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Peter CollinsPeter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.


Part IV - Hidden Strings of Reality

Peter Collins

The major problem with present string theory is the lack of an adequate intuitive explanation of the nature of its concepts. Rectifying this problem requires the incorporation of a complementary qualitative view of strings where each vibration of the string represents a unique holistic interpretation of physical reality. In this context the present scientific paradigm represents but the lowest possible (psychological) energy level of the string!


We have already dealt with the two great revolutions in physics of the 20th century, relativity theory and quantum mechanics. Though each of these theories has proved incredibly successful within its own frame of reference, at a deeper level it has long been recognised that - in their present form - they are incompatible with each other. Though this problem is usually expressed in terms of the difficulties in integrating gravity with the three other known physical forces, ultimately it relates to the reduced manner in which the relationship between whole and part is conventionally interpreted in science.

Properly understood the whole is distinct from the part with the relationship between both as quantitative to qualitative (and qualitative to quantitative). Though in an isolated context, it is indeed possible to treat wholes and parts with respect to their mere quantitative aspects, this is no longer strictly valid where they are considered as interdependent.

So both relativity theory pertaining to the macro whole universe and quantum mechanics pertaining to the micro particles of reality respectively, are quantitatively consistent in isolation. However in relationship to each other they are inconsistent. So the clear implication before proper integration can take place is that both quantitative (analytic) and qualitative (holistic) aspects of interpretation must be incorporated.

Now it might be initially thought that string theory - or more correctly superstring theory - is designed to provide a solution to this problem that enables consistent integration of relativity theory and quantum mechanics (within a merely quantitative framework).[1] However I hope to amply demonstrate as we proceed, that this in fact is not the case and that fundamental problems remain with present interpretation that can only be resolved through incorporation of the - as yet unrecognised - qualitative aspect of string behaviour.

String theory itself dates to the late 1960's and the attempt to explain certain mathematical data with respect to the strong (nuclear) force. These seemed to fit well with a 200 year old mathematical formula (the Euler-beta function). Attempts to give a physical meaning to this result then required viewing the strong force between particles as due to extremely thin elastic like strands. These strands were called strings leading to the birth of string theory. It must be remembered that this represented a very radical departure at the time from accepted treatment of the smallest particles as points (rather than lines)!

However as new data on the strong force appeared, string theory proved less successful in explaining results, causing a rapid loss of interest in its potential. However a dramatic switch in focus led later to it being considered as a quantum mechanical explanation of the gravitational force. Though little interest was shown initially, when important difficulties with the emerging theory were resolved (largely due to Schwarz and Green), by the mid 80's the first string revolution was born.

And because string theory now provided the prospect of consistently unifying the gravitational with the other forces it became the logical candidate for the long sought for TOE.

Problems remain, with many criticising string theory on the grounds that it cannot be experimentally verified due to the present inability to detect strings (which are postulated to be extremely small in length).

However, much more profound issues surround string theory, which have not yet been properly addressed.

The Problem with a String

I have always had philosophical issues with the definition of a string. Indeed my interest in the area originally arose out of an attempt to redefine some of the key concepts of string theory in a way that I - at least - could find intuitively satisfying.

This relates to a deeper problem with the 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 reality is normally understood).

At the moment string theory resembles a kind of secret code that undoubtedly has generated 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 qualify as a true physical explanation of reality.

So the problem with string theory not alone relates to present difficulties in finding ways of experimentally testing its findings; perhaps more significantly, it relates to even greater difficulties in providing a satisfactory explanation of the true nature of its concepts.

From my own particular perspective the root of this problem is easy enough to state. Once again conventional scientific method is based on mere 1-dimensional interpretation (which conforms best to everyday intuitive notions of the nature of the physical world).[2]

However when we probe at the extreme margins, linear interpretation breaks down badly as a means of adequately conveying the overall nature of reality investigated.

What we need in effect are progressively higher dimensional interpretations in qualitative terms that incorporate both linear (analytic) and circular (holistic) notions in ever more refined fashion.

So my own investigations with respect to integral science arose with a view to providing - in basic fashion - these much needed holistic dimensions.

The integral aspect of science 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 even to conventional physicists that the attempted definition of a string is somewhat 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 this does not stand up in physical terms.

From a conventional perspective we cannot conceive of an object that is 1-dimensional. Once again, though in geometrical terms, we can represent a line as 1-dimensional this simply is an idealisation that does not strictly hold in the physical world. So any line, that we actually try to represent, must inevitably possess a degree of width (for its identity to be demonstrated).

So if a string has length (without width) how can its phenomenal identity be established? Also what about time in such a characterisation? Surely, again to have phenomenal meaning a string must possess an 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, the string must in some way embody these very dimensions (as contained in the same entity)!

One way of moving towards a more satisfactory explanation is - perhaps surprisingly - through consideration of a 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 a 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 aspects. Whereas in a direct sense the conscious (analytic) aspect corresponds to real rational understanding, the unconscious (holistic) aspect properly relates to intuitive appreciation, which then can be given an indirect imaginary rational interpretation in a holistic mathematical manner.[3]

Therefore in corresponding physical fashion we must conceive of reality itself as entailing the interaction of two distinct modes that are real and imaginary with respect to each other.

Thus a string (and indeed combined group of strings) must be rooted in a dynamically conceived complex notion of reality.

In this way both the (whole) dimensional and (part) object aspects of the strings can be viewed - without gross reductionism - in an interacting real and imaginary fashion (which continually alternate).

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 it is confined formally to merely real modes of interpretation. Indeed this is the very reason therefore why physicists find it so difficult to properly incorporate dimensions within string theory![4]

To deal with such dimensions satisfactorily will require a radical new scientific approach that combines both conventional (linear) and new integral (circular) modes of interpretation.

Defining the Holistic String

The very notion of a string can now 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.[5]

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 interpretation of reality (as with conventional science). However when such understanding dynamically vibrates through contemplative type awareness, it can then potentially generate all the "higher" dimensions of understanding that comprise ever more refined interpretations of reality.

Conventional scientific appreciation is decidedly 1-dimensional in nature. Thus, in terms of our second notion of a string, it simply represents its lowest possible psychological energy state, where very little dynamic vibration is allowed. In other words from this perspective, scientific interpretation is viewed formally in a merely rational manner (without consideration of intuitive interaction).

One of the big "eureka" moments for me with respect to my complementary approach to strings 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. Just as the different vibrations 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 correspond 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 for the integral scientific approach!

Then in Figure 6.3 on P. 145 he gives another example that geometrically corresponds with the 16 roots of unity.[6]

As Greene explains, the greater the frequency (i.e. the shorter the wavelength) the more physical energy is associated with the vibration of the string. In corresponding holistic fashion the higher the dimension of understanding, the greater the degree of (intuitive) spiritual energy associated with one's manner of scientifically interpreting reality.

So the point once again is that in a more comprehensive treatment, 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 analytic understanding of physical reality i.e. the real aspect of interpretation as separate from psychological.

However the alternative definition, that I am here providing, corresponds - by contrast - with holistic appreciation of both physical and psychological reality as complementary (i.e. the imaginary aspect of interpretation).

And then this is the crucial point!

Just as the relationship 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 Conventional (Analytic) Science comprises its real aspect properly geared to interpretation of particles in the context of string behaviour.

Integral (Holistic) Science comprises its imaginary aspect properly geared to holistic interpretation of corresponding dimensions.

Radial (Comprehensive) Science can then fruitfully combine both aspects in a complex rational approach.

Analytic and Holistic Strings: Complementary Aspects

It might be helpful here to further extend the complementary connections as between the two notions of strings, 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 an infinite number of particles can be generated.

Now so far, only a relatively small finite number of distinct particles have been found. The explanation is due to the fact that actual particles discovered correspond to extremely low energy levels of string vibration. These likewise entail less energy (corresponding to relatively low masses).

New particles entailing greater vibration of the string would require extremely high amounts of energy (well beyond the present technological capacity of generation).

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 number of integer (whole) dimensions of understanding that unfold with contemplative development, therefore we can likewise provide an infinite set 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 to the infinite range of potentially valid 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 appreciation).

Though higher energy states are associated with authentic contemplative development, it would be quite rare - in terms of corresponding cognitive progress - for it to grow significantly beyond the lower number dimensions.

And I have persistently argued that these lower dimensions - certainly 2, 4 and 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 physically sets a certain limit to the possibilities of phenomenal investigation, we have in a sense a corresponding Planck length, in psychological terms, that sets a certain limit to the possibilities of rational type interpretation. What this means in effect is that the dynamic interaction of polarities (such as internal and external) becomes so fluid through advanced contemplation 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 below the Planck length, they remain merely implicit (without being capable of measurement in explicit terms). 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). Here both the knower and what is known are finally united in an undivided experience with no gap remaining as between actual manifestations of physical reality and the psychological interpretations with which they correspond.

Dimensions in String Theory

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.

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).[7]

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 a 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 interpretation).

So, one who remains firmly rooted in linear understanding - quite literally - will not be able to intuitively "see" from the perspective of the "higher" contemplative vision. Rather, interpretation will be reduced to the understanding of the lower 1st dimension. Thus, development, with respect to these new circular dimensions of spiritual "seeing", will 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, 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, an entirely distinctive approach 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). And this relationship as between linear and circular undergoes continual refinement at progressively higher dimensions!

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 holistic notion of dimension, a 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 interpretation.

Once again, the notion of dimension that is appropriate to string theory has very little to do with conventional concepts 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 quantitative appreciation of its corresponding roots.

New Understanding of Dimensions

The basic rationale for this new understanding of dimensions - which applies equally to both physical and psychological reality - is that it is based initially on the two key polarities sets, which necessarily underlie all phenomenal experience.[8]

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).

We can represent these polarities as four equidistant coordinate points (like a compass) on the circle of unit radius with internal and external the end points along the horizontal 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 here is that wholes and parts are combined in a static reduced manner as whole-parts and then considered as external to the observer. This results in the typical 3-dimensional spatial rigidity which objects thereby possess. The final 4th dimension is then separated from the spatial and interpreted as time.

So, standard linear interpretation is associated with an asymmetrical appreciation of both space and time.

When we adopt a 2-dimensional interpretation (in this newly defined circular holistic manner), space and time now appear as paradoxical and symmetrical both having complementary positive and negative aspects. 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) corresponding to the 2 roots of unity.

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 in structural terms directly with the 4 roots of unity.

So space and time 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 also 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 in terms of 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).

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 100-dimensional reality, we obtain the corresponding 100 roots of 1 (in a quantitative manner).

Now these roots 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 interpretation that an attempt is made to separate both linear and circular notions (that is so 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 in physical terms (as do conscious and unconscious interact in corresponding psychological fashion).


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 strings 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 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, 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 number 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.[9] 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 a strong similarity with the corresponding analytic treatment in string theory!

Each of the integral interpretations therefore offers a certain unique qualitative view 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 physical perspective of what their worldview 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 relates 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 either the analytic or (unrecognised) 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. Though phenomenal appearances will always remain necessary in dynamically approximating this reality, ultimately they cannot deal with its essential nature.


One of 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 reveal 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, with appropriate transformations we can show that they are equivalent (through using a different language of expression).

However the really interesting area is to establish much 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 conventional scientific interpretation.

However though this Type is best in terms of detailed rational appreciation (of an analytic kind) it is least satisfactory in terms of providing the necessary 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 of quantitative interpretation. 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.

Brane Power

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 = number of dimensions involved.

Then 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, which could be identified with the psychic/subtle realm, one now acquires the default understanding of that dimension (based on the dynamic complementarity of opposite real poles).

Then as further development occurs, 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).[10]

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 that takes 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. In like manner - as would be conventionally explained - what is properly a membrane (2-brane) will be interpreted as a string (1-brane) when the additional (circular) curled-up dimension is not recognised.

Now this important finding - regarding a whole range of enhanced 1-dimensional interpretations (from the perspective of higher dimensions) - seems to me to equate very 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 other possible 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 string could appear as a p-brane (where p >1) !

Thus, higher brane objects would appear as lower branes (in the first case) and in the 2nd, lower brane objects appear as higher.

So for example a 4-brane could appear as an enhanced 2-brane and a 2-brane object could possibly appear as a (diminished) 4-brane.

Likewise we could in psychological terms obtain an enhanced view of 2-dimensional interpretation from the attained perspective of the 4th and in turn a diminished view of 4-dimensional interpretation from the 2nd dimension (as the highest yet properly attained in experience).

Recently, there has been speculation regarding another class of "object" referred to as a zero-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 holistic counterpart to a zero-brane would relate to understanding that is 0-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 appreciation amenable to intuition.

So the paradox for present physics is that true appreciation of zero-brane reality can only come through the extreme expression of qualitative - rather than quantitative - type interpretation. So, whatever the precise merit of such ideas, string theorists need to accept that the current language of interpretation is largely meaningless from any coherent philosophical - or indeed physical - 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 interpretative 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.

String Theory - A New Myth for Our Age

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 influenced the nature of analytic investigation.

In this context it is interesting to compare an earlier story of the nature of creation with 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 is greatly lacking from an analytic perspective. However it is really designed to convey a deeper qualitative (holistic) meaning.

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 used are designed to convey holistic - rather than strict analytic - meaning.

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 reduced fashion by explaining the whole in terms of its partial elements.

Because of the ambitious designs of string theory to obtain a whole explanation of reality through formulation 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 principally provide a holistically satisfying explanation (conveying qualitative meaning), the purpose of a modern scientific account - such as string theory - is to provide an analytically exact explanation (of a quantitative nature).

However just as the former accounts were greatly lacking from an analytic perspective, string theory is also greatly lacking in holistic terms.

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 addressed.

Even with respect to standard mathematics, mere rational explanation does not constitute meaningful understanding.

For example with any mathematical proof, actual understanding does not materialise until the appropriate intuitive connections are likewise made. Only then can one literally "see" what the rational logical deductions entail. Fortunately however, at the macro level of appreciation, intuitions are designed to conform to accepted common sense notions of reality (dictated by linear type interpretation).

Likewise with respect to string theory, mere rational explanation does not constitute meaningful understanding. However the big difference here is that the confirming intuitions 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 interpretation 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 gravely exposed.

Therefore I would see that we are perhaps approaching the dawning of a great new stage in scientific understanding where we can now begin to deal with universal holistic type issues in a more 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 both specialised analytic and holistic type appreciation).


I will seek to briefly summarise my findings.

Though science properly should combine two aspects that are analytic (quantitative) and holistic (qualitative) with respect to each other, present science is based formally on sole interpretation with respect to its analytic aspect.

So expressed in holistic mathematical terms, scientific understanding should properly be of a complex nature (combining both “real” quantitative and “imaginary” qualitative aspects). However, conventional science is formally based on sole use of the “real” aspect.

Thus for every scientific explanation in the accepted “real” manner, a parallel holistic explanation can be given in an “imaginary” fashion.

And in this article with respect to developments is string theory, I have sought to demonstrate the basic nature of such parallel “imaginary” understanding.

The clear implication therefore is, that as presently constituted, string theory is greatly lacking any adequate holistic dimension to properly interpret the nature of its quantitative findings.

The deeper implication is that - while readily admitting that further exciting developments are likely to unfold with respect to the (conventional) development of string theory - any hopes in the unveiling of a final TOE are very much misguided.

Properly understood, findings with respect to physical reality always mirror the nature of our psychological means of interpreting such reality.

The ultimate question therefore does not relate to the nature of what is known but rather more subtly to the nature of the relationship as between the knower and what is known. And this can only be finally resolved not in any theory but through direct experience (where phenomena of form lose any residual meaning). So the proper goal of science in this context is to provide a coherent means of facilitating this ultimate experience!

The great mystics down through the ages have already realised this; perhaps it is time now for scientists to do likewise!

And rather than constituting a defeat, such a realisation would then open the way for an extraordinary enlargement and enrichment of the true nature of science.


1. Original string theories were mainly concerned with explaining the operation of forces (bosons). To include fermions (matter particles) in string theory a certain kind of symmetry must be maintained i.e. supersymmetry, and superstring theory is the name given to this theory. However string theory is still generally used as convenient shorthand for - what is properly - superstring theory.

2. In this context, 1-dimensional is equivalent to one-directional. So for example the essence of scientific observation is to give phenomena a single unambiguous direction i.e. as external to the observer.

3. In holistic mathematical terms what is “real” corresponds with what can be known in a direct conscious manner. What is “imaginary” directly relates to holistic unconscious meaning that is then indirectly translated in a rational manner. So in qualitative terms the great problem with conventional science is that it attempts to confine itself to the “real” aspect though - properly appreciated - both “real” and “imaginary” aspects are necessarily involved. In the terms that I have defined it therefore, holistic mathematics is concerned in qualitative terms with the “imaginary” aspect of mathematical understanding; Integral science is likewise concerned with the “imaginary” aspect of scientific understanding.

4. I have demonstrated before that in a mathematical context, when we raise a number quantity to a dimension (or power) that the number representing the dimension is properly qualitative (with respect to the base number quantity). So for example if 1 is raised to 1/3 then the fractional number 1/3 is qualitative (representing the dimension) with respect to the base number 1 (as quantity). This thereby causes a shift from linear to circular mode with the root lying on the circle of unit radius (in the complex plane).

Correctly understood it is likewise true in a physical context with the dimensions to which object phenomena relate properly qualitative (with respect to the quantitatively perceived objects).

The clear implication is that we should be using a circular rather than linear notion of dimension in physics. Though the linear notion - which dominates conventional science - works well as a limiting approximation at the macro level of physical reality, it breaks down badly at other levels. In fact in a correct qualitative manner, conventional science employs a linear notion of dimension. So not surprisingly in such an approach 3 of the 4 dimensions relate to the quantitative characteristics by which objects are spatially measured, with the remaining dimension treated as time. And this qualitatively linear approach then carries over to string theory where time still remains 1-dimensional with an attempt then made to extend quantitative spatial characteristics to the remaining “dimensions”.

So the remarkably conclusion is that the true notion of dimension, on which physical reality is structured, is directly of a holistic mathematical nature. Though string theory does indeed point to the need for a new interpretation of physical dimensions, the deeper implication has not been recognised in that adequate recognition of its nature requires direct incorporation of the (unrecognised) qualitative integral aspect of science.

5. The standard definition of “the string” attempts to deal with it in a merely “real” quantitative manner. However “the string” has an alternative definition in an “imaginary” qualitative fashion. And this qualitative aspect relates - not directly to physical phenomena - but rather to the fundamental psychological means through which we can attempt to interpret such phenomena. And just as the “real” aspect is translated through linear (either/or) logic based on the separation of polar opposites, the “imaginary” aspect is translated through circular (both/and) logic based on the complementarity of these same opposites. Again the great weakness of the conventional linear approach to science is that it assumes that there is just one (default) way for attempting to interpret reality whereas correctly understood, there are an infinite number of such possible interpretations. So again in indirect terms I would see string theory as pointing to the need to include (qualitative) interpretation alongside (quantitative) observation. Indeed properly understood, science should really be concerned not with observation or interpretation per se but rather the varying dynamic configurations as between both aspects. The key inference of what has been stated here is that the present inability to properly include dimensions (in the definition of string) is due to the fact that the alternative “imaginary” aspect of strings (which directly relates to such dimensions) is not recognised.

6. This explanation of the 16-dimensional nature of reality is extracted from my blog on integral science.

7. Earlier 26 dimensional string theories were based on the importance of 24 (with two added) and more recently 10 dimensional theories based on 8 (with two again added). I have long believed that this use of dimensions reveals a very important link with Jungian type personality theories. For a recent explanation of my thinking here see Personality Types.

8. There are connections with the 4 quadrants in that my two key polarity sets (internal/external and part/whole) are similar to Wilber's interior/exterior and individual/collective distinctions. However the key value of the holistic mathematical approach is that the dynamic structure of these 4 polarities (as dimensions) is given with respect to the 4 corresponding roots (of 1). Thus in 4-dimensional qualitative terms, phenomena keep switching in a dynamic interactive manner as between both positive and negative and real and imaginary identities.

Though conventional string theory still tries to envisage dimensions in solely real terms, we can perhaps see why the theory requires so many dimensions. At the most fundamental level of matter organisation it is not realistic to consider dimensions as separate. So if we now consider a dimension - not as independent - but representing a certain configuration of partially formed entangled dimensions, then we would obtain 24 possible permutations. And other permutations would also be possible. So the key finding is that a dimension would be looked on here as a certain dynamic configuration of what are 4 independent dimensions at the normal macro level. So just as in human terms, we can look on each personality type as possessing a unique way of configuring the experience of space and time (representing a distinct dimension), in corresponding physical terms each basic “impersonality type”, representing a unique (asymmetric) vibration of the string, likewise represents a distinct dimension.

9. I have already given a simple holistic mathematical justification for the importance of these 5 Types in "The Number 24".

10. There are strong connections here with an important paper by Polchinski which is discussed by Brian Greene in the “Fabric of the Cosmos” (pages 388 – 391). The gist of this paper is to show that the study of higher dimensional branes in M-theory can effectively be reduced to the analysis of strings. And this is exactly what one would expect within the interpretative model of conventional science that is strongly 1-dimensional! So the qualitative corollary of Polchinski's paper is that the higher dimensional levels of understanding (such as unfold with contemplative type development) can effectively be reduced to standard linear interpretation. However once again this represents reductionism of an unacceptable kind.


Penrose, Roger: The Road to Reality: A Complete Guide to the Laws of the Universe: Vintage, 2007

Greene, Brian: The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory: Vintage Books, 2000

Greene, Brian: The Fabric of the Cosmos: Space, Time, and the Texture of Reality: Vintage (Trade Paperback Edition), 2005

Kaku, Michio: Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension

Peat, David F.: Superstrings and the Search for the Theory of Everything: McGraw Hill, 1st Edition, 1989

Davies, P.C.W. and Brown, Julian (Editors): Superstrings: A Theory of Everything?: Cambridge University Press, 1992

The Official String Theory Web Site

Susskind, Leonard: The Cosmic Landscape: String Theory and the Illusion of Intelligent Design: Little, Brown and Company; First Edition edition, 2005

Hawking, Stephen: A Brief History of Time: Bantam 10th Anv. Ed., 1998

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