journey that are continually mediated and refined through the central religious quest.

Thus from a scientific standpoint there is a desire to realise ultimate knowledge; from an artistic perspective there is the desire to realise ultimate beauty; and finally from a religious standpoint there is a desire to realise ultimate essence in our own true eternal identity.

Even if present science could provide - which by its very nature is actually not possible - final answers to the biggest problems, it would still fail to satisfy our deepest needs.
For science can only seek truth through attempting to separate the self from the environment. However as the deepest longing is to end all such separation, science cannot directly satisfy this longing.
Art then seeks personal truth though the affective aspect. However again without constant appeal to the complementary cognitive aspect (bringing a more universal dimension) this truth will necessarily be pursued in an unduly self centred fashion.

Finally religion seeks truth more directly in a spiritual realm. However because in experience this necessarily interpenetrates with material form, it too is in need of "right science" and "right art" if it is to realise its destiny.

However in conclusion I come back to my basic point. The biggest single restriction in terms of potential integration of the three great domains is the unduly limited nature of present science.

Not alone is a greatly enlarged perspective on science required so as to properly pursue its own aims, but ultimately such a perspective will be required to enable its true integration with both religion and the arts.

Integrating Science and Religion

2010-07-09T06:44:00.000-07:00

In terms of successful psychological development we can identify - for convenience - three key personality functions in need of integration.

These are 1) the cognitive function related to the pursuit of knowledge and order; 2) the affective function related to love and beauty and finally 2) the (primary) volitional function related to motivation and fulfilment.

Corresponding to these functions are the three great domains of science, art and religion respectively.

Therefore for successful integration in society, ultimately science, art and religion must be reconciled.

However a great limitation at present is the manner in which science and religion especially - as presently understood - seem in many ways to be mutually in conflict with each other.

As I have always been greatly interested in both domains, a key issue for me has related to attaining their mutual compatibility.

Basically I would see that such compatibility requires two major revolutions with respect to current understanding.

The first requirement relates to the recognition that present interpretation of the nature of science is greatly restricted.

Though in truth there are two key aspects to comprehensive scientific understanding, at present only one of these aspects is formerly accepted.

So conventional science - using linear logic - is based on a merely analytic understanding of phenomena geared to their quantitative interpretation. Present science therefore can only deal with qualitative issues through a continual process of reductionism!

However there is an equally important dimension to science (using circular logic) that is based on true holistic understanding of phenomena geared directly to their qualitative interpretation.

So before we can hope to successfully integrate science with religion, a major revolution is required with respect to the nature of potential scientific understanding leading to a much more comprehensive approach.

And of course this blog is designed precisely to address the nature of this hidden scientific dimension!

However - again for successful integration - an equally important transformation is required with respect to our interpretation of religion. Though it may cause convulsions among many committed followers of the various traditions, all of these offer but secondary expressions of a primary universal reality that is spiritual.

In other words the major traditions, in a commendable attempt to disseminate spiritual meaning to followers, inevitably employ a variety of culturally determined mythical symbols so as to suitably convey the desired message.

For example this is very true with respect to my own tradition of Roman Catholicism. Here there is strong emphasis on God as One (rather than both One and Many) on God as male (rather than female) on God sending His only Son into the World (which again is very restrictive when everything in creation comes from God) on Christ being born of a virgin (though normal birth cannot occur in this way) on Christ being conceived without sin (though it is existentially meaningless for human life to be without sin) and so on.

So what we have here are various attempts to convey universal spiritual truths in the form of important myths (where however the meaning becomes unduly restricted through overidentification with the particular phenomenal symbols used). This then leads misleadingly to "magical" interpretations represented as the core of religious beliefs.

So a considerable need remains to demythologise such symbols thus allowing a much better appreciation of the key spiritual truths which they contain.
So there is a universal spiritual reality of which all religions convey valid - but necessarily limited - phenomenal interpretations.

Until this point is clearly realised not alone will the various religious traditions find unnecessary sources of conflict between themselves, but equally it will remain extremely difficult to find any common language for discourse with science.

Properly understood the three domains of science art and religion should be pointing (through differing functions of understanding) to the same great secrets of life.

However once again before this reconciliation can be achieved - especially with respect to science and religion - we need to first "spiritualise" science (through proper recognition of its hidden qualitative dimension). We then equally need to demythologise the various great religious traditions recognising them all as but important secondary - and necessarily restricted - expressions of a primary universal spiritual reality. And this spiritual reality exists not just here on planet Earth but everywhere throughout the created Universe.

Monstrous Moonshine - mysterious connections

2010-04-21T04:18:00.000-07:00

Sometimes a degree of ignorance can be a virtue.

I am by no means properly acquainted with the quantitative mathematical intricacies of modular functions (and the important expression relating to the j-function).

However in attempting to look at issues from a holistic mathematical perspective, I am not surprised that intimate connections have been demonstrated as between dimensions in the Monster Group and coefficients of the j-function.

After all the Monster Group relates to an amazingly symmetrical object as group of rotations in 196,883 (Euclidean) dimensional space with group order:

|M| = 808017424794512875886459904961710757005754368000000000

= 2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71

Modular functions likewise relate to objects with supersymmetrical properties that can be transformed in an (infinite) variety of ways while remaining unaltered.

It is intriguing however from a holistic mathematical perspective that such functions are defined in complex terms with respect to hyperbolic space.

What immediately strikes me here is that - whereas the Monster is defined with respect to linear (i.e. Euclidean) dimensions - modular functions are defined by contrast with respect to circular notions (interpreted in a quantitative manner).

We have seen that in holistic mathematical terms that the unconscious direction of experience is negative (relative to the rational conscious).

Hyperbolic space can be understood as having negative - as opposed to positive - circular curvature. This is therefore extremely difficult to visualise from a rational perspective (suggesting that its real significance relates to intuitive holistic meaning of a qualitative kind).

Therefore though not directly accessible in quantitative circular terms, it would however be accessible - when appropriately understood - in a qualitative manner.

Also it is vital to recognise that even though i (representing the square root of - 1) can indeed be given an indirect quantitative interpretation in Conventional Mathematics, that it properly relates to holistic intuitive meaning that is indirectly expressed in a linear rational manner.

Roger Penrose for example keeps pointing to the magic of complex numbers in that they can so frequently be demonstrated to possess amazing holistic aspects. However the key to deeper appreciation of why this is the case requires corresponding qualitative understanding of a complex kind i.e. combining both real (quantitative) and imaginary (qualitative) interpretation.

The famous Taniyama-Shimura conjecture - the proof of which opened up the way for Andrew Wiles to solve Fermat's Last Theorem - establishes a key link as between modular forms and elliptical curves (that every elliptical equation with integer solutions has a corresponding modular form).

An elliptical function corresponds to the expression

y^2 = ax^3 + ax^2 + bx + c (with a, b, c and d integers).

Now in quantitative terms, solutions to such an equation will be two-valued (with positive and negative results). This suggests in corresponding qualitative terms the need for a two-valued logic using 1-dimensional (linear) and 2-dimensional (circular) interpretation.

So in this context the modular represents an appropriate circular quantitative expression (i.e. in complex space) of the elliptical equation.

Though such conversions are quite frequent in Mathematics from a quantitative perspective, the huge missing ingredient is the lack of any coherent qualitative (holistic) mathematical means of interpreting what is involved.

Therefore though Conventional Mathematics is truly ingenious in deriving and proving results (from a quantitative perspective), in many important cases it lacks the means to qualitatively explain why such results can in fact arise!

And Monstrous Moonshine is just one very important example of this problem!

However the deepest clue as to what is qualitatively involved with Monstrous Moonshine is provided by the Fourier expansion of the j-function (pardon the notation!).

j(r) = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + .....

r here is known as the half period ratio which itself expresses the ratio of two complex numbers.

q is especially interesting from a holistic mathematical perspective as it directly involves - what I have referred to as - the fundamental Euler Identity, e^(2pi*i) = 1

So q = e^(2pi*i*r)

Now the significance of this is of paramount importance (from the holistic mathematical perspective) as the fundamental Euler Identity provides the very basis for the derivation of all circular dimensions.

So in qualitative terms the dimension k (as qualitatively understood) is given by the identity

1^k = e^(2pi*i*k) which has the same structure as

1^(1/k) = e^[2pi*i*1/k)] as quantitatively interpreted.

So again the qualitative structure of the 2nd dimension (k = 2) is given

as 1^2 = e^(2pi*i*2) which has the same structure as the quantitative expression:

1^(1/2) = e^[2pi*i*(1/2)] = e^(pi*i) = - 1;

And we have already provided the qualitative explanation of this 2nd dimension as the dynamic negation of (unitary) rational form (of a positive conscious nature), which like the fusion of particle and anti-particle in physical terms, is the very means through which holistic energy of an unconscious nature is generated in experience.

With respect to the j-function therefore q = e^(2pi*i*r), implies that,

1^r = e^[2pi*i*r] in quantitative terms.

Therefore the actual qualitative dimension involved here is 1/r (the inverse of the half period ratio) with the same structure resulting.

Thus the important conclusion then follows that the j-function actually involves a mathematical expression involving the use of circular - as opposed to Euclidean - dimensions.

So from this perspective the j-function could be simply seen as yet another of these fascinating mathematical conversions whereby a linear quantitative is transformed into an equivalent circular expression.

As John McKay first noticed, there is a direct link as between coefficients in the j-function and corresponding dimensions in the Monster Group.

Thus the first relevant coefficient 196884 = 196883 + 1 (i.e. just one more than the smallest numbers of linear dimensions in which the Monster exists).

Then all other coefficients can be derived from simple linear expressions of corresponding dimensions in which the Monster can be constructed.

So for example the 2nd coefficient 21493760 = 21296876 + 196883 + 1. (In this context 21296876 represents the next lowest number of dimensions in which the Monster can be constructed).

The Monstrous Moonshine quest was therefore to prove that there was indeed a solid (quantitative) mathematical basis for such connections (rather than just mere coincidence) and Conway and Norton were subsequently able to achieve this while Richard Borcherds then established connections with string theory.

However once again, so even though Conventional Mathematics can indeed demonstrate such quantitative links, it cannot provide a satisfactory qualitative explanation as to why such links arise (due to the lack of formal development of any holistic mathematical aspect).
So what I have been briefly attempting to demonstrate is that the two areas, the Monster Group and the j-functions, actually employ complementary notions of dimensions (that are linear and circular with respect to each other).

Indeed just as there is strong link with 24 in the construction of the Monster, likewise this is also the case with the j-function.

The j-function is defined as:
j(r) = 1728 * J(r) where r again is the half period ratio and J(r) an expression involving this half period ratio and - what is referred to as - the elliptic lambda function. This in turn relates to an expression involving the ratios of that ubiquitous Fundamental Euler Identity (which as we have seen is the very basis for defining circular dimensions).

Now 1728 = 12^3 and is thereby closely related to 24.

So 1728 = 24 * 24 * 3.

Other fascinating connections have equally been made. For example Andrew Ogg seems to have first noticed that the very numbers used in the construction of the Monster Group are the supersingular primes which form the basis for the construction of many j-functions. And there also appears to a connection as between the Monster Group and the strange and mysterious number 163!

However this would still be consistent with my qualitative notion of circular dimensions as the number 163 famously arises in the expression e^{[(163)^1/2]*pi} which is very close to a whole number.

From a holistic mathematical perspective, prime numbers represent an extreme both in linear and circular terms. They are the most independent in linear terms (as the building blocks of the number system) and likewise the most interdependent in circular terms (with respect to their general distribution). 163 especially embodies these extremes being famously involved in the solution to the quadratic expression x^2 - x + 41 (where x = 1 to 40, generates prime numbers).

Also I note that the fraction 4/27 appears in the elliptic lambda function, J(R).

Well 163 - 1 = 162 = (27/4) * (24/4). So could there even be in 163 some link with the j-function?

Monstrous Moonshine - 24 dimensional space

2010-04-16T01:47:00.000-07:00

In the early 1980's I was very interested in Jungian psychology (and especially with respect to his theory of Personality Types). Part of the attraction arose from the fact that implicitly Jung formed many of his key notions in a manner amenable to holistic mathematical interpretation.

I mentioned in the last post the (true) qualitative circular notion of dimension and contrasted this with the merely reduced quantitative linear interpretation that predominates in conventional mathematical and scientific understanding.

As four dimensional space-time is so important (as conventionally) understood it is only reasonable to assume that an important circular interpretation of such dimensions can be equally given.

It was here that familiarity with Jungian concepts proved valuable.

Jung organised his understanding of Personality Types around 4 key functions that are often shown as equidistant points on the circle. So this bears obvious comparison with the four roots drawn on the circle (of unit radius) in the complex plane, which when qualitatively interpreted give the four circular dimensions of space-time. Indeed in his description of these functions, Jung used language, amenable to mathematical interpretation, in defining two (conscious) rational and two (unconscious) irrational functions respectively. A key breakthrough for me was the realisation that in more correct holistic mathematical terms, these functions constituted two real and two imaginary functions with complementary positive and negative aspects in each case.

So right away we can see that the (true) qualitative notion of dimension is mathematical in a holistic sense. Furthermore both physical and psychological behaviour dynamically conforms (in complementary fashion) to such dimensions.

Thus from the circular qualitative perspective, reality as we know it, in dynamic experiential terms, is based on four fundamental dimensions of space-time that are real and imaginary with respect to each other (with positive and negative polarities).

Once again, "real" in this context relates to distinctive phenomena that can be consciously understood. Positive and negative polarities imply that all these have both an external and internal direction (that are - relatively - positive and negative with respect to each other).

So for example when a mathematician attempts to understand any phenomenon an interaction inevitably arises as to between the (subjective) knower and what is (objectively) known. And when one reflects on it, such a two-way dynamic interaction is unavoidable even at the most abstract level of interpretation.

Thus - strictly - all truth is relative even in mathematics. The reason therefore that mathematical truths can appear absolute is because of the interpretation involved. Thus linear interpretation always entails reducing the internal mental contribution of understanding to what is objective.
So for example we view a mathematical proof as "objectively" true (in absolute terms). Alternatively it can involve reducing the objective aspect to its internal mental interpretation as "subjectively" true (again in absolute terms).

In mathematics general hypotheses relate to the latter aspect; the empirical testing of such hypotheses relates to the former. However because of the absolute nature of interpretation involved, in both cases a direct correspondence is assumed (which in dynamic interactive terms is not strictly valid).

In other words conventional understanding here is one-dimensional i.e. as consciously understood in absolute positive terms.

So allowance for the true dynamic interaction of both real polarities implies that mathematical truth is merely relative.

Thus at the two-dimensional level of understanding all mathematical truth has both positive and negative polarities that continually interact (and thereby change with respect to each other in experience).

Actually this is deeply relevant with respect to the present status of the proof that all finite simple groups have now been classified.

This proof is far from absolute and really represents - as correctly is true of all mathematical proof - an existing consensus among the mathematical community involved - that all the major issues have been successfully dealt with. However this entails that we must remain open to the possibility that such a consensus could conceivably break down in the light of new discoveries.

So interestingly in this important respect the present "proof" with respect to finite simple groups is ultimately pointing to the need for a new qualitative interpretation - that goes beyond the merely linear - of dimensions!

The 3rd and 4th dimensions in this dynamic circular context relate to the crucial relationship as between whole and part that ultimately is imaginary (of an unconscious nature). Now again familiarity with Jungian concepts can help to see what is involved here.
Jung realised well that the unconscious when unrecognised always projects itself into consciousness in a hidden manner (as the shadow side of personality).

Now because in formal terms Mathematics is understood in a merely rational manner, this implies from a Jungian perspective that the unconscious aspect remains completely unrecognised.

And this imaginary (unconscious) aspect is - as I consistently stated - the qualitative holistic aspect of mathematical understanding, that is vitally important yet completely ignored in conventional terms.

Probably the most prevalent form of reductionism, that pervades all conventional mathematical understanding, is that between whole and part.

When one attempts to understand the relationship between whole and part in merely reduced conscious terms, whole are thereby reduced to part notions (with the whole simply viewed as the sum of its constituent parts!).

So for example when we add two numbers say 2 + 3 the (whole) total = 5 is viewed in merely quantitative terms as the sum of the constituent part numbers.

Though of course such a reduced approach is extremely useful in a wide variety of contexts it cannot properly preserve true qualitative (i.e. holistic) meaning.

To preserve the true qualitative distinction as between whole and part, imaginary (qualitative) notions must be used; in this way the (material) part is seen in some measure as reflecting the universal whole (that properly is of an empty spiritual nature); in turn the universal whole reflects in turn the various parts. Therefore in this way the parts can be related to the whole and the whole to the parts without being directly confused with each other.

Now the ability to properly preserve such distinctions depends on the quality of the imaginary intuition generated (pertaining to the unconscious). So seen from this qualitative perspective, mathematical activity is no longer in formal terms just "real" (i.e. pertaining to the rational conscious) but also "imaginary" (i.e. pertaining to the intuitive unconscious).

Once again, the formalised understanding of the rational aspect (which must necessarily be indirectly fulled by intuition) comes from Conventional Mathematics.

However the formalised understanding of the intuitive aspect (which indirectly must be conveyed through appropriate rational forms) comes from Holistic Mathematics.

So seen from the qualitative holistic perspective the four dimensions dynamically relate to the most fundamental polar distinctions that can be made with respect to all phenomena i.e. internal and external distinctions on the one hand and whole and part distinctions on the other.

Indeed we cannot even begin to understand in any phenomenal context without implicitly making these fundamental distinctions!

Coming back to Jung he went on to use his functions (and two attitudes) to define 8 personality types. Subsequently this work was to be significantly extended in the Myers Briggs classification of 16 Personality Types.

Though I found the Myers-Briggs Typology very useful, I gradually came to the conclusion that my own Personality Type did not readily fit into the system and that indeed there were 8 missing Personality Types.

The key problem here is that in the Myers Briggs approach, each Type is defined with respect to either/or distinctions. For example one is defined as either E (extrovert) or I (introvert); as either S (sense oriented) or N (intuitive); as either T (thinking) or F (feeling) and finally as either J (judgement) or P (perception).

However I could see that there was another class of Personality Types that was inherently based on the complementarity of these opposite poles with 8 more resulting.

So in a more comprehensive system 24 distinct Personality Types would exist.

It then struck me that these 24 Types could be derived from the various configurations of the original 4 dimensions. In other words each Personality Type could be fruitfully interpreted to combine these four fundamental polar co-ordinates in a unique manner.

On further reflection I then realised that the holistic mathematical configuration for the 24 types could be given as 24 dimensions (corresponding structurally to the 24 quantitative roots of 1).

So viewed from this holistic mathematical perspective, each Personality Type represents a unique manner in which space and time can be experienced with 24 different dimensions resulting.

Thus the key point about this appreciation of "higher" dimensions is that dimensions are no longer understood as separate but combined in varying configurations with each other.

Later using this approach, I made a direct connection with String Theory. In holistic mathematical terms, psychological and physical aspects of reality are complementary. Therefore the 24 Personality Types (as the fundamental dimensions in which the psychological experience of space and time is organised) have a complementary explanation as 24 Impersonality Types (as the fundamental dimensions in which the physical nature of space and time is correspondingly organised).

And this latter interpretation I could see as relating directly to the vibration of a string in 24 dimensions!

When one reflects on the matter it makes little sense at the level of string reality to attempt to conceive of space and time as separate dimensions. Rather in varying ways they remain entangled with each other. So each dimension here actually relates to a distinctive configuration with respect to such entanglement!

However we cannot really grasp this while attempting to stick to a linear notion of dimension; rather in this context the true circular qualitative notion is more appropriate.

Thus conventional understanding of space and time in physics again reflects - from a qualitative perspective - merely 1-dimensional appreciation!

So without any knowledge at this stage of the Monster Group, I had already come to see, using a distinctive holistic mathematical approach, that 24 dimensional reality was of special significance and could see a clear connection as between this new interpretation of dimensions and string theory. In particular, it provided an explanation, through using the true qualitative interpretation, of how "higher" dimensions can be shown to conform to actual experience.

When interpreted in a linear (Euclidean) manner 24 dimensional space has an important connection with the Monster Group.

We can perhaps appreciate its significance by initially considering the efficient packing of (circular) coins in 2-dimensional space. Now if we attempt to lay out the coins in touching rows (with the position of coins in each row matching those above) we will not get the most efficient packing arrangement. The most efficient solution - thereby minimising the space between the coins - can in fact be achieved by arranging the coins in alternative fashion with the RH axis - drawn from the centre of the coin - in one row matching the LH axis of the coin immediately above (or below).

This most efficient arrangement implies that each coin will touch 6 others (whereas in the previous example only 4 would be touched).

Now one could imagine a somewhat similar arrangement for packing spherical objects (approximated by oranges) in 3-dimensional space. Here in fact the most efficient arrangement would lead to each object being touched by 12 others!

One could pursue such packing arrangements into higher dimensional space. Though one cannot properly visualise these arrangements, the mathematical properties entailed can be clearly articulated.

The importance of 24 in this context is that it lends itself - unlike other dimensions - to an amazingly efficient packing arrangement (i.e. with respect to 24-dimensional hyperspherical objects).

Indeed the most efficient arrangement possible (in work that also has strong connections with efficiency in sending communication signals) was provided by John Leech in what has become known as the Leech Lattice.

So Leech was able to demonstrate that with 24-dimensional space, the most efficient packing arrangement would entail that each hyperspherical circular object would be touched by 196,560 others.

Now if we consider for the moment the packing in two dimensions of (linear) square objects or in 3 dimensions cubes, this could be efficiently done without any space at all between the objects. And we could extend this thinking to hypercubes in 24 dimensions!

So seen from this perspective the efficient packing of hyperspheres represents the attempt to accommodate as closely as possible circular with linear quantitative notions (which is achieved when as little free dimensional space as possible is left over). In other words - though a most valuable exercise - it actually represents an attempt to reduce the quantitative dimensions as much as possible to the objects (thereby contained).

Now in fascinating reverse fashion, when one views the 24 Personality Types (representing the corresponding qualitative circular notion of dimensions) once again we have the attempt to accommodate linear and circular aspects. In other words the system of the 24 Personality Types, as outlined, is actually the attempt to harmonise as closely as possible the (linear) rational conscious with the (circular) intuitive unconscious.

So again with respect to the linear aspect we defined 8 "real" Personality Types (with an orientation primarily to conscious reality); then we also had 8 "imaginary" Personality Types (defined by a corresponding orientation to unconscious reality; finally we had eight complex types (defined primarily by the need to reconcile both conscious and unconscious).

However whereas in the quantitative case, we attempt to reduce any free dimensional space, in this qualitative treatment of dimensions, the goal is the opposite so as to free up as much space and time as possible (through minimising rigid attachment to quantitative phenomena).

The successful minimisation of object attachment in psycho spiritual terms, requires that all 24 Types be successfully harmonised in personality. In other words psychological integration requires identifying strongly not with just one Personality Type but in being able to recognise the equal contribution of all (with each providing a unique valid perspective on reality)!

Furthermore applying this to string reality we can equally say that 24-dimensional circular space provides a particularly suitable environment for successful physical vibrations with respect to both material and dimensional aspects of the string (in a manner that literally frees up space and time so as to facilitate such dynamic interaction).

Indeed, as I have mentioned in an earlier blog, a fascinating holistic mathematical explanation can be given for the importance of the 24th dimension (in qualitative mathematical terms).

Once again the nth dimension in this qualitative context is structurally related to its corresponding nth root (i.e. (1/n)th dimension in quantitative terms.

We can obtain any nth root of unity as:

Cos (2pi)/n + i Sin (2pi)/n where 2pi = 360 degrees (derived in turn from the fundamental Euler Identity, e^(2i*pi) = 1.

(Indeed this same Euler Identity plays an important role in the j-function to which unexpected numerical connections with the Monster Group have been demonstrated to exist resulting in the term "Monstrous Moonshine"!)

Now when we concentrate on the absolute magnitude of both Cos and Sin values the sum of such values when squared will always fall within a range between 1 and 2.

For example when n = 1, n = 2, n = 4 the extreme minimum value will result = 1.

In a psychological qualitative context this is due to the fact that these dimensions themselves represent respective extremes of pure rational linear understanding (n = 1), pure intuitive understanding (n = 2) and pure imaginary understanding (n = 4).

Then when n = 8, the squared total = 2. This in turn represents the fact that the 8th dimension represents another extreme expressing an attempted holistic balance as between both real and imaginary (conscious and unconscious) understanding.

This would suggest that the best radial position (allowing for the maximum in terms of balanced analytic and holistic understanding) would occur at 1.5 (as the mean of both extremes) and this indeed occurs when n = 24!

So with no reference whatsoever to the conventional mathematical treatment of dimensions, I had already discovered a compelling reason in holistic mathematical terms as to why 24-dimensional reality is important from a circular qualitative perspective! And once again such reality needs to be properly considered in a dynamic interactive manner where a very close relationship exists as between all 24 qualitative dimensions (that in turn bear a direct structural correspondence with the 24 quantitative roots of unity).

The Monster Group was first constructed by Bob Griess in 196,884 dimensions.

This in turn split into 3 subspaces (all with an intimate connection with 24)

So 98,304 + 300 + 98,280 = 196,884

The first of these numbers 98,304 = (2^12)*24

The second number 300 = the sum of the 1st 24 natural numbers (1 + 2 + 3 +....+ 24).

The final number 98,280 = 196,560/2 (i.e. half of the number that appeared in the Leech Lattice for touching hyperspheres).

Interestingly 98,280 = 98,304 - 24 (thus establishing a connection between 1st and
3rd numbers).

We have also already seen that 1^2 + 2^2 + 3^2 +....+ 24^2 = 70^2 (1)

Also 196,883 the minimum no. of dimensions in which the Monster can be constructed = 196,883 = 196,884 - 1.

And 196,883 = 47 * 59 * 71

So this last prime is just one greater than the number 70 appearing in (1).

Finally these 3 primes 47, 59 and 71, have a range of 24 (71 - 47), and can be further shown to be related to 24 in an interesting manner.

47 = (2 * 24) - 1

59 = (2.5 * 24) - 1

71 = (3 * 24) - 1

Monstrous Moonshine - clarification of dimensions

2010-04-14T04:54:00.000-07:00

Perhaps the single greatest problem with respect to conventional mathematical understanding relates to the standard treatment of dimensions, which is both highly reduced and unsatisfactory.

From my current perspective I see this problem as so fundamental that it is truly remarkable how it is persistently overlooked.

I generally describe the standard method of mathematics as the linear rational approach.

Now the line is literally 1-dimensional; so alternatively from a qualitative philosophical perspective, Conventional Mathematics is based on 1-dimensional rational understanding.

To see what this actually entails we can consider for a moment the natural number system.

Now, when we list the natural numbers i.e. 1, 2, 3, 4,...they are commonly defined - merely - with respect to their quantitative characteristics.

However implicit within such understanding is a (default) qualitative aspect (as dimension 1).

In other words implicitly each number as quantity is defined qualitatively with respect to the first dimension.

Thus in conventional terms whenever a number is raised to a dimension (other than 1) its ultimate value is defined with respect to the (default) 1st dimension.

For example we raise 2 to the power of 2 i.e. 2^2, the resultant value is given in reduced quantitative terms as 4 (i.e. 4^1).

Even as a child I could see that there was something wrong here. For example in geometrical terms we could represent 2^2 as a square (perhaps representing the area of a table with sides measured in metres). However quite clearly the 2-dimensional expression (4 square metres) is qualitatively different from the reduced 1-dimensional expression (i.e. 4 linear metres).

Thus whenever one raises a number to a dimension (or power) other than 1 - or alternatively multiply or divide two numbers - a qualitative as well as quantitative transformation takes place in the units involved.

However quite remarkably this qualitative transformation is conventionally ignored with numerical results expressed merely with respect to their reduced (i.e. 1-dimensional) quantitative characteristics.

So the first thing to clearly recognise here is this: even though - relative to the quantitative aspect of number - its corresponding dimension is correctly of a qualitative nature, a merely reduced (i.e. quantitative) notion of dimension is employed in Conventional Mathematics.

Though I did therefore recognise that there was a major problem here, it took me a long time to articulate - at least to my own satisfaction - the true mathematical nature of this alternative qualitative (i.e. dimensional) aspect of Mathematics.

Eventually I was to realise that this alternative system was of a circular - rather than linear - nature bearing a close structural relationship with the quantitative notion of number roots.

Of course, I do not for a moment question the great value of the conventional (quantitative) approach to Mathematics. My point is simply that there is - literally - an equally important qualitative dimension that is almost completely unrecognised.

So with respect to natural numbers we start by defining two separate systems.

In the first i.e. conventional approach, the quantitative aspect ranges over the natural numbers (with the qualitative dimensional aspect fixed as 1).

i.e. 1^1, 2^1, 3^1, 4^1,....

In the second i.e. holistic mathematical approach, the quantitative aspect now remains fixed as 1 (with the qualitative dimension ranging over the natural numbers).

i.e. 1^1, 1^2, 1^3, 1^4,....

Clearly from the conventional linear perspective, this second number system appears quite uninteresting (with the reduced quantitative value in each case = 1).

However from the correct qualitative perspective it is quite different.

Now the initial clue as to the nature of this new system is the realisation that in structural terms a unique inverse relationship exists as between the qualitative dimension and its corresponding root. So if D is the (qualitative) dimension is question, 1/D represents the corresponding (quantitative) root.

Therefore for example to find the form of the 2nd dimension we thereby raise in quantitative terms 1 to the power of 1/2.

In other words as regards the structural form of the relationship,1^2 (qualitatively) = 1^1/2 (quantitatively).

So in general 1^k (qualitatively) = 1^1/k(quantitatively)

We then use the expression derived from - what I term - the fundamental Euler Identity i.e. e^2ipi = 1,

So 1^(1/k) = cos(2pi/k) + i sin (2pi/k) where 2pi = 360 degrees.

So the structural form of dimension 2 is thereby given as

1^(1/2)= cos 180 + i sin 180 = - 1

So whereas 1^(1/2)= - 1 can be given a quantitative interpretation, 1^2 = - 1 can be given a corresponding qualitative (i.e. holistic mathematical) interpretation.

Now this number, - 1 lies on the unit circle in the complex plane.

Therefore the qualitative aspect of Mathematics is based ultimately on a circular - rather than linear - logical appreciation.

(Of course as I have continually stated, the most comprehensive form of Mathematics - which I term Radial Mathematics - combines the full interaction of both quantitative (linear) and qualitative (circular) understanding.
Put another way this entails recognition of an alternative binary system (linear = 1 and circular = 0), that potentially can encode in mathematical terms all transformation processes!)

Now perhaps we can appreciate clearly how Conventional Mathematics represents just a special case, where no distinction exists as between the (whole) qualitative dimension and its (part) quantitative root.

In other words when D = 1, both D and 1/D are thereby identical. So once again Conventional Mathematics represents the special case where no distinction can be made as between the quantitative and qualitative aspects of a relationship,

However with respect to all other dimensions a unique distinction does indeed exist as between (whole) dimensional numbers (as qualitative) and their reciprocal part expressions (as quantitative).

Put another way, to avoid the continual reduction of whole to part notions in Mathematics we must incorporate fully this (unrecognised) holistic aspect.

So we have seen that Conventional Mathematics is defined in linear terms by the 1st dimension.

However we have now uncovered the 2nd dimension (existing on the unit circle as - 1) which thereby has a circular logical meaning.

So in this context, we need now to holistically define in qualitative what is - 1.

Quite clearly we cannot just maintain the standard linear interpretation (which is merely quantitative!)

+ 1 in a holistic context relates to (unitary) form that is consciously posited in experience (in linear rational terms)

In corresponding terms - 1 in a holistic context relates to negation of such unitary form (which is the very means through which experience switches from conscious to unconscious appreciation).
So just as in physics when a particle particle fuses with its negative counterpart it leads to creation of material energy, likewise in psychological terms the dynamic negation (in unconscious terms) of what has been consciously posited, leads to the generation of spiritual intuitive energy.

In rational terms this leads to a new circular logical system that is inherently dynamic in nature and based on the complementarity of opposites.
Thus rather than the either/or approach of linear logic (where opposite polarities are unambiguously separated) we now employ a both/and circular logic (where opposite polarities are always paradoxical with precise meaning in any context depending on the arbitrary relative context employed.

Therefore the key implication of now incorporating this 2nd (with the 1st dimension) is the formal recognition that mathematical activity itself comprises both rational (conscious) and intuitive (unconscious) aspects.

So Mathematics is no longer formally defined in merely rational terms (which befits the 1st dimension) but now entails distinctive rational (conscious) and intuitive (unconscious) aspects which cannot be reduced in terms of each other. And once again this latter unconscious aspect, that is primarily of a holistic intuitive nature, is translated indirectly in rational terms through a circular logic of the complementarity of opposites (both + and -).

Holistic Mathematics represents the specialised development of the intuitive aspect of mathematical understanding just as Conventional Mathematics represents the specialised aspect of rational understanding.

However just as good mathematical work at the conventional level must be inspired by appropriate intuition, likewise in reverse manner, good mathematical work at the holistic level must preserve an appropriate manner of rational interpretation (in the form of mathematical symbols that are qualitatively understood).

Thus ultimately the two aspects are complementary (with enhanced appreciation of both arising from their mutual interpenetration).

Now a unique holistic mathematical interpretation is associated with every dimension (the structural nature of which is identical with its corresponding root).

Of particular relevance here is the holistic mathematical interpretation of the 4th dimension. Structurally this is identical with the 4th root of unity which is i.

The qualitative mathematical interpretation of i is of paramount significance and basically provides an indirect means of translating holistic appreciation (of an unconscious nature) through linear type symbols.
Put another way, it provides the rational means for mathematically incorporating qualitative understanding in an acceptable manner.

Indeed in this context whereas Conventional Mathematics represents the real aspect, Holistic Mathematics represents the corresponding imaginary aspect of a more comprehensive mathematical understanding.

So this comprehensive understanding i.e. Radial Mathematics incorporates in qualitative terms a complex rational approach (combining both real and imaginary aspects).

Once again, though it is of course true that Conventional Mathematics employs complex numbers (with real and imaginary parts) it does so within a merely reduced quantitative context.

Such an approach attempts to deal with complex quantities from a merely real qualitative perspective!

To sum up therefore the qualitative approach leads to the clear recognition that all mathematical understanding represents varying configurations with respect to real (analytic) and imaginary (holistic) understanding. And a key task of Holistic Mathematics is to provide the appropriate interpretation of each dimension (representing such complex configurations).

Now when we return to the Monster Group we can see that it is based on a merely reduced quantitative notion of dimension (as befits the Euclidean exploration of space).

However for every dimension that is quantitatively understood in this manner, there is a corresponding circular notion (existing on the unit circle in the complex plane) which can be given a coherent holistic qualitative interpretation.

Thus once again The Monster is at present defined analytically with respect to its component building blocks (as parts). However such understanding in itself does not facilitate the understanding of the qualitative relationship as between these parts in holistic terms. So ultimately an entirely different alternative mathematical approach has to be incorporated with present understanding.

In the next contribution I will suggest in the context of the Monster how this alternative understanding of dimensions can be incorporated!

Monstrous Moonshine - role of the unconscious

2010-04-12T07:07:00.000-07:00

As is well known the term "Monstrous Moonshine" was coined by John Conway with respect to an unsuspected mathematical connection demonstrated as between the Monster Group and the j-function (an important part of number theory related to modular functions).

The Monster Group itself appeared somewhat like the final piece in a massive jigsaw puzzle.

The whole area of Group Theory - that has now assumed great importance in Mathematics - arose out of the attempt to classify the different types of symmetrical objects that can arise.
A simple geometrical example of a symmetrical object is provided by the 3-dimensional cube which has six similar faces. Now there are many different ways of rotating the cube (48 in all) so that its symmetry remains unchanged. So the quest that arose in the 19th century was to classify the "atoms of symmetry" or basic building blocks as it were for all symmetrical objects, in what were termed finite simple groups.

In some respects this parallelled the similar quest to find the basic building blocks of the natural number system (i.e. the prime numbers).

Thus a great number of symmetrical objects are based on the different rotations possible with respect to various prime number permutations.

However other simple groups exist where the number of indivisible permutations do not correspond to prime numbers!

In the classification of the various simple groups that subsequently arose, most conveniently fell into four coherent families. However like separated islands, many exceptions remained (26 in all) that were lumped into a Sporadic Groups category, the largest of which is the Monster.

The total number of symmetries in this group - which cannot be decomposed into smaller sub-groups - is truly enormous and greater than all the quarks in the Universe. The lowest number of (non-trivial) dimensions in which they exist is 196,883 (47 * 59 * 71). It is not surprising therefore that this has been termed the Monster Group.

Then by surprise it was discovered that the number 196,883 and other larger dimensional numbers associated with the Monster bore a close relationship with coefficients of the j-function (a seemingly unrelated area of Mathematics).

It is in this connection that the term "Monstrous Moonshine" was coined by Conway.

"The stuff we were getting (i.e. the unexpected connections) was not supported by logical argument. It had the feeling of mysterious moonbeams lighting up dancing Irish leprechauns"

From my own holistic mathematical perspective as an "Irish leprechaun", I find this comment very interesting as it actually points to a deep unrecognised limitation of Conventional Mathematics.

The moon is commonly used to signify unconscious life. Indeed one remnant of this association is in the term lunacy (derived from the Latin word for the moon). So lunacy or madness would reflect a deep disorder with respect to the unconscious.

Then in the spiritual life "dim contemplation" is often used to refer to the faint passive light (like reflected moonlight) that greatly facilitates understanding of a universal holistic quality based on pure unconscious intuition.

As I have continually maintained there are in fact two equally important aspects in balanced (i.e. psychologically symmetrical) understanding of Mathematics.
One is the quantitative rational aspect which obtains its specialised expression in Conventional Mathematics. However the other - largely unrecognised - is the qualitative intuitive aspect which obtains its specialised expression in what I refer to as Holistic Mathematics.

As we will once again see the mathematical notion of Euclidean dimensions (on which the classification of the finite simple groups is based) is but a reduced rational notion based on merely linear logical concepts.

Indeed there is a strong paradox in this mathematical approach to symmetry based - as it is - on classification of irreducible building blocks (i.e. the finite simple groups).

Symmetry in any context entails the holistic relationship as between the various part aspects involved. However such holistic appreciation (literally of the whole context) should not be confused with successful analysis of the various parts. Though both aspects (whole and parts) are necessarily interrelated they should not be identified with each other. However, because of sole recognition of the quantitative aspect (suitable for partial analysis), in any relevant context Conventional Mathematics can only proceed by attempting to reduce the whole to the parts.

So the point I am making - which will be developed in further posts - is that true holistic understanding is needed to qualitatively appreciate what is entailed by "Monstrous Moonshine".

Another obvious paradox about the mathematical drive to understanding symmetry comes through recognition of the extremely important role played by many of its leading proponents who display classic symptoms of Asperger's Syndrome. For example John Conway, Simon Norton and Richard Borcherds would seem to fit readily into this category.

Indeed one could go further and argue that success in the highly specialised abstract task of finding the unique building blocks for all symmetrical objects has been greatly facilitated in this context through Asperger's Syndrome.

And I do not wish by this to question the merit of this achievement which has been truly outstanding; rather I am suggesting that from an overall comprehensive mathematical perspective it is quite unbalanced. Thus a much more rounded appreciation (requiring qualitative rather than quantitative mathematical notions) likewise needs to be provided.

Some Clarifications

2010-01-22T06:32:00.000-08:00

It is important to distinguish linear and circular, in holistic mathematical terms, from real and imaginary (which in many ways are closely related).

Initially, linear relates - in a scientific context - to understanding according to either/or logic. Here the crucial polar opposites, which condition all phenomenal experience of reality (internal/external, whole/part and form/emptiness) are clearly separated in interpretation. This then enables rational linkages to be made in an unambiguous sequential fashion. So Conventional Science is characterised by linear (1-dimensional) interpretation.

Such linear understanding is directly posited in a real manner. To posit in this dynamic context (which entails the holistic meaning of +) simply means to make conscious. Real in a holistic mathematical context means understanding that is of a direct conscious nature.

So Conventional Science is linear, corresponding to real rational interpretation in a positive manner.

However in dynamic experiential terms one can negate (as well as posit) phenomena. The very significance of such negation (-) is that it thereby opens the door to unconscious (intuitive) type appreciation.

Now, such intuitive appreciation strictly relates to the generation of spiritual energy (necessary to fuel all conscious dynamics). Just as the positive and negative polarities (in the form of matter and anti-matter) annihilate each other on contact, creating physical energy, likewise the positive and negative polarities of phenomenal understanding likewise "annihilate" each other creating spiritual (intuitive) energy.

Intuition is generated directly therefore in a non-rational manner (through negation of phenomenal constructs).

However when one then later rationally attempts to clarify the nature of such understanding it appears as two-dimensional. This new two-dimensional understanding is based on an alternative circular logical system of interpretation relating to the complementarity - and ultimate identity - of the fundamental polar opposites in experience.

Therefore though we separate polar opposites such as internal and external in (rational) linear interpretation, we recognise these same opposites as complementary, and ultimately identical, from an (intuitive) circular perspective.

So linear understanding is 1-dimensional; circular is 2-dimensional. The structure here of the 2nd dimension of understanding corresponds to the 2nd root in quantitative terms of 1 (i.e. 1 raised to the power of 1/2).

So there is an intimate link as between a qualitative dimension (as D) and its corresponding quantitative root (as 1/D).
Now when the dimension is 1, both D and 1/D are identical. This implies that with linear (1-dimensional) interpretation, no distinction is made as between quantitative and qualitative interpretation!

However for all other integer values of D, a clear distinction is preserved as between quantitative and qualitative (implying the combined use in interpretation of both linear and circular appreciation).

So both/and logic is inherently circular in nature. It is also, as we have seen 2-dimensional (when confined to two opposite poles).

The question then arises as to how such understanding might be converted into linear expression.

Thus to reduce the 2nd to the 1st dimension, we obtain the square root of the 2-dimensional expression (- 1). And this represents the precise holistic mathematical equivalent to the notion of an imaginary number in analytic terms.

So imaginary understanding represents circular type intuitive understanding, indirectly expressed in a linear manner. What this means in effect is that such intuitive understanding therby becomes embodied in conscious phenomena (that are understood in a linear manner).

Properly understood, all phenomena are complex in nature (with real and imaginary aspects).

The real aspect relates to direct conscious appreciation of a local kind.
However the imaginary aspect by contrast relates directly to unconscious appreciation of a holistic nature (indirectly mediated through the local object).

For example if one was to purchase a house, both aspects of understanding would be important. Real conscious appreciation would be used to identify and precisely locate the house. However such a purchase could also fulfil unconscious longings of a holistic kind. So one might well - in this context - refer to one's "dream" house - reflecting such unconscious meaning (which is mediated through the same object).

So real and imaginary, implicitly, always interact in understanding (even in the most rigorous scientific interpretation). When the focus is on analysis, the real aspect is of paramount significance. When the focus by contrast is on qualitative synthesis, the imaginary aspect is now critical. Unfortunately - as we have seen - because of the current solely conscious focus of scientific interpretation, this latter "imaginary" aspect is thereby reduced to the real.

Now if we go back to the definition of the string we can precisely illustrate what is involved.

The conventional (linear) real definition relates to the fact that the patterns of vibrations of the physical 1-dimensional string (representing the fundamental stuff of matter) create all possible matter particles.

However the new (linear) imaginary definition relates to the fact that the patterns of vibrations of the same string (representing the fundamental stuff of psychological interpretation) create all possible dimensions for such interpretation. And as physical and psychological notions are complementary in holistic terms, this equally implies that it is the vibrations of the second aspect of the string that create all possible physical dimensions of reality (i.e. spacetime).

So the string has in fact two intimately related aspects (that are real and imaginary with respect to each other). Indeed we could rightly refer to this as the complex string!

Now if we follow this a little further we will see why differing dimensions of qualitative interpretation are so important.

At the linear 1-dimensional level, only the real positive aspect of the string (as the fundamental "stuff" of all matter) will be recognised.

Then at the 2nd dimension of interpretation the real negative aspect now also becomes apparent as one recognises the role of both conscious (reason) and unconscious (intuition) in the dynamics of understanding. The means - in complementary physical terms - that the string is now given both a specific (local) and a holistic (collective) identity.

Then at the 4th dimension of interpretation, the imaginary aspect of the string unfolds. So at this level of appreciation, one understands the string in complex terms (with both real and imaginary aspects). The imaginary aspect here corresponds to the holisic (circular) identity of the string indirectly expressed in a linear manner.

So as we reach higher dimensions in qualitative terms, actual interpretation with respect to physical reality becomes ever more refined with new aspects of understanding unfolding (which remain hidden at lower dimensions). And it is not just psychological interpretation that changes in this manner but likewise physical reality itself (which directly corresponds to such interpretation).

Thus one could validly say: not alone do we have a uniquely different psychological interpretation of reality corresponding to each new "higher" dimension (the range of which is infinite) but equally we have a uniquely different physical reality!

However, to be even more precise, we should recognise that all these interpretative dimensions (with their corresponding physical realities) are but appearances, or - if one prefers - secondary realities.

True primary reality is non-phenomenal and ineffable, existing continually in the absolute present moment. Phenomenal reality (of matter and spacetime) always serves therefore as a relative expression of what is primary.

So a key insight once more is that the basic relationship as between part (as physical particles) and whole (as the dimensions to which they are relate) are as real to imaginary.

This brings us back to the the crucial point.

To properly investigate physical reality, we need to incorporate a complex rational approach. Such an approach combines real interpretation (suited for analytic interpretation of a quantitative kind) with imaginary interpretation (suited for corresponding holistic interpretation of a qualitative nature).

And because real and imaginary are so closely related to linear and circular, we can also express this complex rational approach in another way (which is highly revealing).

As we know the (analytic) binary system that is so important in IT is based on just two digits 1 and 0 that potentially can be used to encode all information.

However there also exists a corresponding (holistic) binary system that potentially can be used to encode all transformation processes.

So a comprehensive binary system - with potential power to encode with respect to both information and transformation - has both analytic (linear) and holistic (circular) components.

Once more, the fundamental relationship as between whole and part is based on the distinction as between linear and circular understanding (or alternatively real and imaginary).

Put another way the basic relationship as between matter and spacetime requires such a distinction!

The reason why present String Theory is finding it so difficult to properly incorporate spacetime in its approach is because of its inherently reductionist nature. In other ways it is still attempting to explain the whole (dimensions) in terms of the parts (strings) as physical particles. And as long as it persists with this approach it will continually fail to achieve a satisfactory resolution.

So the quest for a coherent TOE along the present reductionist lines is not attainable.

However a much greater prize might yet be won if it leads finally to incorporation of a true holistic aspect in physics based - literally - on recognition of its hidden qualitative dimensions.

With such a realisation, science could then be truly revolutionised in a manner greatly exceeding all previous developments.

Another Interesting Connection

2010-01-19T12:10:00.000-08:00

I will comment here on one more interesting connection that I can make as between the analytical findings of string theory and my parallel findings in holistic terms.

As the energy of a string increases, its vibration intensifies. However this process does not continue indefinitely. Beyond a certain point additional energy with respect to the string may lead - rather than a further increase in vibration - to the lenghtening of the string. And theoretically its length can then continue without limit!

Now there is a fascinating possible holistic counterpart which had puzzled me for many years.

Once again, this can initially be expressed with respect to psychological development. However - as always - because physical and psychological reality are complementary, it has - I believe - an intimate bearing on the holistic interpretation of strings.

As we have seen normal development in our Western culture largely plateaus with specialisation of 1-dimensional (linear) understanding. However where authentic contemplative development occurs, understanding can potentially reach several higher dimensions.

What happens therefore is than an intensification in spiritual energy takes place with respect to the psychological string (of 1-dimensional understanding) which leads to the generation of new interpretative models of reality (of an increasingly circular nature).

Now if this process continues in a pure contemplative direction, then we thereby have an intensification in spiritual energy greatly increasing vibration (in the dynamic interaction of dynamic polarities).

However there is a limit to such pure contemplative development - which - ultimately becomes unbalanced and unhealthy if one does not learn to return to dealing appropriately with the world of form.

So for development to continue in a sustainable fashion, at a certain point the desire for emptiness must give way to a realisation of the need to equally embrace the world of phenomenal form. And when this occurs there is a considerable restoration of linear type understanding.

Thus with mature radial development a spiritual contemplative does not remain detached from the world of form, but engages with it in an ever more committed manner (demanding specialised linear understanding).

Recently when dealing with the Riemann Hypothesis, I made the holistic connection as between the values of the zeta function (for negative whole integer values of s) and corresponding states of psychological development.

Interestingly the behaviour of the function in this respect supports my position here. In fact the value of the function for negative odd values of steadily diminishes up to s = - 5) and then gradually starts to increase, gradually at first, then more rapidly in a process that continues without limit.

The holistic connection I made here is that this provides evidence for the initial shrinkage with respect to linear understanding (at higher dimensions of understanding) before then changing in the opposite direction.

So I would see the holistic physical counterpart to this psychological finding relating to the behaviour of strings.

Remember strings naturally relate to the prime behaviour of matter. And as the Riemann Hypothesis is intimately connected with prime number behaviour, it therefore seems justifiable that the holistic interpretation of the Riemann Hypothesis would have deep relevance for the dynamic interpretation of strings.

Cornucopia of Riches

2010-01-19T01:27:00.001-08:00

I would like to briefly extend further here the holistic mathematical approach to the qualitative interpretation of strings that has the scope to open up new areas of refined understanding (not possible through the conventional approach).

We will start with the very notion of a p-brane. In conventional terms this is defined in a (merely) positive manner. So for example when we speak of a string, it is defined positively in 1-dimensional terms.

Now, in holistic mathematical terms every possible mathematical relationship that we define in conventional terms can be given a holistic (mathematical) interpretation with application to both physical and psychological reality (in complementary fashion).

This implies therefore that a string can be holistically defined with respect to the negative (as well as positive) 1st dimension

Well what does this all mean!

It perhaps is easier to approach the manner initially from a psychological context.

In holistic mathematical terms, to posit is to simply make conscious; to negate - by contrast - is to make unconscious. Now the important thing to remember is that - from a dynamic experiential perspective - both conscious and unconscious are necessarily involved in all understanding. Translated into scientific appreciation this entails that rational interpretation corresponds directly with the positive direction, whereby intuitive appreciation by contrast corresponds - relatively - with the negative.

Though conventional science formally ignores this point, mere rational explanation - of itself - does not constitute meaningful interpretation. Without corresponding ability to - literally - (intuitively) see what is implied by such rational linkages, understanding cannot arise.

Thus in the dynamic holistic interpretation of the string, the positive dimension (+ 1) corresponds directly with the rational aspect and the negative dimension (- 1) directly with the corresponding intuitive aspect of understanding respectively.

Thus for full appreciation of the string - from the interpretive standpoint - both the positive and negative linear understanding i.e. of the 1st dimension is required.

However in dynamic interactive terms both physical and psychological aspects of reality are necessarily complementary. Therefore the clear implication is that a proper definition of the string (in physical terms) likewise requires both the positive and negative signs with respect to the 1st dimension.

What this implies is that string cannot be properly understood, as if it is somehow independent of the rest of reality. In other words while maintaining a certain independence (with respect to the positive sign) this is likewise negated through establishing a necessary interdependence with reality (in overall terms).

Put another way - whereas the string as object is directly defined through the positive aspect - the string as dimension is defined through the negative aspect.

In this way, it can be perhaps appreciated that to give a string a proper dynamic interactive meaning that we must necessarily define it with respect to both positive and negative directions (of the 1st dimension).

So the standard linear interpretation of conventional science (which recognises solely the positive direction) is necessarily therefore of a reduced nature!

Actually this can be highly instructive in thereby arriving at the true meaning implied by a 0-brane (zero brane).

It is now postulated in a new development of string theory (Matrix Theory) that zero branes are in fact the most fundamental entities (from which all strings, membranes and higher branes are ultimately derived).

In dynamic experiential terms, the 0 dimension is best expressed through the equal positing and negating of any higher dimension.

So if we confine ourselves to strings, we can define a zero brane as the equal positing and negating of the 1st dimension.

So 1 - 1 = 0.

Again what does this actually imply?

Well! Once more it is perhaps easier to initially approach the matter from a psychological perspective.

In contemplative terms we cannot associate spirit with either rational (+) or intuitive (-) understanding separately. Rather the possibilities for pure spiritual awareness are maximised where an equal balance as between both is maintained.

We can see this best in geometrical terms as the reconciliation of line and circle (relating to linear and circular understanding). This reconciliation takes place at the midpoint of the circle (which is also the midpoint of the line diameter). So when this state is attained (whereby linear and circular understanding are now perfectly integrated in ineffable manner) one experiences reality in zero dimensional terms (which rightly represents a state of pure spiritual emptiness).

Now if we apply this understanding in complementary fashion to physical reality, it implies that the fundamental starting point of reality occurs when both linear and circular aspects of reality are indistinguishable. What this means in effect is that matter (as potentially existing) has yet no independent (linear) or equally interdependent (circular) capacity. Put another way matter yet has no existence as discernible form; likewise dimensions with respect to physical reality have no meaning.

Thus the zero brane actually implies a state of pure physical emptiness (as potential for existence of phenomenal form).

However this physical understanding of emptiness is indistinguishable in experience from the corresponding experience of spiritual emptiness (both relating to the present moment).

So one can only truly understand the origin of the physical universe as nothing (with potential for all form) through the corresponding realisation of the psychological destiny of this universe (as the realisation of form).

So properly recognised both of these aspects comprise the pure mystical experience of spiritual union.

Once again a merely linear interpretation leads to a reductionist notion of zero-branes (whereby the illusion is maintained that material notions can somehow grasp the essence of nothingness).

String Theory - a new myth for our age

2010-01-18T02:09:00.000-08:00

It is amazing how in life if one waits long enough that things tend to come full circle. And this is true of the development of science itself.

In earlier times, science was much influenced by the intrusion of confused holistic notions of a spiritual nature that significantly impeded proper analysis of physical behaviour.

For example in the middle ages, largely to serve theological notions, the Earth was believed to be the centre of the solar system. Therefore when Galileo supported an alternative viewpoint (based on objective empirical analysis), he was forced to detract his opinions so as to preserve religious orthodoxy.

So it is only in the last 400 years or so - largely as a result of the monumental contribution by Newton - that science has successfully differentiated itself from subjective beliefs based directly, or indirectly, on religious notions.

Indeed one could argue that the prevalence of the notion of the aether (which Einstein finally discarded in the early 20th century), represented an important remnant of such confused understanding i.e. where a holistic spiritual notion influences the nature of analytic investigation.

In this context it is interesting to compare an earlier story of the nature of creation from that emerging from modern science (in the form of string theory).

In the Christian Bible we are told in Genesis (the first book of the Old Testament) that God created the world in 6 days. Now as an acceptable scientific account, amenable to modern tastes, this account is greatly lacking from an analytic perspective. However it is really designed to convey a deeper qualitative (holistic) meaning.

6 was well known in the ancient world as the first perfect number. It is no accident therefore that God creates the world in 6 days as this is designed to convey the deeper significance of His creation as perfect.

So the story of creation in the Bible properly constitutes a myth (though admittedly of a significant nature). And it is the very nature of myths that explanatory symbols that are used are designed to convey holistic - rather than strict analytic - meaning.

Now when we contrast the Bible story with a modern attempt - such as string theory - to explain the nature of creation, we are presented with the opposite extreme.

In other words - precisely because of its analytic exactness - science has now managed to divest itself almost completely of holistic qualitative notions.
However this had led to a significant new problem (which is not properly recognised).

Analysis by its very nature is suited to partial investigation i.e. where part of a system can be investigated (in isolation from the whole).

Thus when we attempt to investigate the whole framework (as distinct from its parts), scientific analysis loses its effectiveness. Indeed - by its very nature - it can only attempt this task in a reductionist fashion i.e. by attempting to reduce the whole to its partial elements.

Because of the ambitious designs of string theory to obtain a whole explanation of reality through formulatioon of a Theory of Everything (TOE) it is especially open to this reductionist charge.

Thus what we are seeing now with string theory is precisely the opposite problem that affected earlier mythical accounts of creation.

Whereas the design of these accounts was to to principally provide a holistically satisfying explanation (conveying qualitative meaning), the design of a modern scientific account - such as string theory - is to provide an analytically satisfying explanation (of a quantitative nature).

However just as the former accounts were greatly lacking from a satisfactory analytic perspective, string theory is also greatly lacking from a satisfactory holistic perspective.

In other words string theory as it stands cannot provide any coherent intuitive account of the nature of its concepts and this is a truly major issue that has not been properly faced.

Even with respect to standard mathematics, mere rational explanation does not constitute meaningful understanding.

For example if I rationally demonstrate the Pythagorean Theorem to a student, actual understanding does not materialise until the appropriate intuitive connections are likewise made. Only then can then student literally "see" what the Theorem is about.
Fortunately however, at the macro level of appreciation, intuitions are designed to conform to accepted common sense notions of reality (dictated by linear type understanding).

Likewise with respect to string theory, mere rational explanation does not constitute meaningful understanding. However the big difference here is that the intuitions necessary for confirming intuition to literally "see" what is meant by the theoretical explanations are not actually provided by linear type understanding.
And string theory - as presently constituted - has no means of providing this qualitative intuitive component.

So for example when Edward Witten announces that string theory fundamentally operates in 11 - rather than 10 dimensions - this strictly represents an abstract mathematical finding (based on linear understanding). However it lacks any true qualitative coherence with the holistic nature of physical reality (which at the required level of appreciation requires going substantially beyond 1-dimensional interpretation).

At a deeper level, the implication is that science has now come full circle.

Initially - as we have seen - analytic investigation was greatly impeded through being confused with rigid holistic notions (based on religious beliefs).

Especially since the arrival of Newton, science has made enormous strides with respect to analytic understanding through successfully differentiating itself from confused holistic notions. And of course this aspect of science will continue to flourish for some considerable time to come!

However in the attempt to deal with universal type explanations (pertaining properly to the whole of reality), the limitations of analytic science are exposed.

Therefore I would see that we are at the dawning of a great new stage in scientific understanding where we can now at last begin to deal with universal holistic type issues in a successful manner. However this will require an entirely different qualitative approach that is utterly distinct from present appreciation.

Then eventually when this holistic side of science has been properly appropriated, a truly comprehensive approach to science can then at last unfold (combining specialised analytic and holistic type appreciation).

Prime Time (and Prime Space)

2010-01-16T08:55:00.000-08:00

An important part of my integral approach to science relates to the fundamental role that holistic appreciation of the various number types can play.

In this holistic classification of numbers, I refer firstly to - what I call - the original numbers (1 and 0), then 2 and the other prime numbers, followed by the natural numbers (and integers) then the rationals, followed by the irrationals (algebraic and transcendental), the imaginary and complex and the transfinite.

Of course all of these are well recognised in standard analytic terms. However their true holistic significance has been greatly missed.

In what follows we will be confining ourselves to the holistic interpretation of the original, prime and natural nos.

The original numbers 1 and 0 have their holistic equivalents as oneness and nothingness respectively. Now, we must realise that we are here referring to dynamic - rather than static - notions. Nothingness in this context thereby refers to a physical (and spiritual) void as nothing (in phenomenal terms) which equally implies the potential for all phenomena of form. In the Eastern tradition this notion of nothingness is generally referred to as emptiness.

The critical point is that the ultimate origin (and destiny) of the Universe is often described in terms of the holistic interpretation of the original numbers.

Physical creation emerges from nothingness (holistic 0); likewise its ultimate goal and fulfilment is generally referred to in terms of spiritual oneness (holistic 1).

And as these notions relate to a continual present moment, they both ultimately coincide. So pure contemplative attainment can equally be referred to as spiritual emptiness (of nothingness) or alternatively as spiritual oneness.

What is fascinating is that the very symbols we use to refer to 1 and 0 respectively are the line and the circle. Thus in Western culture - where rational (linear) notions hold greater sway - the ultimate state is generally referred to as oneness; by contrast in Eastern traditions - with a greater recognition of intuitive (circular) notions, the ultimate state - by contrast - is generally referred to as a void (or emptiness).

So the pure ultimate state of the universe - either in terms of its physical origins - or alternatively its psychological destiny (in spirit) relates to a present moment (continually renewed) that is best referred to as the coincidence of oneness and nothingness.

In other words in this ultimate state, form (as independent phenomena) cannot be distinguished from emptiness (as the interdependence of all phenomena).

Thus using geometrical notions, we have here the pure coincidence of both linear and circular notions.

We can then learn a great deal about the possible origins of the physical universe with reference to corresponding development of the psychological universe in a baby infant.

In the beginning the (linear) conscious is not differentiated from (circular) unconscious; therefore neither the experience of phenomena (nor equally of dimensions of space and time) is possible.

Now because of the complementarity of physical and psychological, this likewise means that with the Universe, in the beginning (linear) form through distinctive phenomena, is not distinguishable from (circular) dimensions through interdependent links as between phenomena. So in the original state, one cannot make a distinction as between matter and spacetime (as neither enjoys an independent identity).

In psychological terms, experience of reality commences through the initial separation of conscious and unconscious. This then leads to the birth of duality (i.e. holistic 2).

In corresponding physical terms, actual phenomena come into being through the separation of linear and circular aspects of reality. Here, the most primitive phenomena attain a certain limited independence (while retaining an extremely high level overall of confused interdependence). This likewise creates a starting dimensional context (of space and time) for emerging phenomena.

So the birth of space and time in the Universe (as the holistic context for phenomena) coincides with the birth of distinct phenomena.

However, it is vital to realise that both phenomenal and dimensional characteristics of reality always have a merely relative validity, emerging necessarily from a continual absolute present moment.

Now the earliest infant experience of form (and dimensions) is - literally - of an extremely primitive kind. And in the understanding of "primitive", remarkably we find the true holistic qualitative context for appreciation of prime numbers.

With primitive infant understanding, conscious and unconscious - though not entirely confused - still remain greatly enmeshed with each other.

So, the primitive experience of an object (as conscious) immediately becomes identified with its holistic dimensional background (as unconscious). In this way the infant is unable to maintain a detached view of any object. Rather, because dimensions become immediately confused with objects, a quick collapse takes place in phenomenal events. So initially, experience of phenomena is extremely short-lived with no object constancy possible. This only materialises when the conscious achieves a greater degree of separation from the unconscious!

Again it is remarkably similar in physical terms. The first existence of objects is literally - in holistic mathematical terms - of a prime nature; likewise the dimensional context in which these objects exist is also prime. So the first phenomena exist in prime spacetime (with both aspects inextricably intertwined).

So the very essence of prime behaviour in qualitative terms is that both object phenomena and related dimensions (of space and time) remain tightly interwoven with each other. Therefore phenomena cannot maintain any constancy (which would require a stable background of space and time).

Now all this is deeply relevant to the reality we associate with string theory, for in a precise holistic mathematical sense, strings comprise the prime constituents (of both matter and dimensions).

If we look briefly now at the conventional understanding of prime numbers in mathematics it is quite revealing, for once again only the linear component is evident.

In other words prime numbers are looked on as the basic building blocks of the natural numbers. In like manner in physical terms, strings (as prime components) are looked on as the basic building blocks of all matter.

However there is an equal circular aspect to prime numbers. This relates to the fact that in their general distributional behaviour, they display a remarkable interdependence with the natural nos.

So from one valid analytic (linear) perspective, the prime nos. are the most independent structures, serving as the building blocks of all natural numbers.

However from the equally valid holistic (circular) perspective, the prime nos. are the most interdependent, being totally dependent on the natural nos. for their precise distribution. So as well as each prime number anjoying indidividual structure (as independent), overall prime numbers share a general state (as interdependent)

This then implies that there is a key ingredient missing from the conventional physical perspective on strings. Whereas the linear perspective is very prominent (i.e. with strings as structure forming the basic building blocks of matter) there is no equal appreciation of their important circular aspect as state. Here the overall behaviour of strings (relating directly to their dimensional characteristics) intimately depends on natural phenomena (to which they are related).

Thus a two-way relationship therefore necessarily exists as between strings (as prime constituents of matter) and dimensions (in the general relationship of strings to matter). Whereas it may be valid to say that (manifest) physical particles result from unique dynamic vibrations of a string, the actual dimensional properties of strings (in general) result in turn from their overall relationship to such physical particles.

All along I have maintained that one cannot divorce the (physical) properties of nature from the (psychological) dimensions employed for its interpretation.

As I have repeatedly stated, conventional science is based on 1-dimensional interpretation and is thereby - literally - linear in nature.

And in this view the basic ingredient or "stuff" of everything in nature is a string which again is - literally - envisaged as a 1-dimensional line.

So we can see here a remarkable correspondence as between physical matter and its corresponding psychological interpretation both of which are 1-dimensional!

Thus the starting point for a more balanced dynamic treatment (with respect to both physical understanding and its psychological mode of interpretation) is to explicitly combine linear and circular aspects.

And this is done through combining the present conventional understanding of strings (linear) with several higher qualitative dimensions of understanding (circular) thereby providing many unique interpretations of string reality.

Brane Power

2010-01-15T10:59:00.000-08:00

Another of the findings of M-theory is the realisation that 1-dimensional strings are not the only fundamental "objects". Rather a whole host of extended objects in higher dimensions also exist called p-branes, where p = no. of dimensions involved.

Also in 11 dimensional M-theory, where we have 10 spatial dimensions, the number of possible branes ranges from 0 - 9 dimensions.

In parallel fashion, I have long been aware of a similar situation with respect to its holistic counterpart, where we allow for differing dimensions of interpretation with respect to string concepts.

Because this understanding initially arose with respect to clarification of the stages of psychological development, I will discuss my findings in this context.

When understanding of a particular dimensional level unfolds, strictly one should refer to it as the default level of understanding.

So for example, when the rational linear stages properly unfold in experience, one thereby obtains default 1-dimensional understanding.

Then when - as with authentic contemplative progress - development evolves to the 2nd dimension, one now acquires the default understanding of that dimension.

Then as futher development unfolds, a new perspective unfolds by which its links with earlier stages can take place. Therefore as development passes beyond 2-dimensions, one now can establish an enhanced appreciation of earlier 1-dimensional understanding (from the perspective of the now attained higher 2nd dimension).

So strictly speaking, not alone can we have (default) 1-dimensional interpretation, but also a whole range of enhanced interpretations from the perspective of higher dimensions attained.

This was a revelation to me at the time, as I now realised that linear understanding was rooted - not only in its own dimension - but also in each of the higher dimensions (excluding the highest yet attained).

Now it would not be strictly possible for someone who has reached the 8th dimension - which coincides with pure contemplative awareness - to obtain an enhanced appreciation of earlier 1-dimensional interpretation (from the 8th dimension). While this (default) dimension is being consolidated in experience, one is unable to obtain an enhanced perspective of earlier dimensions! Put another way, while the default dimensional experience is unfolding, it remains somewhat closed (to other dimensions). However with further development it becomes more open. Thus the enhanced appreciation of the 1st dimension (from a higher dimension) reflects the fact that it thereby becomes temporarily constrained to this 1st dimension (providing that dimension with its enriched perspective).

Just to clarify what is involved here, I will briefly illustrate - with respect to the stages of development - the distinction as between (default) 1-dimensional interpretation (from the perspective of the 1st dimension) and enhanced 1-dimensional interpretation (from the perspective of the 2nd dimension).

In the former default case, one views the stages of development as unfolding essentially in a linear fashion (so that for example the lower necessarily unfolds before the higher stage).

However, in the enhanced view (from the perspective of the 2nd dimension) one realises that there are in fact two aspects (internal and external) involved. So one now envisages development taking place in linear terms with respect to (internal) stages relating to psychological development of the self and (external) physical stages relating to understanding of the world.

The value of this enhanced perspective is that in dynamic interaction both aspects will then operate - relatively - opposite to each other, introducing a circular paradoxical element to development. By contrast, in the default linear interpretation of stages, no such subtlety is involved.

Now this important finding - regarding a whole range of enhanced 1-dimensional interpretations (from the perspective of higher dimensions) - seems to me to equate pretty well with the corresponding discovery of extended objects within string theory.

Thus the 2-brane in string theory is the analytic correspondent of enhanced linear interpretation (from the 2nd dimension) in holistic terms.

Likewise, for example, a 4-brane would be the analytic correspondent of enhanced linear interpretation (from the perspective of the 4th dimension)!

However the holistic approach I have outlined would suggest that there are several more elements (that perhaps have not been generally recognised in M-theory).

For example, just as one can obtain enhanced appreciation in psychological terms of a lower dimension of understanding (from the perspective of a higher) likewise one can obtain a diminished appreciation of a higher dimension (from the perspective of a lower).

Now, if we were to try and suggest what this might imply for the analytic theory of strings, perhaps it would mean that what is - in truth - a p-brane (where p >1) would appear as a string!

Also in holistic terms, we can get an enhanced perspective of any lower dimension from a higher e.g. enhanced appreciation of 2-dimensional interpretation from the perspective - say - of the 4th. Likewise we can get a diminished appreciation of any higher dimension from a lower e.g. diminished interpretation of the 4th - say - from the perspective of the 2nd.

What the correspondent of this for analytic strings would be is hard to say! Perhaps it would imply in some way lower brane objects being transformed into higher (in the first case) and in the 2nd higher brane objects being transformed into lower.

So for example a 2 brane object could possibly transform into a 4-brane and in turn in other circumstances a 4-brane into a 2-brane!

Recently, there has been speculation regarding another class of "object" referred to as a 0-brane (wittingly referred to as a no-braner) which it is hoped will throw light on the fundamental origins of the Universe. However this for me is where M-theory - or more properly in this case Matrix theory - finally strays completely outside of its interpretative limits. Defining an object ultimately in terms of a point represents but another idealised conception that has no direct relationship to physical reality.

The corresponding psychological counterpart to a 0-brane would relate to understanding that is 0-dimensional i.e. non-dimensional.

However such understanding represents purely contemplative (empty) understanding of reality that is totally ineffable (and thereby incomprehensible in rational terms).
Though such experience indeed represents the zenith of qualitative type appreciation of reality it has no direct correspondent in linear understanding.

Once again whereas 1 represents linear understanding (amenable to reason), 0 by contrast represents circular understanding (amenable to intuition).

So the paradox for present physics is that true appreciation of 0-brane reality in physics, can only come through the extreme expression of qualitative - rather than quantitative - type interpretation.

However - whatever the specific merit of such ideas - string theorists need to accept that the current language of interpretation is largely meaningless from any coherent philosophical perspective.

I accept that - when interpreted in a 1-dimensional manner - the findings of string and now M-theory, portray reality accurately as it appears (through the interpretative lens of conventional science).

However the key point again is that this interpretation model is quite inadequate to properly describe the dynamic nature of reality involved.

Therefore, in my opinion, an enormous amount of interpretative decoding (of a holistic kind) is required to convert string theory into an acceptable physical model of reality.

Dualities

2010-01-15T08:19:00.000-08:00

One the key ideas that Witten proposed in 1995 was the idea of dualities.

Thus the five recognised string theories - which initially were believed to be separate - were in fact connected by duality transformations, so that by using the appropriate key one theory could be mapped on to another (revealing precious new insights in the process).

Now interestingly my own holistic approach is based directly on polar dualities i.e. internal/external, whole/part and form/emptiness and immanence/transcendence.

In the Type 0 approach - which concurs with conventional scientific interpretation - these dualities are considered as separate; however the four other Types revealing increasing levels of complementarity (and ultimate identity) as between these opposites.

So in the (conventional) analytic approach the five recognised string theories provide unique perspectives on the ultimate nature of reality. Then through appropriate transformations they can be seen as in fact equivalent expressions.

Likewise from the holistic qualitative aspect each of the five dimensional Types (singled out as especially relevant) provide unique perspectives for attempted interpretation of this reality. Also through appropriate transformations we can show that they are equivalent (though using a different language of expression).

However the really interesting area is making closer links as between the analytic and holistic perspectives in the hope of ultimately applying intuitively satisfying explanations for all key string notions.

Now the first of the holistic models (Type 0) in fact represents the conventional scientific interpretation.

However though this Type is best in terms of detailed rational interpretation (of an analytic kind) it is least satisfactory in terms of providing the appropriate qualitative understanding to enable its concepts to resonate intuitively with (appropriate) psychological experience of reality.

So there is a huge role - of which my own limited attempts represent but a beginning - in terms of transforming present Type 0 understanding through the interpretative lenses of the other four Types.

Thus in a more complete understanding we should be able to immediately provide the equivalent qualitative interpretations for the Type 0 findings of conventional science through each of the four other models.

And of course the process would work both ways as one could move from qualitative appreciation at higher dimensions to its implications for the Type 0 level. In this way, highly creative new thinking could be better incorporated in standard string theory development.

However the key point must be repeated continually until its significance is eventually absorbed.

The time has clearly come for science to greatly enlarge its possible scope by at last admitting its - hidden - holistic dimension.

Whereas the standard conventional approach is directly suited for quantitative analysis of reality, the complementary holistic approach is best suited for qualitative synthesis of that same realty.

Like two blades of a scissors both aspects need to be incorporated in a more comprehensive vision of science (which I term radial).

However before such a comprehensive approach can prosper, considerable attention must be first given to development of the (neglected) holistic aspect.

What is M-Theory?

2010-01-15T04:05:00.000-08:00

What is now known as "the second superstring revolution" took place in 1995 following publication of a paper by its acknowledged leading proponent, Edward Witten.

Up to this, a well recognised problem with string theory was the lack of one unique theory. In fact five - seemingly - separate theories existed referred to as type I, types IIA and IIB and two heterotic (hybrid) string theories.

The hope at the time was that just one of these would eventually win out to qualify as the long sought for TOE.

However Witten was to surprise the string community by demonstrating very close links as between all the existing theories and postulated that they were in fact part of on an even grander synthesis (M-theory).

He also argued that this overarching theory would find expression in an 11 - rather than 10 - dimensional framework.

Now of course, I am by no means qualified to deal with the analytic issues involved.

My purpose is rather to point to the integral (holistic) scientific significance of such developments.

From the holistic perspective, no single unique interpretation of reality can be given. Rather - as we have seen - a distinct interpretation can be associated with each number (as dimension) so that potentially an infinite no. of possible interpretations exist.

However, we have argued that in practice only a small finite number of such explanations are especially relevant.

Indeed remarkably - though perhaps coincidental - we can distinguish 5 such interpretative models (for which I have already given a simple mathematical justification in "The Number 24").

These five models coincide with 1, 2, 4, 8, and 24-dimensional holistic interpretation respectively.

Even before I heard anything regarding M-theory, I had referred to the first 4, as Type 0 (this 1st type - which represents linear understanding - is inherently unsuited to true integral appreciation; hence the use of 0), Type 1, Type 2 and Type 3 Integration respectively. Only recently - largely through making connections with string theory - have I given proper consideration to the 5th which could be designated as Type 4. So even my very designation of the various holistic interpretations bears similarity with the corresponding analytic treatment in string theory!

Each of the integral interpretations therefore offers a certain unique qualitative perspective with respect to the "same" reality. Though the higher dimensional perspectives are by their very nature more refined in holistic terms, they also suffer in that they are likewise further removed from analytic type consideration.

So there is an inevitable "Uncertainty Principle" at work. In other words the more accurately defined one's qualitative holistic interpretation of reality, the less amenable it becomes to any specific kind of phenomenal analysis.

Of course the reverse is true with respect to conventional string theories. Therefore the more exact such analytic theories become in terms of successfully explaining the nature of physical reality, the less amenable they become to any kind of holistic intuitive understanding.
Indeed this is already readily apparent with string theorists quite unable to provide in qualitative terms an adequate philosophical perspective of what their world view entails.

This therefore would lead one to be very sceptical regarding the true nature of this so-called M-theory. In many ways the M most accurately stands for mirage for it is pointing to a reality (which greatly exceeds the explanatory limits of the standard 1-dimensional model of interpretation).

Even the finding that we now need 11 rather than 10 dimensions points to this fact. As I have stated in corresponding holistic terms, the need for an extra dimension implies an admission that the current level of psychological intuition is inadequate to appropriate the nature of the reality investigated.
So likewise in analytic terms the need for 11 dimensions implies that some key element is missing in terms of conventional scientific understanding.

Reality at its most fundamental physical level requires going below the Planck length.

However though enormously important in terms of any ultimate understanding, eventually it becomes impossible to meaningfully deal with such reality in a detached objective manner.

Likewise from the holistic perspective at very high levels of spiritual integration, dynamic interaction with respect to phenomena becomes so rapid that they no longer even appear to arise in experience.

And this is the essential point! Science - which always requires a degree of detachment from the environment - can only deal with phenomenal appearances (from both analytic and holistic perspectives).

So in the end science (with respect to either aspect) can only deal with secondary phenomenal appearances and thereby cannot directly probe essential reality.

We can only approximate experience of such reality (that ultimately is experientially attained in complete unity and emptiness). However appearances will always remain necessary in dynamically approximating reality and thereby realising the true secret of this all important game (which we know as life itself).

New Explanation of Dimensions

2010-01-14T08:19:00.000-08:00

The basic rationale for this new understanding of dimensions - which applies equally to both physical and psychological reality - is that is based initially on the two key polarities sets which necessarily underlie all phenomenal experience.

Now these two polarity sets can be referred to as horizontal and vertical with respect to each other.

The horizontal relates to the basic distinction as between internal (subjective) and external (objective); the vertical then relates to the further critical distinction - in any context - as between whole (general) and part (specific).

Now we can represent these polarities as four equidistant coordinate points on the circle of unit radius with internal and external the end points along the horizontal line and whole and part end points along the vertical line respectively (dividing the circle).

In physical terms, the dimensions of space and time arise through the interaction of these polarities; likewise in psychological terms, experience of space and time arises through a similar interaction.

Conventional notions of the 4 dimensions (3-space, 1-time) stem directly from the linear manner of interpretation adopted. What essentially happens is that whole and part are combined (in a static reduced manner) and then considered as external to the observer. This then results in the typical 3-dimensional spatial rigidity which objects thereby possess. The final 4th dimension is thereby separated from the spatial and interpreted as time.

So standard linear interpretation is associated with an asymmetrical understanding of space and time.

Now when we adopt a 2-dimensional interpretation, space and time now appear as paradoxical and symmetrical having complementary positive and negative aspects with respect to both space and time.
This represents (integral) holistic as opposed to (differentiated) analytic understanding.

So space and time have now both two dimensions (that are positive and negative with respect to each other)

When we adopt a 4-dimensional interpretation, space and time again appear as paradoxical and symmetrical having complementary real and imaginary as well as positive and negative aspects with respect to both space and time.

This qualitative interpretation corresponds directly with the 4 roots of unity.

So both space and time now have both real and imaginary aspects (in positive and negative terms); now this might suggest 8 - rather than 4 - dimensions. However it has to be remembered that real and imaginary keep switching in dynamic interactive terms (and positive and negative). So when time is real, space is imaginary; likewise when space is now real, time is imaginary; also the positing of one aspect with respect to either space or time requires the negation of the other and vice versa.

From an equivalent perspective, in dynamic interactive terms, the whole (in any context) is imaginary with respect to the part and vice versa; and internal and external are positive and negative with respect to each other.

When we adopt an 8-dimensional interpretation we generate four additional dimensions that have a complex structure (with equal real and imaginary aspects).

As explained in previous blogs these provide the holistic mathematical structure of the four forces (in both physical and psychological terms).

All other (qualitative) dimensions are obtained with reference to the corresponding root structure of 1 (in quantitative terms).

So to obtain for example the qualitative structure of the 100th dimension, we obtain the corresponding 100th root of 1 (in a quantitative manner).

Now these additional roots will all have a complex structure (with real and imaginary aspects).

This entails a corresponding complex structure to the nature of space and time expressing in turn a highly dynamic configuration of the four fundamental polarities (internal, external, whole and part).

In corresponding psychological terms this entails a highly refined interaction as between conscious (rational) and unconscious (intuitive) aspects of experience.

One final point! it is only in the 1-dimensional linear interpretation that an attempt is made to separate both the linear and circular notion of dimension (which is very evident in terms of the standard explanation of dimensions in string theory).
This in turn reflects the fact that conventional interpretation is based formally on merely (conscious) rational notions of understanding.

In all other dimensional interpretations, it is assumed that both linear and circular notions necessarily interact with each other (as in turn does conscious and unconscious).

Linear and Circular Dimensions

2010-01-14T04:05:00.000-08:00

I fully accept that enormous mathematical strides have been made with respect to the development of string theory. In many cases these resonate strongly with physical understanding of the Universe apparently offering the hope that many key difficulties (such as the integration of Quantum Mechanics with the Theory Of Relativity) can at last be successfully achieved.

However from a philosophical perspective I would find that the attempted explanation of many of the fundamental concepts of string theory (such as the nature of a "string" and the meaning of "dimensions) remains very confused.

Once again I would see the reason for this problem as representing the limits of what can be achieved through conventional scientific method (based on the linear method). Strictly speaking the philosophical limitations of this approach had already been exposed through the paradoxical findings of Quantum Mechanics (though in practice largely ignored). Though clearly these philosophical reservations would also naturally apply to string theory, we witness instead a renewed hope in the reductionist approach (in its ability to finally arrive at a TOE). It is almost as if string theorists believe that by going beyond quantum mechanics they can somehow bypass all its mysteries by rooting it in even more fundamental reductionist interpretation of nature.

My basic contention with respect to science (which of course includes physics) generally is that it needs to be widened to include three (rather the present one) areas.
So as stated on several occasions previously we need:

1) Conventional (analytic) science - representing the present understanding of science - geared directly to quantitative type interpretation of reality;

2) Integral (holistic) science - representing the approach pursued on this blog - geared directly to qualitative type appreciation. From this perspective the integral scientific approach is especially geared to deal with philosophical issues with respect to string theory concepts;

3) Radial (comprehensive) science - representing the mature interaction of specialised analytic and holistic understanding. Needless to say there is little possibility of this 3rd aspect being developed at present due to the - almost - complete lack of recognition of the integral (holistic) approach!

Using the integral scientific approach, I have already suggested that the very notion of a "string" needs to be considerably widened to include both an analytic and holistic aspect.

Whereas the analytic aspect concurs with present definition, the holistic aspect relates directly to the psychological manner of interpretation (of physical data and concepts). So rather than just one default interpretation being available for physical reality i.e. 1-dimensional rational, potentially an infinite number are available (though in practice I would expect that a relatively small number would suffice).

The fascinating thing is that the holistic aspect literally relates to the qualitative mathematical notion of dimension and it is only in this context that "dimensions" in string theory can be properly interpreted.

Put another way, present developments with respect to the analytic understanding of string theory are being severely impaired due to lack of corresponding lack of development with respect to the qualitative notion of dimensions.

Indeed I believe that the key insight with respect to string theory (akin to the equivalence principle of General Relativity) is much more revolutionary than at present realised! In other words it really is pointing to the need for two distinctive aspects of scientific understanding i.e. analytic and holistic respectively, as the fundamental relationship as between object phenomena and dimensions (of space and time) requires this distinction.

In this context it is fascinating to examine how the notion of "dimensions" is dealt with in present string theory.

Conventional macro understanding of 4-dimensions (3 of space and 1 of time) simply reflects (in qualitative philosophical terms) the linear approach.

Thus though 4-dimensions are recognised (in quantitative terms) this qualitatively reflects 1-dimensional understanding. In fact this directly concurs with treating time as 1-dimensional.

Now when speaking of 10 - or now perhaps 11 - string theorists have an obvious problem in dealing with "higher" dimensions (which do not conform to linear interpretation).

So in the linear (qualitative) approach time is still treated as 1-dimensional with the remaining dimensions spatial.

However - say in a 10-dimensional framework - we thereby have 9 spatial dimensions (only 3 of which are recognised according to linear understanding).

So the standard representation of the remaining 6 is to treat them as rolled up circular dimensions that are so small (as to be invisible to normal linear detection).

Now there is fascinating counterpart issue with respect to the holistic dimensions of psychological understanding.

Moving qualitatively into higher dimensions in this context entails developing an increasingly refined intuitive type appreciation of reality (based on circular notions).

In other words as contemplative development proceeds - rather than viewing the physical world in a detached objective manner - one increasingly realises the complementary nature of all physical and psychological understanding (which qualitatively represents circular understanding).

Now one who remains firmly rooted in linear understanding - quite literally - will not be able to intuitively "see" (from the perspective of the "higher" contemplative perspective). Rather interpretation will be reduced to the understanding of the lower 1st dimension. So again, the development in these new circular dimensions of spiritual "seeing" will thereby remain so limited as to be invisible.

What this therefore entails is that the very explanation that is offered for the extra "dimensions" in string theory in itself reflects a merely linear manner of interpretation (which is quite inadequate for the kind of reality involved).

Thus when one qualitatively attempts to interpret these extra "dimensions" - not from the conventional linear standpoint but rather - from the intuitively refined higher dimensions of understanding, an entirely distinctive perspective unfolds.

So from the perspective of 2-dimensional understanding, one already understands the notion of a dimension in an inherently dynamic interactive manner (where both linear and circular aspects interact).

It cannot be stated too strongly that the (conventional) scientific manner of interpreting relationships, reflects merely - in qualitative terms - the linear (1-dimensional) approach.

Thus when we adopt the alternative understanding of a different dimension, an uniquely distinct interpretation unfolds.

The linear approach is inadequate to properly convey the nature of "higher" dimensions. Therefore we need to incorporate the holistic integral aspect (based on higher dimensional appreciation) into overall appreciation.

Once again the notion of dimension that is appropriate to string theory has very little to do with conventional notions of space and time.

Remarkably however it is directly related to the mathematical notion of dimension (when given a coherent qualitative interpretation).

And structurally this qualitative notion of dimension is directly related to the corresponding quantitative notion of its corresponding root.

Other Interesting Connections

2010-01-13T09:13:00.000-08:00

It might be helpful to further extend the complementary connections as between the two notions of strings i.e. analytic and holistic, respectively.

From the analytic perspective it is readily admitted that, as the string can vibrate in an infinite number of ways (with each unique vibration corresponding to a distinct matter particle) that potentially therefore we can generate an infinite no. of particles.

Now so far, only a relatively small finite number of distinct particles have been found. The actual explanation for this fact is due to the fact that actual particles discovered correspond to extremely low eneregy levels of vibration of the string. These likewise entails less energy (corresponding to relatively low masses).

New particles entailing greater vibration of the string would require extremely high amounts of energy (well beyond present technological capacity).

The reassuring fact therefore is that in terms of "normal" energy states we need only concern ourselves with a relatively small group of particles (corresponding to low vibrations of the string).

There is a fascinating counterpart in terms of the holistic notion of the string (corresponding to qualitative interpretation of reality).

Potentially as there are an infinite no. of possible (whole) dimensions, therefore we can likewise provide an infinite no. of possible interpretations for any physical event (with each interpretation relating to a unique dimensional number).

So the situation here might therefore seem hopeless for science (due an infinite range of possible interpretations).

However, given the present level of human evolution, we are capable of exploring only a very limited range of these dimensional interpretations.

As we have seen Conventional Science is still firmly rooted in the lowest psychological energy state (associated with 1-dimensional interpretation).

Though higher energy states are associated with authentic contemplative development, it would be quite rare - in terms of corresponding cognitive development - for it to grow significantly beyond the lower dimensions.

And I have persistently argued that these lower dimensions - certainly up to 8 - are of special significance in terms of integral appreciation.

It is possible however to describe in general terms what happens to interpretation at much higher (spiritual) energy states. Here phenomenal understanding becomes so transparent that it no longer even appears to arise in experience (with a person continually absorbed in a deep contemplative state of union).

And just as mass becomes significantly greater in physical terms (at extremely high energy levels) in reverse complementary fashion, the psychological experience of mass becomes significantly lower (as phenomena acquire a purely transparent existence).

So just as the Planck length sets a certain limit to the possibilities of phenomenal investigation (in physical terms), we have in a sense a corresponding Planck length (in psychological terms) which sets a certain limit to the possibilities of rational type interpretation. What this means in effect is that the dynamic interaction of polarities becomes so fluid that it is no longer possible to explicitly preserve any rigid linear element!
So even though the range of possible dimensional interpretations of reality is potentially infinite, explicitly in rational terms, meaningful interpretation could only take place within a - relatively small - finite number of these dimensions.

Thus in both physical and psychological terms, though phenomenal interactions still take place, they remain at a merely implicit level.
So from the physical perspective we can no longer experimentally investigate such interactions; equally from the psychological perspective we can no longer formally give them rational interpretation (even in an extremely refined intuitive manner).

However the actual unification of both aspects of strings (analytic and holistic) paradoxically can only take place through penetrating those very regions (below the Planck length).

In other words, ultimate unification with respect to strings comes neither at the level of detached (physical) verification nor (psychological) rational understanding. Rather it takes place in pure ineffable mystery (as the ever present source and goal of all life, both physical and psychological).

More on Dimensions in String Theory

2010-01-13T02:03:00.000-08:00

As stated so frequently in these blogs, the standard linear (1-dimensional) approach employed in Conventional Science is unable to maintain the key qualitative distinction (in any appropriate context) as between whole and part. It can only operate therefore by essentially reducing whole to part notions (in real conscious terms) or alternatively part to whole notions (in a likewise real conscious manner).

Fundamental therefore to maintaining the proper qualitative distinction as between both aspects is the recognition that unconscious (circular) and conscious (linear) notions must be formally incorporated with each other in interpretation.

As we have seen, whereas in a direct sense the conscious aspect of understanding relates to (real) rational interpretation (corresponding to either/or logic)), the unconscious relates directly to corresponding intuitive recognition (relating to complementary both/and logic).

However this latter intuitive aspect can be given an indirect rational interpretation in - what in precise holistic mathematical fashion is - an imaginary fashion.

So Conventional Science deals essentially with "reality" that is amenable to real rational interpretation. Though it must be readily admitted that complex mathematical notions are also widely used, these are restricted to merely quantitative type analysis. Thus the qualitative method of interpretation remains decidedly real.

However to deal adequately with "reality" requires that we incorporate likewise in qualitative terms an approach that is both real and imaginary.
Here essentially in terms of rational understanding we give expression to two distinct types of logical understanding. The real aspect requires standard either/or logic (based on the clear separation of polar opposites in experience); the imaginary aspect uses an alternative both/and logic (based on the dynamic complementarity of these same polar opposites in experience).

Thus, whereas the analytic aspect of scientific understanding (suited to differentiated interpretation) is directly associated with the real aspect, the corresponding holistic aspect (suited to integral interpretaton) is associated with the imaginary aspect.

So in qualitative terms, science needs to incorporate a complex rational approach (combining both real and imaginary aspects of understanding).

Now this is all deeply relevant to the very understanding of dimension in science.
Current accepted understanding - which is heavily based on solely real interpretation - is but a reduced notion that does not represent its true meaning.

Quite remarkably the true notion of dimension (with respect to both physical and psychological reality) corresponds exactly with the holistic mathematical notion (when interpreted in its qualitative fashion).

The very relationship as between object phenomena and dimensions relates in turn to the fundamental relationship of whole and part.

Now to properly preserve the true qualitative distinction as between whole and part (thereby avoiding reductionism) we must understand both aspects as real and imaginary with respect to each other.

In the very dynamics of experience, our actual appreciation of object phenomena and related dimensions (of space and time) requires the interaction of conscious and unconscious. Whereas the appreciation of objects relates directly to conscious recognition, dimensional appreciation (in providing a whole background context) is by contrast provided directly by the unconscious. Unfortunately, subsequently in rational interpretation, dimensions are simply reduced in terms of the parts (so that the whole - in any context - tends to be viewed as the sum of its quantitative parts).

To appreciate the true relationship of objects and dimensions, we need to consider the simple mathematical relationship whereby 1 is raised to a rational fractional power (say 1/4).

So if start with the base (object) number quantity as 1 and the dimension (i.e. power or exponent) as 1/4.

Now both of these numbers are linear (i.e. lie on the straight line).

However when we allow for interaction as between object quantity and dimensional quantity (through raising the former to the latter) a remarkable transformation takes place.

So when we raise 1 to 1/4 we obtain the result i (the square root of -1).

Now this new number is not linear (i.e. does not lie on the straight line). However it is circular (i.e. lies on the circle of unit radius drawn in the complex plane).

Thus whenever we raise a rational number to a rational fractional dimension (provided that the expression cannot be reduced further) a transformation takes place from linear to circular (in quantitative terms).

(In reverse manner when we now raise 1 to the circular dimension i, once again a transformation takes place this time from circular to linear.
So the result .207879... is linear lying on the straight line number scale).

Now what is completely missed in conventional mathematical understanding is that when we now raise the same number to a whole number dimension that a transformation likewise takes place from linear to circular (but now in qualitative terms).

So in holistic mathematical terms when we raise 1 to 4 i.e. express the number 1 with respect to 4-dimensional interpretation, the result is imaginary (corresponding to circular understanding).

Likewise when we interpret any relationship in an integer dimension (other than 1) a transformation takes place in the nature of understanding from linear to circular.

Now the very reason why this important finding is missed in conventional interpretation it solely operates in a linear manner (and thereby unable to preserve the uniqueness of qualitative distinctions).

So when we raise 1 to 4, though the qualitative nature of the relationship has now changed, in quantitative terms the result remains the same. In other words 1 (raised to the power of 4) is indistinguishable from 1 (raised to the power of 1) in standard linear interpretation. This is why - in a very precise manner - standard mathematical - and by extension standard scientific - interpretation is literally 1-dimensional!

So the true nature of object phenomena to dimensions (both with respect to physical reality and corresponding psychological interpretation) is as linear to circular which in rational translation is as real to imaginary.

Thus when "reality" is properly understood in dynamic interactive terms it is necessarily of a complex nature (with interacting real and imaginary components).

Now you may wonder where this all gets us with respect to string theory!

In the last blog, I made the important observation - that properly speaking - the very notion of a string can be given two distinct interpretations (which ultimately are complementary).

Once again in standard physical terms the 1-dimensional string is looked on as the fundamental building block of matter. The dynamic vibrations of the string then lead to the physical particles that comprise reality.

However there is equally another (unrecognised) qualitative psychological notion of a string. Here the 1-dimensional string (i.e. linear type understanding) is looked on as the basic building block for understanding of reality. However when such understanding dynamically vibrates it can then generate all the "higher" dimensions of understanding that comprise our potential interpretation of reality.

In this context in terms of our second notion of a string, conventional scientific appreciation (which is decidedly 1-dimensional in nature) simply represents the lowest possible energy state of the string (where very little dynamic vibration or interaction) is allowed.

One of the big "eureka" moments for me with respect to my complementary approach to stings came when reading Brian Greene's excellent book "The Elegant Universe".

On P. 144, in Figure 6.2 he gives some examples of vibrating strings, where just as the different vibrational patterns of a violin string give rise to different musical notes, the different patterns of a fundamental string give rise to different mass and force charges.

Now the three diagrams presented in this figure resemble pretty exactly the geometrical interpretation of the 2, 4 and 8 roots of unity respectively (which in holistic mathematical terms corresponds with the 2, 4 and 8 dimensions of scientific interpretation). And these are the very dimensions of interpretation that I have long advocated as essential in the integral scientific approach!

In "The Fabric of the Cosmos" Greene then repeats this diagram on p.357, now including an extra diagram (geometrically resembing the 16 roots of unity). And recently I have started to qualitatively incorporate 16-dimensional understanding in my interpretative model!

So the point once again is that in a more comprehensive understanding, the notion of a string needs to be defined in two complementary ways (corresponding to two distinct logical systems).

Once again the existing definition corresponds to attempted analytic understanding of physical reality (i.e. the real aspect of interpretation).

However the alternative definition, that I am providing, corresponds - by contrast - to attempted holistic understanding of both physical and psychological reality as complementary (i.e. the imaginary aspect of interpretation).

Adn then this is the crucial point!

Just as the relationship as between object phenomena to dimensions is correctly complex (as real to imaginary) then this clearly implies that we cannot hope to properly incorporate dimensions in string theory without likewise incorporating this alternative holistic understanding.

So to sum up, in qualitative holistic mathematical terms:

Conventional (Analytic) Science comprises the real part of (scientific) interpretation;

Integral (Holistic) Science comprises the imaginary part;

Radial (Comprehensive) Science combines both parts in a complex rational approach.

The Problem With A String

2010-01-12T06:54:00.000-08:00

I have always had philosophical problems with the definition of a string. Indeed my interest in the area originally arose out of an attempt to redefine some of its key concepts in a way that I - at least - could find intuitively satisfying.

This relates to a deeper problem with the whole area in that theorists and practitioners seem unable to bridge the gap as between mathematically satisfying results and a coherent explanation of the physical concepts involved (that can evoke any acceptable resonance with the manner in which people understand reality).

So at the moment string theory resembles a kind of secret code that undoubtedly can generate many exciting mathematical results (suggestive of deep physical connections). However without a much more coherent philosophical decoding of the nature of its concepts it cannot really qualify as a true physical explanation of reality.

So the problem with strings not alone relates to the present difficulties in finding ways of experimentally testing its mathematical findings; perhaps more significantly, it also relates to even greater difficulties in providing an intuitively satisfying means of the nature of its concepts.

From my own perspective the root of this problem is easy enough to state. Once again conventional scientific method is based formally on mere 1-dimensional interpretation of reality (which conforms best with everyday intuitive notions of the nature of macro reality).

However when we probe at the extreme margins of reality, 1-dimensional interpretation breaks down as a means of adequately conveying the nature of reality investigated.

What is needed in effect are progressively higher dimensional interpretations that incorporate both (analytic) linear and (holistic) circular notions in ever more refined fashion.

So my own investigations with respect to Integral Science are designed with a view to providing - in basic fashion - this much needed holistic dimension. It is not properly geared to developing quantitative theories (that can in principle be tested experimentally). Rather it is designed to provide the holistic framework within which concepts - such as in string theory - can be given an appropriate philosophical explanation (that intuitively concurs with the new understanding adopted).

It must be apparent to conventional physicists that the attempted definition of a string is unsatisfactory.

We are led to believe that a string is an object with one spatial dimension of length (but lacking any other dimensions).
Though it might indeed be useful in mathematical terms to characterise a string in such a fashion, clearly it does not stand up in philosophical terms.
In conventional terms we cannot conceive of an object that is 1-dimensional. Once again though in geometrical terms we can represent a line that is 1-dimensional, this is simply an idealisation that does not strictly hold in physical terms. So any line that we actually try to represent must inevitably possess a degree of width (for its phenomenal identity to be established).

So if a string has length (without any thickness) how can its phenomenal identity be established? Also what about time in such a characterisation? Surely, again a string to have phenomenal meaning must have some sort of existence in time!

Also to even talk about a string is to attempt to give it a distinct identity (which is a merely linear notion of interpretation). And this presupposes that we can somehow place the string in a pre-established dimensional framework (of space and time). But even physicists will accept that realistically speaking, that the string must in some way embody these very dimensions (as contained in the same entity).

The best way of moving towards a more satisfactory explanation is initially with reference to the conventional piece of string. As we know we can stretch a string out in a linear manner. We can also if we wish fold it around in circular fashion. Or we can equally bend and twist the string in variety of configurations (in both linear and circular fashion).

So in philosophical terms the basic starting point for a coherent definition of strings is the realisation that we must combine both linear (analytic) and circular (holistic) notions. From a psychological perspective this entails that interpretation must necessarily entail meaning pertaining to both conscious and unconscious. Whereas in direct sense the conscious (analytic) aspect corresponds to rational understanding, the unconscious (holistic) properly relates to intuitive appreciation.

Therefore in corresponding physical fashion we must conceive of reality itself entailing the interaction of two distinct modes that are real and imaginary with respect to each other.

Therefore a string (and indeed combined group of strings) must be rooted in a dynamically conceived complex notion of reality (with - relatively - switching, real and imaginary aspects)
In this way both the (whole) dimensional and (part) object aspects of the strings can be viewed in dynamic interacting real and imaginary fashion (which continually altenate).

However once again this dynamic view of physical reality must be complemented by an equally dynamic interpretative mode (that combines real and imaginary aspects).

The huge problem with present attempted understanding is that is confined formally to merely real modes of interpretation. This is the very reason therefore why physicists finds it so difficult to properly incorporate dimensions within string theory.

To do this satisfactorily will require a radical new scientific approach that combines both conventional (linear) and new integral (circular) modes of interpretation.

Ramanujan and Strings

2010-01-12T03:17:00.000-08:00

In the previous post, I demonstrated the interesting fact that with respect to root quantitative values (and corresponding dimensional qualitative values) that both n = 8 and n = 24 play a special role.

Here with respect to the relationship cos(360/n) + i sin(360/n) the square of sum of (absolute) real and imaginary values = 2, for n = 8, and = 1.5, for n = 24 (where all possible values range from 1 to 2).

So the former represents the maximum that can be attained whereas the latter represents the mean average (of all possible values).

Let us remind ourselves of the context out of which this has arisen!

Earlier I explained how the original Jungian Theory of Personality Types was based on 8 distinctive types.

Then in the holistic mathematical extension of this approach (based on permutations the four key coordinate points on the complex plane) 24 distinctive types could be defined.

One way of looking on each Personality Type is that it provides a unique way of configuring characteristic experience of the 4 dimensions of space and time.

So in this dynamic interactive sense, a dimension is now understood as representing - not separate entities - but rather uniquely distinctive arrangements of all 4 dimensions.

However the holistic mathematical approach that yields fundamental "Personality Types" (in psychological terms) equally yields corresponding "Impersonality Types" (in physical terms).

Now this is where the connection with string reality can be made. As - properly speaking - at this level, physical phenomena are inseparable from the dimensions with which they interact, we cannot view the notion of dimension (as at the macro level) as independent entities but rather as a configuration of - as yet - entangled embryonic dimensions (where matter cannot be properly distinguished from its holistic background environment).

Some years ago I was reading "Hyperspace" by Michio Kaku. In a Chapter on "Superstrings" to refers to Ramanujan Functions and how the number 24 repeatedly appears in his work. Also the number 24 is also the origin of the mysterious cancellations occurring in string theory (thus enabling its consistency).

So in string theory each of the 24 modes in the Ramanujan function corresponds to a physical vibration of the string.
We could of course equally say that each of the 24 modes (with respect to Personality Types) correspond to a unique psychological vibration relating to a distinct personality type.

Now the very rationale of Integral - as opposed to Conventional - Science is that we can always establish a complementary psychological interpretation for every physical notion (or alternatively physical for every psychological notion). So the very means of achieving integration in experience derives from successfully establishing at the deepest levels such complementary links!

So we now are we seeing a way here through which the very notion of "a string" can be given both complementary physical and psychological interpretations.

So once again in physical terms each of the 24 modes (in the Ramanujan Function) corresponds to a unique vibration of the string!

Now in corresponding psychological fashion, we see that each of the 24 modes corresponds to a unique vibration of the string!

Thus whereas in the former, a string is designed to represent a basic physical entity, in the latter a string now represents a basic means of psychological interpretation of reality!

The significance of this is quite momentous.

Conventional Science is conducted within just one standard mode of interpretation i.e. the default linear (1-dimensional mode). The very essence of this mode is that qualitative is reduced to quantitative meaning.

However once we allow for true qualitative - as opposed to strict quantitative - meaning we then see that parallel psychological holistic meanings exist for all analytically defined concepts (in physical terms).

When the Ramanujan function is generalised, the number 24 is replaced by the number 8. Now physicists customarily include two additional dimensions.

Thus earlier superstring theories were based on 24 + 2 = 26 dimensions, and in generalised terms 8 + 2 = 10 dimensions.

Now Kaku gives an explanation at the end of the Chapter for the inclusion of these two additional dimensions. Basically the extra dimensions are included to allow for symmetrical (as well as asymmetrical) states so that the 8 and 24 respectively relate to the asymmetrical vibrations of the string.

In corresponding psychological terms we could equally include two additional symmetrical states (relating to attainment of a pure state of spiritual unity and alternatively pure state of nothingness respectively).

Recent developments with M-theory would suggest 11 dimensions (9 + 2) as the ultimate arrangement.

Interestingly, perhaps the best known Personality System (The Enneagram) is based on 9 distinctive types. So this would complement the new physical understanding.

However this is not the important issue. What I am anxious to demonstrate here is that we need to use a distinctive notion of dimension (in string theory) as the dynamic configuration of - yet - not properly formed dimensions (where matter particles cannot be clearly distinguished from these dimensions).

Secondly the very notion of a string needs to be significantly extended to include complementary physical and psychological aspects.

Whereas in physical terms the vibration of a string results uniquely in distinctive matter particles, in corresponding psychological terms the vibration of a string results uniquely in distinctive qualitative interpretations of this reality. Linear interpretation (i.e. 1-dimensional) simply characterises the lowest mode of vibration of the psychological string (where dynamic interaction as between psychological and physical aspects of reality is ignored). This then results in the illusion that we can somehow "objectively" understand physically reality in a comprehensive manner.

As Conventional Science by its very nature simply reduces the qualitative to the quantitative aspect, it can never (by itself) unravel the true mystery of string reality. Indeed, even the very hope in an eventual quantitative TOE reveals its reductionist nature.

The Number 24

2010-01-10T12:30:00.000-08:00

24 in its own right is a fascinating number.

Firstly it represents all the permutations of 4 (containing 4 elements) that can be made from 4 which is 4 * 3 * 2 * 1.

However there is another very interesting property that if we add up the squares of the consecutive numbers from 1 to 24 (inclusive) that the result 4900 will be the square of an exact whole number i.e (70).
This is the only case known where the sum of squares of successive natural numbers is equal to the square of another whole number!

Interestingly the sum of 1 + 2 + 3 +....+ 24 = 300, while the sum of the prime numbers between 1 and 24 = 100!

24 - as we shall see - plays a key role in Ramanujan functions, which in turn provides a direct link to the number of dimensions in one of the earlier string theories.

24 also plays a crucial role in the search for the Monster Group (the largest known symmetrical object) which again provides a direct connection with string theory.

As we know, if the proper divisors of a perfect number are summed, the total is the same number. So for example 1, 2 and 3 are the proper divisors of 6 and the sume of 1 + 2 + 3 = 6. So the ratio here of proper divisors to the number = 1

Now 24 is not a perfect number. However interestingly the ratio of its proper divisors to the number = 1.5.

Another highly interesting relationship to 24 that equally results in 1.5 is worth commenting on in greater length.

I mentioned in an earlier contribution that I had found a mathematical justification as to why the 1, 2, 4 and 8 dimensions play an especially important role in an overall integral approach.

Once again the holistic qualitative notion of dimension is directly linked to the quantitative structure of its corresponding root (of unity).
So 1 qualitatively is related directly to the 1st root (of unity) which is identical. Thus in effect, no clear distinction exists as between qualitative and quantitative interpretation with respect to the 1st dimension.

To obtain the nth root of any number (which in qualitative terms corresponds with its nth dimension) we simply obtain cos (360/n) + i sin (360/n)

The value of both cos and sin will range in absolute terms as between 0 and 1.

Then when we add both terms and square the result, the absolute value will range between 1 and 2.

Now when the root and dimension is 1 the value is 1 + i (0) = 1.

So here we have a maximum in terms of the real part representing the absolute dominance of rational understanding according to one dimensional i.e. linear interpretation. Equally we have a minimum in terms of the square of sum of sin and cos values = 1.

When root and dimension is 2 the value is cos 180 + i sin 180 = - 1 + i (0) = - 1.
Now we have a maximum in terms of the negative real part. This reflects in turn the dominance of negative linear (i.e. intuitive) understanding.
Equally again the square of the absolute sum of sin and cos values = 1 (which again is a minimum in terms of what can be achieved).

When root and dimension is 4 the value is cos 90 + i sin 90 = 0 + i (1) = i.
Now we have a maximum in terms of the positive imaginary part. This reflects in turn the dominance of imaginary linear (i.e. the indirect rational expression of intuitive) understanding.
Equally again the square of the absolute sum of cos and sin values = 1 (which again is a minimum in terms of what can be achieved).

Finally when root and dimension is 8, the value is cos 45 + i sin 45 = 1/(sq. root of 2) + i/(sq. root of 2).
Now we have a balanced equality in terms of real and imaginary parts. This reflects in turn the harmonisation of rational (conscious) and intuitive (unconscious) understanding.
Equally again the square of the absolute sum of cos and sin values = 2 (which now is a maximum in terms of what can be achieved, representing the limit in terms of pure contemplative integration of experience).

So what we see is that 1, 2, 4 and 8 dimensional interpretation represent specialised extremes in terms of rational (real), intuitive, rational (imaginary) and empty understanding (as the equality of conscious and unconscious) respectively.

One could perhaps suggest therefore that the ideal radial balance (in terms of overall understanding) should come at the midpoint between the two extreme values i.e. 1 and 2 (for the square of absolute sum of cos and sin values).

Now this in fact happens when n = 24.

Therefore from this perspective, 24-dimensional understanding represents the ideal in terms of overall balanced understanding.

Coming back to the previous post on Personality Types, this would suggest that for truly balanced radial understanding of reality, one would need to successfully combine attributes of all 24 Personality Types.

So when we look at development from a radial perspective , the key goal is to successfully differentiate (to a degree) traits associated with each Personality Type before then integrating all in a simultaneous manner.

Personality Types

2010-01-10T04:30:00.000-08:00

During the early 80's, perhaps the deepest influence on my thinking at the time was Jung.
Happily the college library where I worked in Dublin had the "Complete Works of C.G. Jung" in stock which I regularly read in preference to standard Economics material.

It was the volume on "Personality Types" that chiefly engaged my attention.
In his theory Jung distinguished as between two attitudes (introversion and extraversion) and four functions (sense, intuition, thinking and feeling).

What fascinated me from an early stage however was the degree to which Jungian concepts were amenable to the holistic mathematical interpretation that I had been developing for some years.

For example, Jung classified his functions as rational (thinking and feeling) and irrational (sense and intuition).

Now rational and irrational are also used in a mathematical context with respect to differing number types.

Since the early 70's I had been using a four-dimensional classification (which in geometrical terms yielded a four quadrant approach) as the basis for the most fundamental interpretation of the key polar aspects underlying all phenomenal reality. Now these four dimensions corresponded in structural terms with the four roots of unity yielding - in relative terms - two real and two imaginary dimensions with respect to all experience with in each case one positive (+) and one negative (-) respectively.

It readily occurred to me that extraversion and introversion would constitute - in holistic mathematical terms - polarities that are positive and negative with respect to each other.
In then dawned on me that the functions (that Jung was addressing) were in fact - correctly - real and imaginary (with respect to each other). In other words when a function - say - sense was dominant in personality it was thereby given a real identity i.e. in conscious terms. Therefore the other function - in this case intuition - remained largely imaginary i.e. unconscious. So the true - and vital - relationship as between conscious and unconscious was as real to imaginary.

So I could now see clearly that once we accept actual experience of reality as necessarily involving both interacting conscious and unconscious aspects, that then we need to move, in holistic mathematical terms, to a complex rational approach (entailing both real and imaginary components).

And as I have been stressing all along the root problem with Conventional Science is that it is confined in formal terms solely to real rational interpretation.

So once again Integral Science relates to the corresponding specialised development of the imaginary aspect and finally Radial Science to the mature interaction of both (specialised) real and imaginary interpretation.

Now the Jungian classification of Personality Types (with 8 main categories) was later extended to 16 in the Myers-Briggs system. Here two additional attitudes (perception and judgement) were included.

However though in general I personally found the Myers-Briggs revised typology very useful, I could see a fundamental difficulty (based initially on examination of my own personality). In other words classification in the Myers-Briggs is basically made on the either/or treatment of opposite characteristics. However for certain personalities, it was my belief that both/and classification had also to be included.

For example, the Myers-Briggs system would classify a person as either (mainly) extrovert or introvert! In a primary sense I felt that certain personalities - e.g. with pronounced mystical tendencies - should be more rightly classified as centrovert (though in a secondary sense exhibiting either extravert or introvert tendencies). So in my own case though in a secondary sense I had no problem in being classified as introvert, in a primary sense I saw myself as centrovert.

Likewise, this approach would apply to the other Myers-Briggs categories. So again instead of approaching reality through sense or intuition, one could primarily operate on the basis of their mutual fusion in seeking what is truly essential with respect to reality (though once again at a secondary level sense or intuition might appear relatively dominant).

Again in relation to thinking and feeling (the 3rd category on the Myers-Briggs) one could primarily have a personality based on the volitional function (of will); finally in relation to the 4th category (perception or judgement) again one could correspond to the mixed type primarily seeking to achieve a middle way (through discernment).

So the four primary categories I define for the new mixed type are centrovert (introvert/extravert), essence (sense/intuition), volition (thinking/feeling) and finally discernment (perception/judgement). Again for the truly mixed type the primary categories would be CEVD (i.e. centrovert, essence, volition and discernment) with a secondary manifestation relating to the four either/or categories (introvert/extravert, sense/intuition, thinking/feeling and perception/judgement)

Then based on the four (holistic) dimensions I then sought to define each Personality Type as a unique permutation (or configuration) of these dimensions.

Now if for example we represent these dimensions as a.b.c and d then 24 different permutations are possible (corresponding to 24 basic Personality Types).

16 of these will correspond pretty exactly with the corresponding 16 types in the Myers-Briggs system. However this leaves 8 additional types corresponding to the new mixed group.

So now instead of 2 broad groups (of 8) we have 3.

In holistic mathematical terms the first are the "real" types (based on predominance of conscious understanding).

The second are the "imaginary" types (based on predominance of unconscious understanding).

The third are "complex" types (based on predominance of what is central to both conscious and unconscious). Paradoxically when integration is successfully achieved, these are also the most "simple" (due to a high degree of spiritual attainment).

Now one intriguing aspect of this modification of Personality Types is that it is essentially based on a new holistic mathematical definition of dimension (where a dimension is now defined as a certain permutation of - what in separate polarised terms are understood as - the four dimensions of space and time).

It also struck me at the time that this interpretation - applied to fundamental classification of psychological interpretation - could equally be applied in complementary fashion to the fundamental nature of physical reality.

In other words just as we can classify basic Personality Types (through which structural interpretation of reality takes place in psychological terms) equally we can classify basic Impersonality Types (as the fundamental structures of material reality).

This was then to lead later to an intimate connection with the dimensions of String Theory (to be dealt with in a future post).

16 Dimensions

2010-01-01T06:39:00.000-08:00

As stated here on many occasions my integral approach to development is largely based on holistic mathematical consideration of 1-dimensional, 2-dimensional, 4-dimensional and 8-dimensional interpretation in development.

However it may help a little to say a little in this context about my Stages of Development approach.

Overall I include 7 Bands in development (each of which comprises 3 main Levels, comprised of a unique holistic mathematical structure). So we have 21 main Levels in all.

The first Lower Band is largely concerned with overcoming confusion with respect to the three central polarities in experience. L3 (Lower 3) relates to initial differentiation of the diagonal polarities (form and emptiness); L2 then entails additional differentiation with respect to the vertical polarities (whole and part); finally L1 relates to further differentiation with respect to horizontal polarities (internal and external).

Then with preliminary successful differentiation having taken place with respect to the three polarities the next Band is concerned with specialisation of linear (1-dimensional) interpretation (where the three polarities are clearly separated in experience).

As we have seen this is the very approach that characterises conventional scientific understanding.

In truth development largely plateaus with the Middle Band in Western culture.
However where authentic contemplative awareness unfolds, further Bands are required.

The third Band (Higher Band) relates to the unfolding of circular (intuitive) awareness.

The first level (H1) is concerned with 2-dimensional interpretation (where both internal and external aspects are now seen as dynamically complementary - and ultimately identical - in experience) The positive aspect is associated with periods of illumination whereas the negative (where substantial dynamic negation of phenomena occurs) is associated with corresponding periods of purgation!

The second level (H2) is concerned with the unfolding of 4-dimensional interpretation. Here whole and part in any context are qualitatively distinguished enabling their proper integration in experience. This leads to the realisation that all reality is properly complex (in holistic mathematical terms) with both real (analytic) and imaginary (holistic) aspects.

The third level (H3) relates to 8-dimensional interpretation (where now both form and emptiness are eventually united). This requires a very subtle form of appreciation where form is equally interpreted with respect to real and imaginary aspects (which has an alternative empty expression as null lines!)

For some time my mapping of development largely culminated at this point. Though I did include a later Radial Band, it was largely for completion purposes without being fully integrated with the earlier Bands.

However gradually I came to realise - that just as linear understanding has its own specialised expression with the Middle Band - that circular understanding has its own specialised expression with a further 4th Band (that I sometimes refer to as the Higher Middle Band). As well as allowing for a further refining in spiritual contemplative awareness, this Band also addresses a typical imbalance as between form and emptiness. Frequently the drive to unity is driven by the transcendent spiritual aspect in the desire to achieve emptiness (beyond all form). However the immanent aspect is equally important as the desire to see such emptiness as inherent in all form.

Once again a holistic mathematical appreciation of the Riemann Zeta Function led me to the realisation that the completion of this Band (with immanence and transcendence in equal balance) coincides with 16-dimensional development. This would also perfectly equate with the inclusion of immanence and transcendence as two additional fundamental polar opposites in experience.

So Radial Development proper only commerces in a mature sustainable manner (with 16-dimensional development). In my latest work, I include three additional Radial Bands (so that we have 21 main Levels in all).

To see a fuller explanation of this model, dealing in detail with the structures and states characteristic of each main level
See Stages of Development; Chaps 8 - 12.

Fractal Dimensions

2009-12-31T12:02:00.000-08:00

One of the great contributions of Fractal Geometry is that it leads to consideration of the corresponding notion of fractal dimensions. For example this is beautifully illustrated with Koch's Snowflake.
See Mathworld.

So to construct this Snowflake we start with an equilateral triangle. Then marking each line into 3 equal divisions we take the middle third and erect another equilateral triangle on each side. Then we continue to proceed in the same manner (constructing a new equilateral triangle on the middle third of each exposed side).

Theoretically, we can continue in this manner an infinite number of times.

The implication of this is that the perimeter boundary of the Snowflake thereby increases without limit. In fact we can easily see that the initial construction of equilateral triangles on the middle third of the original 3 sides of the starting equilateral triangle increases the perimeter length by a factor of 4/3. Thus as we can keep repeating this procedure indefinitely (in each case increasing the perimeter by 4/3) the line can grown without limit even though the area of the entire figure clearly is of a finite magnitude.

In effect the perimeter line (through this process) encloses to a degree the overall 2-dimensional area.

This leads to a new notion of dimension for this line. To acquire it we basically find in this case what root of 4 (= the dimension) gives an answer of 3, the answer which is 1.261859507..

Strangely enough such a dimension (which necessarily falls between 1 and 2) is generally referred to as a fractional dimension.

However this conceals a very important conceptual difficulty.

We commonly think of fractions as rational numbers. However such a dimension (as in this example) will always be of an irrational nature (which thereby cannot be represented as a rational number).

Indeed the only way to get a rational dimension is to attempt to divide the line with irrational number sections!

This is all very interesting from a qualitative perspective.

In proper qualitative terms, finite notions are associated with (linear) reason and infinite notions with (circular) intuition respectively.

Now the irrational notion relates therefore to a situation where the infinite in some ways becomes embedded in the finite.

So corresponding to irrational understanding in qualitative terms is the derivation of an irrational number dimension (reflecting the paradoxical interplay of finite and infinite notions).

The clear implication of all this is that a corresponding qualitative interpretation attaches to the mathematical notion of the irrational.

In psychological terms this becomes equated with the transition from 1 to 2-dimensional understanding. Initially before two clear polar directions of phenomenal understanding crystallise, one goes through a linear phase where rational understanding increasingly interpenetrates with intuitive understanding (with finite phenomena now in some way genuinely embodying spiritual infinite notions).

This also provides an answer to the clarification of dimensions. So now we can look at irrational dimensions as basically representing transition states as between the whole number dimensions. Basically each new higher dimension (as whole number) represents a new more refined form of rational appreciation (with the corresponding number of directions). Therefore the irrational dimensions in between represent various states (whereby reason and intuition are combined in a - as yet - somewhat confused manner).

Once again in quantitative terms a "fractal dimension" between 1 and 2 represents the extent to which the 1-dimensional line can fill in a 2-dimensional area.

In corresponding qualitative terms, a "fractal dimension" (that is indeed of an irrational nature) measures the extent to which one can embody as it were 2-dimensional appreciation coming from a 1-dimensional perspective. This can never be achieved fully. However it can be approximated more closely through a progressively greater degree of intuitive understanding informing rational interpretation.

Return to Chaos

2009-12-31T11:23:00.000-08:00

Like so many, my first introduction to the new science of Chaos and Complexity came from the book "Chaos" written by James Gleick back in 1987.

Recently I returned to read this book wondering as to whether the intervening years had changed my perception or sharpened understanding of the issues raised. As on the first occasion of reading, I was left however with a vaguely dissatisfied feeling, rather like the various courses of of a meal that promise a great deal in preparation, yet somehow fail to deliver on eating.

So I stayed a little with this feeling so as to get to its roots.

Basically I would summarise the position as follows. Chaos Theory and all the various versions of Complexity arise from the recognition that so much behaviour in nature is of a nonlinear nature. However Conventional Science, through its linear method, tends to approach nature through modelling it in the form of linear equations (frequently adjusted finally in some way to approximate living conditions).

However this greatly limits the range of possible enquiry while in many ways misrepresenting the true nature of natural behaviour that is studied.

Chaos Theory therefore arises from the commendable insight that nonlinear behaviour needs to be accommodated in the true study of nature.

Admittedly impressive advances have been made in some fields from this insight. For example fractal geometry, not alone reveals many new fascinating insights but also lends itself to a new form of computer art (where self similar patterns endlessly repeat without exactly replicating themselves).

However one great limitation of these new studies is that they are still attempting to approach scientific problems from within the conventional scientific approach (which by its very nature is linear).

So we have the attempted study of nonlinear behaviour (within the confines of a decidely linear paradigm).

What this means in effect is that the very intuition that is required to - literally - see the important holistic relationships in nature is not actually provided through the new approaches. In fact it leads directly to an equivalent form of reductionism. Whereas the "old" standard approach is criticised as attempting to reduce the whole behaviour of systems to (isolated) constituent parts, the "new" approach actually suffers from the opposite form i.e. of attempting to reduce the rich diversity in nature to some general forms of behaviour.

It seems therefore quite obvious to me that the study of nonlinear systems (in quantitative) terms, properly requires a nonlinear manner of approach (from a qualitative perspective). However as of yet there is no real recognition - either among the "old" standard practitioners or the "new" chaos merchants - of such a nonlinear qualitative approach (which would require a radical revision of what is meant by science).

In the end it is only through this new qualitative approach - which I term integral - that the ability to "see" the important holistic connections in nature (in any required context) can actually emerge.

The fact is that physical and psychological reality are dynamically complementary. Nonlinear behaviour in nature above all requires recognition of this simple fact. In the end the same nonlinear behaviour that governs nature, likewise governs mental life. Though proper appreciation of this basic point, each can thereby operate as a mirror to the other's true identity.

Clarifying Dimensions

2009-12-31T05:52:00.000-08:00

As one can see, my integral approach to science (and development) is heavily based on holistic mathematical interpretation of the 1st, 2nd, 4th and 8th dimensions respectively.

However a key insight here is that corresponding to every number (in holistic terms) is a unique dimensional interpretation (with application to both physical and psychological reality).

In psychological terms, this entails that all possible understanding entails both conscious and unconscious aspects. So a dimension in this sense corresponds to a unique configuration of these two aspects.

In scientific physical terms, this equally entails that reality entails both specific (manifest) phenomena related to an underlying (hidden) ground of reality. So here, each dimension relates to a unique configuration connecting such phenomena to this underlying ground.

And of course in holistic mathematical terms, a dimension (in either its psychological or physical state) is given as a unique mapping of real and imaginary coordinates (on the complex plane).

What I want to do briefly here however is to extend somewhat the significance of dimensions (beyond 1, 2, 4 and 8) already mentioned.

Firstly it has to be recognised that every number can be given a positive (+) as well as (-) sign. The question then arises as the negative meaning of the dimensions already mentioned.

Quite simply, whereas the positive understanding of a dimension relates to conscious rational interpretation, the corresponding negative interpretation relates to its unconscious intuitive recognition.

In spiritual terms, whereas development of the positive understanding is generally associated with periods of illumination, the corresponding negative understanding is associated with periods of purgation.

Thus development of of the negative first dimension literally relates - in psychological terms - to the active negation of linear understanding (in what St. John of the Cross refers to as the active nights of sense and spirit). Here sense is identified with more superficial perceptual phenomena whereas spirit relates to deeper conceptual and volitional awareness!

There is in fact a very close relationship as between a negative dimension and the next positive number dimension. So in this context what is unconsciously understood at the first dimension (in the negation of - formerly - posited linear understanding) is then consciously appropriated at the next positive dimension (i.e. the 2nd, as combining both positive and negative aspects). Once again there is a remarkable connection here with the Riemann Zeta Function where the corresponding relationship is maintained in terms of the famous functional transformational formula. So properly understood the two sides connected through the functional formula relate to specific (conscious) interpretation on one side (for positive values of s) and to holistic (unconscious) interpretation on the other side (for negative values, 1 - s).

Then development of the negative 2nd dimension once again relates to intuitive (rather than rational understanding) of the complementarity of real opposites in experience. St. John refers to this as the passive night (of both sense and spirit).

In general, purely contemplative (intuitive) understanding of an empty spiritual nature is associated with the negative even dimensions (where always an exact complementary pairing as between opposite terms exists). In like manner in the Riemann Zeta Function for all negative even values of s, the result = 0.

This was to prove a key revelation in my examination of this function as it quickly led to the realisation that the Riemann Zeta Function in fact provides numerical results corresponding to both conventional mathematical interpretation (where s > 1) and holistic mathematical interpretation (where s < 0). Thus one cannot properly interpret the Riemann Zeta Function (and the associated Riemann Hypothesis) without including both conventional and holistic modes of interpretation!

There are strong reasons for believing however that in terms of even numbered dimensions, 2, 4 and 8 are the most important. Remarkably there is a very good mathematical reason (holistically interpreted) why this is in fact the case!

However I have also explored - to a certain extent - the nature of both 16 and 24 dimensional understanding (which I may return to in later contributions).

For each whole (integer) number, we also have a reciprocal number (as fraction). So these fractional dimensions (of even numbers) can also be given a qualitative meaning!

The easiest to understand is the reciprocal of 2 (i.e. 1/2). Whereas the second dimension relates to holistic understanding (as one moves away from specific phenomena in understanding), the reciprocal relates to the corresponding movement back to specific phenomena (in the light of this new understanding). In other words it provides a new enhanced form of analytical understanding where all phenomena can now be viewed within two respective reference frames (that are opposite with respect to each other). Thus in terms of analytic understanding each frame provides - literally - 1/2 of a total perspective).

Likewise much more enhanced analytic perspectives open up at the higher dimensions. For example when we come back to linear analytical investigation (after encountering holistic 4-dimensional understanding) we now can provide four distinct frames of reference for any interpretation. (So each interpretation now represents 1/4 of the total linear dimensional perspective).

The corresponding negative fractional dimensions would then relate to additional active purgation (with respect to such enhanced appreciation). So the task of purgation is never complete (either with respect to active or passive understanding) with each new dimension opening up a more refined type of phenomenal understanding (that can only be purified at that level).

As with the Riemann Zeta Function, the odd dimensions prove to be much more problematic. However, having said this, understanding of the holistic meaning of the Function has greatly enhanced my interpretation of the odd dimensions in psychological development (and corresponding physical reality).

What it amounts to is simply this!. Whereas the even number dimensions relate directly to integration, the corresponding odd number dimensions relate to increasingly refined differentiated appreciation of phenomena.

What actually happens in development, is that a certain amount of intuitive confusion remains (as each new odd number dimension unfolds). So in the corresponding purgation with respect to such dimensions, this confused aspect is removed leaving only the purified rational element.

So whereas the negative even number (integral) dimensions are concerned with the purification of confused rational elements, conversely the negative odd number (differentiated) dimensions are concerned with the purification of confused intuitive (i.e. unconscious) elements. Once again this corresponds remarkably with the Riemann Zeta Function where all negative odd values (for s) yield a rational answer.

Once again the fractional use of such dimensions would occur when one returns to the linear level of experience (from the perspective of the higher odd dimension).

For example if one returns to differentiated linear investigation from the 3rd dimension, 3 independent reference frames will now be available (with each corresponding to 1/3 of the total perspective). One of these will concur with the standard linear approach. However the other two will lead to additional complex interpretations (that are equally valid). Thus comprehensive analytic understanding will now require all three interpretations!

We can also give interpretation to imaginary dimensions. It took me a long time to appreciate this (though it is not really so difficult).

When one remains psychologically in real dimensions, it entails a movement away from form in an increasingly unconscious direction (i.e. spiritual transcendence).

However the corresponding reverse direction in development is the movement from the (empty) unconscious back to form in an increasingly conscious manner (i.e. spiritual immanence).

And when this reverse journey takes place, one in fact is operating out of imaginary (rather than real) dimensions.

Again this has a remarkable correspondent in mathematical terms. If one raises 1 to a real rational value) e.g. 1/4, one obtains the imaginary number i. If now in turn one now raises 1 to this imaginary value i, one obtains a real number!
So this demonstrates the extremely important point that object phenomena and spacetime dimensions are real and imaginary with respect to each other in dynamic interactive terms!

In fact this discovery has proven very illuminating in terms of understanding my own development (which has been unduly transcendent).

Thus - though in principle - I would have subscribed to the view that one should return to the 1st dimension (of linear understanding)in an enhanced manner from higher dimensions, this in fact for me always proved very difficult. I found it in fact easier to keep moving on to ever higher dimensions (in a holistic manner) rather than return to the more specific experience of form! Only recently have I been able to realise that this reflected a lack of sufficient immanent development (which inherently requires the use of imaginary dimensions).

In brief the real dimensions are concerned with obtaining an ever more refined conscious appreciation of reality (inspired through intuition); the imaginary dimensions are concerned with ever more refined unconscious appreciation
(controlled by reason).

We can also give a holistic interpretation to irrational dimensions. However I will tie this in later with a brief discussion on fractals.

Integral Theory of Everything (4)

2009-12-31T04:38:00.000-08:00

One way of viewing this integral TOE is in terms of the main contemplative stages required before full union (empty of manifest phenomena) can take place.

Linear understanding (1-dimensional) by its very nature leads to phenomena being given an unwarranted rigidity (by which they are given a seemingly independent existence).

So the first task is to remove such rigidity at a conscious level. This is attained by recognising the inherent complementarity - with respect to all phenomena - of both internal and external polarities (2-dimensional understanding). In holistic mathematical terms these are positive (+) and negative (-) with respect to each other in real terms.
In psychological terms this can be attained at either a rational (cognitive) or sense (affective) level without the full integration of either aspect.
In corresponding physical terms this would relate to a failure to fully integrate both personal and impersonal aspects with respect to scientific phenomena.

The second task is then to remove further rigidity at a deeper unconscious level. This manifest itself in experience when phenomena - representing an overall holistic meaning (pertaining to the unconscious) - are given an indirect conscious
identity. This is in turn attained by recognising the inherent complementarity of part and whole aspects with respect to all phenomena (4-dimensional understanding). In holistic mathematical terms these are real and imaginary with respect to each other (both having positive and negative directions).

In psychological terms this would now entail substantial integration of both cognitive and affective aspects of experience.
In corresponding physical terms it would entail the true marriage - without reductionism - of specific phenomena with dimensions (combining the interaction of both personal and impersonal aspects of phenomena).

The final task (from the purely contemplative view) is to then simultaneously combine both conscious and unconscious in a refined flexible manner (without secondary rigidity arising). This relates directly to volitional intent which serves in turn as the primary basis for continued integration of both cognitive and affective aspects (8-dimensional understanding). In mathematical terms this would combine the 4 additional complex roots of unity with the 4 already attained).

In psychological terms it means that the balance as between conscious and unconscious (in experience) is now so equal (i.e. where real and imaginary parts are of same magnitude) that phenomena no longer appear to arise. This coincides with experience of spiritual emptiness (i.e. null lines). So here the four "spiritual forces" (in the external and internal manifestations of immanence and transcendence respectively) are unified in the one awareness.

In corresponding physical terms, it means that the four physical forces (where wave and particle aspects remain equal) are also unified as the one superforce (existing in the present moment as mere potential for existence).

Finally, as we have seen both physical and psychological understanding fully coincide here in the same contemplative experience in the continuous present moment (as both the Alpha and Omega of all reality).

Of course paradoxically this Holistic Theory - or rather Experience - of Everything is equally an Analytic Theory of Nothing (in that tells us nothing specifically about reality).

So I would now see this purely contemplative stage as but a stepping tone to more advanced radial engagement with reality.

It might seem initially puzzling why this should be in fact the case! I would use the analogy of the lighting fire to illustrate. Pure contemplation is like a fire - say of wood - that is set alight (without an attempt to replenish its material base). So eventually this fire , having reached its zenith blazing state, is destined to slowly die out. In like manner the state of pure contemplation (without active phenomenal involvement) is not possible to sustain.

However if we keep adding fuel appropriately to the fire, not alone can its blazing state be maintained but its very size and intensity can be increased. Here at any given stage we will have a blazing fire that is consistent with an increasing amount of material that has not yet been fully lit.

Likewise in the radial life as one becomes more active in a sustainable manner, not alone does this entail an increasing level of phenomenal experience (not yet fully transformed in spirit) but equally a more intense and more broadly based spiritual contemplative awareness.

So in the end radial activity is not only necessary in order to transform "the world" (as opposed to the self) but equally necessary to properly sustain and enhance true contemplative experience (of spiritual nothingness).

Integral Theory of Everything (3)

2009-12-23T09:14:00.000-08:00

I will elaborate further on the significance of the three key polarities (corresponding to the holistic mathematical interpretation of 2, 4 and 8 dimensions respectively).

The first set of horizontal polarities (corresponding to the relationship between internal and external) as we have seen are positive and negative with respect to each other in integral terms.

The significance of this is that all scientific understanding of reality necessarily entails an internal (subjective) observer in relation to what is observed in external (objective) terms.

Now when we view this relationship in a linear manners, both external and internal frames can be given a merely positive identity (that correspond with each other).

For example I can view the "sun" as in objective terms as an external object.

However equally I can view the "sun" in - relative - subjective terms as relating to its internal perception.

Fortunately for Conventional Science - based on a linear approach - both external object and internal constructs are understood to correspond directly with each other. In this way it does not matter if we designate science in conscious (positive) terms as interpretation of the relationship between object phenomena (external) or alternatively between subject constructs (internal).

However when we attempt to view this relationship in a holistic integral manner, these two frames of reference, external and internal - that seemingly correspond with each other (in linear terms) - are diametrically opposite (in a complementary circular manner).

So now both external and internal are understood as positive and negative with respect to each other (in complementary fashion). Though the rational interpretation of this relationship is paradoxical, the actual reconciliation of both aspects takes place in a directly intuitive (unconscious) manner.

So one clear implication of the Integral Approach is that scientific understanding cannot be conducted in (mere) rational linear terms but rather combines both (linear) sequential and (circular) paradoxical understanding (implying the explicit combination of both reason and intuition).

The refined 2-dimensional appreciation of the role of external and internal aspects of understanding (n both linear and circular terms) corresponds to - what I refer to as - the Integral 1 approach.

The second set of polarities relates to the hugely important distinction as between whole and part.

Once again in linear terms (where both are treated as independent) both of these can be given a rational interpretation in merely conscious terms (i.e. as positive). For example research data would be identified with the part and theory with the whole aspect respectively.

However in circular integral terms (where both are treated as interdependent) the relationship is as real to imaginary. So if we give the part (in any context) a real identity, the corresponding whole aspect is - relatively - imaginary.

So we have the vital insight here that in order to preserve the appropriate qualitative distinction as between part and whole, we must interpret reality in complex rational terms (with both real and imaginary aspects).

This has several important applications (just a few of which I will mention).

The relationship in physical terms as between object phenomena and dimensions (of space and time) is real to imaginary.
In corresponding psychological terms the relationship between perceptions and concepts is also real to imaginary. Furthermore both of these aspects are fully complementary. This - as we shall see - has huge implications in terms of making sense of many of the findings of string theory which really points to the need for an entirely new understanding of the dimensions of space and time.

The relationship as between cognitive and affective in understanding is also real to imaginary. In other words, in Jungian terms, when the cognitive aspect of understanding is real (conscious), the corresponding aspect is imaginary (as the projection of the holistic unconscious); in like manner when the affective aspect is now real the cognitive is then imaginary. So in the dynamics of experience, we keep switching as between real and imaginary (with the revealed aspect conscious and the alternative hidden aspect unconscious).

Finally, a correct appreciation of such complex interpretation exposes the limitations of conventional mathematical proof (which entails a basic form of reductionism where the whole is confused with the part). So, the general proof that applies holistically to all cases cannot be directly identified with any specific instance (applying to the part).
This leads to need for reformulation of the notion of proof in dynamic relative terms where he it is soon as representing but a special form of social consensus.

The third set of polarities relates to fundamental form and emptiness. Again in linear terms these can be treated in an independent manner. However in corresponding integral terms they require 8-dimensional understanding for appropriate interpretation.

In physical terms this leads to the appropriate holistic mathematical way of understanding the nature of the physical forces; in corresponding terms it provides the corresponding means to translate the volitional psycho spiritual forces; and once again both sets of forces are complementary in experiential terms.

We will look at this again in the next contribution.

Integral Theory of Everything (2)

2009-12-20T04:18:00.000-08:00

The Integral Theory of Everything is based on finding a holistic mathematical acceptable manner of interpreting the dynamic relationship as between the 3 fundamental polarities (that underlie all experience of reality).

Once again these polarities relate to (i) the distinction as between internal and external within a given stage of development (horizontal polarities); (ii) the distinction as between whole and part between different stages of development
(vertical polarities); (iii) the ultimate distinction as between form and emptiness both within and between different stages of development (diagonal polarities).

I identify 4 stages of integration corresponding with the qualitative interpretation of 1, 2, 4 and 8 dimensions respectively (relating in turn to the 0th, 1st, 2nd and 3rd powers of 2)

The standard conventional scientific approach - based on linear rational understanding - corresponds with the 1st dimension (Integral 0 approach).

Basically such an approach - which is based on clear separation of the opposite polarities in experience - does nor allow for corresponding complementary understanding. Therefore it necessarily deals with the key polar interactions through a form of reductionism. Thus typically in conventional science, the internal aspect is reduced in terms of the external; the whole in any context is reduced in terms of the part; finally emptiness is reduced in terms of form.

The first truly integral approach (Integral 1) allows for the dynamic complementarity of both internal and external through - what qualitatively represents - 2-dimensional understanding.

Here internal and external (conscious) phenomena are considered in holistic mathematical terms as both positive and negative with respect to each other in real terms. However the Integral 1 approach does not yet allow for appropriate interpretation of the relationship as between whole and part and form and emptiness.

The second approach (Integral 2) allows for additional dynamic complementarity as between whole and part (which are now qualitatively distinguished from each other). In holistic mathematical terms this corresponds with 4-dimensional understanding. Here both real and imaginary distinctions are made with both positive and negative directions. So in any context, if for example the "part" is designated as real (in conscious terms) then relatively the "whole" is designated as imaginary (i.e. pertaining to unconscious meaning).
So the Integral 2 approach allows fully for the necessary interaction of conscious and unconscious in experience; in physical terms this corresponds with the interaction of discrete phenomena with an underlying holistic ground (or field) of matter.

The third approach (Integral 3) then allows for further dynamic complementarity as between form and emptiness which in holistic mathematical terms corresponds with 8-dimensional understanding.

Such understanding is based on the four complex roots (of the 8 roots of 1) where both real and imaginary aspects are equal. The fascinating point here is that these equally constitute null lines (with no magnitude). So in corresponding qualitative terms, these allow for the ultimate unification of form and emptiness (with the complex relating to form and the null designation to emptiness respectively).

In psychological terms this would entail that the interaction as between real (conscious) and imaginary (unconscious) would now be - literally - so equal or harmonious that phenomena (of form) would not even appear to arise in experience this facilitating union with emptiness.