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 I – Setting the Context
Though science should combine both quantitative and qualitative aspects, only the former is presently recognised. The purpose of these articles is to introduce the neglected qualitative aspect illustrating its potential relevance with respect to modern developments in physics.
Modern physics is an extremely exciting field engaging the talents of a great number of gifted researchers and theorists.
Over the last 150 years or so it has brought about immense changes in our fundamental understanding of reality together with many associated technological benefits.
Now, following recent developments in string theory, many believe that we are on the verge of the long sought “Theory of Everything” in the form of a mathematical equation enshrining the original laws explaining the ultimate nature of the universe.
However despite the great achievements of analytic physics, such expectation is very much misguided.
Rather than being on the verge of unveiling nature's last remaining secrets, I believe we are about to gradually discover how limited in fact is the present scientific framework. Indeed an alternative holistic paradigm - with many degrees of ever more refined expression - potentially exists which needs to be employed in attaining a more comprehensive scientific worldview.
In the following articles I wish to demonstrate how we are still greatly lacking any integral notion of physics. Furthermore the quest for such an integral notion requires a very distinct type of understanding that is primarily geared to qualitative - rather than quantitative - appreciation of reality.
So my position here is very much in keeping with my findings in an earlier article on the Riemann Hypothesis, where I argued there, that it was the lack of a qualitative dimension to mathematics that prevented ready recognition of the true nature of this problem!
In parallel fashion I contend that it is the same lack of a qualitative dimension that is preventing recognition of basic relationships that can enable a truly unified appreciation of physical reality.
Further articles in this series will deal with certain key developments with respect to the three great modern revolutions i.e. relativity (special and general), quantum mechanics and string theory. I will attempt to show how present understanding can best be incorporated in a more integral interpretation. In turn I will be using such interpretation to provide an enlarged perspective for appreciation of both the merits and shortcomings of these developments.
Science and the Spectrum of Development
Conventional science is heavily dominated by just one logical system viz. the linear rational approach. It can help to place the nature of this system in context by looking briefly at the various main stages on the spectrum of development.
In my own approach I use the term band to refer to a general group of major stages (or levels) that share the same overall characteristics. In my latest treatment I include 7 major bands on the spectrum.
The lowest band contains 3 main levels (lower 3, lower 2 and lower 1).
These would equate well with the archaic, magical and mythic stages respectively that are outlined well in Ken Wilber's approach.
Now the key characteristic of these stages is that conscious and unconscious aspects of understanding remain confused in various ways with each other.
At the earliest level (lower 3) form cannot yet be properly distinguished from emptiness with phenomena in experience enjoying a merely temporary ephemeral existence. The first successful transition then comes with the differentiation of the bodyself as distinct from the wider psychophysical environment.
However, because the unconscious is still greatly confused with conscious experience, the child cannot yet distinguish wholes from parts thus leading to a somewhat magical appreciation of phenomena. So the second transition (Lower 2) when successfully negotiated, leads to the separation of the individual emotional self from collective reality.
At the third - and most advanced - of the lower Levels (lower 1), experience still suffers from considerable mythic confusion due to the remaining overlap of internal and external aspects of understanding. 
So the third transition relates to the successful differentiation of the mental self (where objective can be clearly distinguished from subjective reality).
Up to the time of the decisive Newtonian revolution the development of science was adversely affected in many ways through the various types of confusion associated with the lower band.
So for example, the failure to properly differentiate form from emptiness was evident in the way religious authorities attempted to control scientific understanding. Because it was deemed of great theological significance by the Catholic Church in the Middle Ages to maintain the Earth as centre of the solar system, this created a considerable barrier in accepting growing empirical evidence against that belief.
Magical and mythical beliefs also heavily influenced earlier views on creation and the nature of evolution. Indeed one could argue that continued acceptance of the aether into the early 20th century - that was only banished by Einstein's contributions - represented an important remnant of such lower band influence. 
Just as a young adult properly comes of age through successfully outgrowing the unconscious confusion of earlier years, science likewise could only truly come of age through the successful differentiation of conscious interpretation of reality from residual primitive influences.
The second major band of development i.e. middle, then relates to the specialisation of understanding so painfully acquired through the earlier stages.
It is such specialised appreciation that characterises the conventional scientific approach.
The first of these stages (middle 1) in this context relates to the development of concrete rational ability that is especially important for empirical investigation and the testing of various hypotheses.
The second stage (middle 2) then relates to the unfolding of formal rational understanding in a greatly increased power of abstraction and deduction.
This is the basis for new theories and hypotheses. For example it is very much to the fore in string theory, which is heavily mathematical in its formulation.
Of course these two stages in practise are usually combined in various ways. However, we can fruitfully distinguish research scientists geared to empirical investigation from theorists who formulate general hypotheses.
The third of the stages (middle 3) then relates to the more intuitive based rational understanding that Ken Wilber refers to as vision-logic.
From one perspective this enables the more fruitful integration of the two earlier stages (data and theory) in an enhanced manner. At a deeper level however it tends to foster that creative insight which is especially important in truly original endeavour. Great pioneers, such as Einstein, clearly operated at an enhanced vision-logic level of scientific understanding!
Key Characteristics of Conventional Science
With respect to phenomena, all reality is governed by the interaction of three key polarities. 
The assumptions we make regarding these polarities then govern the nature of interpretation that consequently emerges, with conventional science based on their clear unambiguous separation.
With respect to the first case, reality entails both internal and external aspects.
In conventional science these are separated from each other with the external identified with physical reality (as observed phenomena) and the internal with the psychological self (as observer of these same phenomena).
So, physical and psychological aspects are clearly divided leading to the view that reality can be successful interpreted in an objective manner independent of our psychological interaction with such reality. Though it may be readily admitted that physical understanding is not possible without reference to the psychological constructs we use for interpretation, conventional science assumes a certain passive correspondence as between both aspects leading to mere objective treatment of such phenomena. 
Quantum mechanics especially raises profound questions regarding this assumption. However the fundamental scientific paradigm remains unchallenged. Likewise the characteristic mindset of the vast majority of physicists has not altered despite the severe challenges posed by such questions.
Indeed the advent of string theory has in many ways strengthened the traditional belief system with ever-greater expectations among many that the ultimate (objective) solution to nature's secrets may be just around the corner.
With respect to the second set of polarities, reality - in every context - entails both (collective) whole and (individual) part aspects in continual interaction with each other.
Though - as we shall see - a comprehensive treatment of this key interaction entails the use of both conscious and unconscious, because of the limited rational nature of conventional understanding, a merely reduced interpretation can be given.
So basically in conventional science the (collective) whole is treated in a quantitative manner as the sum of its (individual) parts.
It has to be readily admitted that at one level this assumption has proved extraordinarily useful leading to an explosion of detailed analytical understanding in many fields. However this very success with respect to quantitative type appreciation has blinded us somewhat to a corresponding great limitation of conventional science.
Quite literally, such science has led to continual fragmentation with respect to current interpretation. Though in analytic terms we know more than ever before, in many ways we can increasingly feel as aliens in our world. Once again conventional science is based on the clear separation of the fundamental polarities of experience.
However it has no means as presently constituted of providing us with a truly integral understanding of this same reality. So the price that has been paid for quantitative appreciation is the loss of a corresponding holistic qualitative dimension. Thus even when science admits that a qualitative element is also involved as for example in the understanding of organic life processes, because of its linear conscious bias, it lacks the means to adequately incorporate this in interpretation.
When I was dealing in an earlier article with the Riemann Hypothesis, I pointed to a fundamental reductionism in mathematics, whereby the value of a number when raised to a higher dimensional value is subsequently expressed in a merely linear (1-dimensional) fashion. 
So, through such linear interpretation the inherent qualitative notion of a dimension is thereby lost.
In fact it is precisely the same in physics. Strange to relate, our customary interpretation of reality as 3-dimensional in space and 1-dimensional in time is based on a somewhat reduced - and ultimately misleading - notion of dimension. While this works well as an approximation at the customary “low-energy” level of macro scientific interpretation, it breaks down at higher energy levels (where greater refinement in understanding is required). 
The third set of polarities i.e. form and emptiness, is the most fundamental with conventional science treating this relationship in a greatly reduced manner where emptiness is reduced to form.
In this way the material realm is clearly separated from the spiritual with physical reality interpreted with respect to its merely phenomenal characteristics (observed in a direct conscious manner).
Though many physicists may be committed religious believers, unfortunately a considerable divide exists as between such beliefs and the actual manner - as scientists - in which they seek to interpret the phenomenal world.
Thus a clear lack of integration is evident in present scientific understanding as between the domains of matter and spirit. Indeed once again, much as scientists may wish to avoid the issue, quantum mechanics emphatically demonstrates that the accepted materialistic view of reality is quite untenable!
So we can sum up the key characteristics of conventional science in the following manner.
1) A dichotomy exists as between the internal and external aspects of experience with physical reality viewed in a somewhat objective impersonal fashion.
So, in this science a person is essentially viewed as a passive observer of a reality that independently exists.
2) A dichotomy also exists as between the (collective) whole and (individual) part aspects with the whole viewed in reduced manner as merely the sum of its collective parts.
The holistic qualitative dimension of experience is therefore essentially removed with such science geared merely to the quantitative interpretation of phenomena.
3) Finally, a dichotomy exists as between the form and emptiness aspects of experience with scientific attention focussed merely on physical phenomena (that can be consciously interpreted).
Therefore, though ultimately untenable, a fundamental separation is created as between the material and spiritual domains of reality.
I can readily accept that the three differentiations listed above have been crucially necessary for science to come of age and have enabled remarkable progress in our understanding and mastery of reality at many levels.
However there is a considerable downside in that emphasising differentiation, it has thereby ignored the equally important complementary aspect of true integration of scientific experience.
Thus carried to excess, conventional science can lead to considerable alienation in terms of our overall appreciation of the world.
Also because of its one-sided focus, even in its own terms it is inevitably doomed to failure in terms of unfolding nature's deepest secrets.
So as I would see it, what we are now witnessing in society is but the peaking of the mature phase of the first great stage of scientific understanding i.e. analytic quantitative interpretation based on clear differentiation of phenomena.
However I would see at least two further important stages that are required before a truly comprehensive scientific vision can emerge (and of which there is scarcely any recognition at present)!
The first of these is integral science (which forms the main topic of these articles).
The second is - what I refer to as - radial science, which represents the most advanced stage. In fact I believe that a truly comprehensive approach can only begin to unfold with radial science. Then, developed understanding with respect to both analytic (differentiated) and holistic (integral) appreciation of reality can finally interpenetrate with each other in ways that can be both highly creative and immensely productive.
Higher Stages of Development
Just as we have demonstrated how conventional science represents the rational linear understanding that characterises the middle band of development, integral science can likewise be shown to represent the more refined intuitively inspired rational appreciation of two higher bands on the spectrum.
As with the earlier bands (i.e. lower and middle) I would see the first of these higher bands as comprising three main stages (higher 1, higher 2 and higher 3). In some important respects these would equate with the psychic/subtle, causal and nondual realms (as outlined in Ken Wilber's approach). 
It is important to stress the point that integral science is based directly on the refined cognitive structures of the higher bands of development.
So before concentrating on the key features of such an integral science, let me briefly outline some relevant features of the higher stages
Just as the last stage before the middle (i.e. lower 1) was mainly concerned with the differentiating the external and internal aspects of experience, in complementary fashion this earliest of the higher stages (higher 1) is now concerned with their proper integration.
Before we can properly integrate in experience we must properly differentiate.
So the first task in development is rightly concerned with the successful differentiation of structures.
However when mastery of this process has been sufficiently attained (with the specialised development of the middle stages) in the maintenance of proper balance, attention should then switch to corresponding integration.
Higher 1 usually commences with illumination in a new outpouring of spiritual intuition, which transforms one's appreciation of nature.
However this is also associated with a more refined form of cognitive understanding.
As we have seen at the middle levels, in rational interpretation the external aspect tends to be clearly distinguished from the internal, so that physical reality is considered to exist “out there” somewhat independent of the enquiring mind.
However at this stage, as both of these aspects are now gradually integrated in experience, external and internal are viewed in an increasingly complementary manner.
This leads to what - in holistic mathematical terms - constitutes 2-dimensional understanding.
Such understanding is of a paradoxical bi-directional nature where every (external) physical event necessarily has a complementary (internal) psychological counterpart and likewise where every (internal) psychological event likewise can be given an (external) physical counterpart.
One deep implication of this is that reality can no longer be understood in merely physical, but rather in dynamic interactive terms as psychophysical.
And properly understood one important message of quantum mechanics is that reality is indeed psychophysical in this sense!
Interpreting a Cat
We will deal with quantum mechanics in more detail in a later article. However, it might illustrate the nature of this new understanding, by showing how an object such as a “cat” is significantly transformed through higher 1 interpretation.
Conventional scientific understanding - as we have seen - is literally 1-dimensional in rational terms. This means in effect giving just one direction to experience.
Thus in scientific terms a cat is interpreted unambiguously as “out there” with an independent objective existence.
However with higher 1 understanding, interpretation now becomes much more refined in a two-dimensional fashion (which thereby implies two directions).
One now realises that the external object of “cat” has no meaning independent of the - relatively - internal mental perception of “cat”.
Thus from one direction we have the external object of a “cat” (to which the - relatively - internal perception of a “cat refers).
Then from the other direction we have the internal perception of a “cat” (to which the - relatively - external object of a “cat” refers).
If we attempt to view both of these directions as relatively independent, then unambiguous type understanding can be maintained.
However it is in the very dynamics of appreciation at h1, through increased interaction of opposite poles, that a high degree of interdependence as between both aspects now takes place.
Thus where opposite directions are now viewed as interdependent the “cat” has a purely paradoxical meaning.
Now this is precisely the nub of the famous paradox of “Schrodinger's cat” (to which we will refer again in a later article). Thus the problem actually arises due to an inadequate means of cognitive interpretation.
In other words we cannot properly resolve such a paradox (or indeed the many more raised by quantum mechanics) while attempting to stick to the linear form of conventional scientific thinking.
The true significance of paradox at the higher 1 level of interpretation is that it can only be resolved in a nondual fashion (though intuitive insight). So keen recognition of the paradox involved at a rational cognitive level thereby facilitates a transformation in understanding to direct intuitive awareness (of a nondual kind). Then in reverse fashion, growth in such intuitive awareness more readily facilitates appreciation of paradox at the rational dualistic level.
The clear implication of this then for science is that quantum mechanics requires both dual and nondual aspects for proper interpretation.
And as conventional macro reality is necessarily built on underlying quantum mechanical relationships then this too ultimately requires both dual and nondual aspects.
So what we know as conventional science based on dual (i.e. local) notions is but a convenient approximation with respect to interpretation of macro reality that breaks down considerably at the sub-atomic regions of matter.
And one should notice in all this a clear example of vertical complementarity.
In other words as we descend “lower” in nature into the minute areas of sub-atomic matter, we require the understanding associated with a complementary “higher” psychological stage for proper interpretation.
So the reason why quantum mechanics appears so weird is that scientists are trying to appropriate a “lower” level of physical reality using the middle stage of understanding, whereas it properly requires a complementary form such as provided by the higher 1 stage.
Using holistic mathematical language, higher 1 is defined in 2-dimensional (bi-directional) terms by the complementarity of real polarities (i.e. that are understood in conscious terms). This then provides the appropriate means of interpretation for the lower 1 level of sub-atomic physical reality (which once again entails the bi-directional interaction of opposite polarities). 
This is what I refer to in my writings as an integral 1 approach, which represents the minimum that is required for cognitive translation of the dynamics of true integral understanding.
Limitations of Integral 1 Approach
However there are limitations with this approach. The continued dynamic negation of (posited) conscious phenomena may indeed lead to considerable growth in nondual spiritual awareness.
However it thereby shifts experience dramatically from conscious to the unconscious regions of personality.
Thus when phenomenal activity is restored at the higher 2 stage, it largely serves as the projected expression of the unconscious mind. Such activity then tends to fall into two categories that tend to increasingly diverge from each other. On the one hand, from the “higher” psychological perspective, we have transparent spiritual archetypes i.e. where phenomena increasingly operate as universal holistic symbols; on the other - from the corresponding “lower” perspective - we have the projection of remaining repressed primitive instinctive behaviour that tends to have a very fleeting existence.
It is the increasingly refined dynamic interaction as between such unconscious projections that reveal the true relationship as between wholes and parts, which in turn has a vital significance for the proper scientific interpretation of both dimensions and physical objects respectively.
Just as mathematics now recognises that the number system is complex with real and imaginary components, the qualitative corollary of this is that actual scientific understanding is also complex with real (conscious) and imaginary (unconscious) elements.
Once again, the real and imaginary aspects of such understanding are translated through the use of linear and circular type logic respectively!
So for example an object such as a house can be given a real conscious existence. However the house also possesses an “imaginary” unconscious identity as indicative of a holistic quest for meaning. For example one might refer to a “dream” house (directly suggesting this unconscious holistic aspect)!
Because of the interaction of conscious and unconscious in experience, all objects necessarily entail both real (conscious) and imaginary (unconscious) aspects with each having - relatively - positive (external) and negative (internal) directions.
This exactly corresponds with the 4-dimensional interpretation of reality, which characterises stage 2 cognitive appreciation that is bi-directional in both horizontal and vertical directions.
Using such understanding we can provide the appropriate cognitive basis for the proper integration (without gross reductionism) of wholes and parts in experience with both - relatively - real and imaginary in qualitative terms with respect to each other. Furthermore, it offers a much deeper philosophical explanation for the use of complex numbers at the quantum mechanical level.
It also provides a compelling explanation as to why relativity and quantum mechanics relating to the whole and parts of the physical universe remain - within their own respective frames of reference - incompatible theories.
In particular as we shall demonstrate, it provides the appropriate means for properly interpreting the complementary wave and particle aspects of sub-atomic particles.
I refer to this 4-dimensional perspective as the integral 2 approach. Because we live in a world of four dimensions, this approach is immensely important in providing a qualitatively consistent account of all dynamic holonic interactions within this framework. Indeed ultimately I believe it is of the utmost importance for physics as it leads to a completely new understanding of the true nature of space and time. Not surprisingly therefore it can also throw considerable light on the need for a greater number of dimensions in string theory.
However, in terms of contemplative type development, problems still remain which can only be resolved at the next stage (higher 3).
At h1, one can attempt to integrate internal and external aspects horizontally within a given stage.
Then at h2 one can attempt additionally to integrate whole and part aspects vertically between the various stages.
Typically however imbalances remain. In transcendent development, more emphasis is placed on top-down integration (the lower from the perspective of the higher). However it is equally important to achieve bottom up integration (the higher from the perspective of the revisited lower).
Also based on personality preferences the external and internal aspects may remain lop-sided.
An introvert for example will typically place more emphasis on the internal side attempting to integrate the external aspect from this standpoint, whereas the extrovert will tend to operate in reverse attempting to integrate the internal from the more developed external side. So h3 is concerned with the attempted integration of the diagonal polarities that is required to achieve greater balance with respect to both sets of aspects (horizontal and vertical). 
Again when successful, to the necessary extent, this enables the greatest possibilities for the unfolding of pure nondual insight. Associated with this is a highly refined rational understanding of the dynamic nature of phenomena with the four additional dimensions ultimately providing - I believe - the ultimate true symmetrical nature of the physical forces.
This is what I refer to as the integral 3 approach, which culminates in a holistic TOE.
Indeed such a paradoxical qualitative TOE which simultaneously pre-exists all phenomena (as potential source of reality) while going beyond all phenomena (as actualised goal of this reality) is the only TOE that can validly exist.
Once we are dealing with the world of phenomena the truth of any interpretation is necessarily of a limited and approximate nature.
And just as analytic interpretation has its specialised development in the unfolding of the middle band, holistic interpretation, as the basis for integral science, has its specialised development in the next major (4th) band on the spectrum.
So the specialised appreciation of integral science would relate to this 4th band of development. 
One might be inclined to think that physics should culminate with the specialised development of the higher stages. But it truth, from a more comprehensive perspective it can only properly begin with radial development.
So there are two major aspects required for radial science.
The first requirement is specialised understanding with respect to the (analytic) differentiation of phenomena in a quantitative manner. This aspect defines the conventional approach to physics representing the rational structures of the middle band of the spectrum.
However the second - greatly neglected - requirement relates to corresponding specialised understanding with respect to the (holistic) integration of phenomena in a qualitative manner. Though in a direct sense this relates to a pure type of spiritual awareness fostering the contemplative vision, indirectly it is mediated through associated refined rational structures of a circular nature with a potentially crucial role to play regarding appropriate interpretation of reality. And as we have seen such understanding slowly unfolds during the three stages comprising the higher band before attaining specialised development with the 4th band on the spectrum.
Only when these two distinctive aspects have attained specialised development can they be most fruitfully combined with each other in an increasingly interactive manner.
So the last great phase of radial science can now only properly begin. Just as natural life begins at birth, this new transformed life - that marries both the human and divine potential of the personality - begins now with a spiritual rebirth.
Once again such science reflects the understanding that unfolds with the radial stages.
Recently I have come to the view that radial development - in its full unfolding -should give rise to 3 further bands on the spectrum (bands 5, 6 and 7). In this way we thus have 7 major bands on the spectrum as a whole (with three main stages in each band).
So with band 5 we have an emerging radial understanding mirroring earlier natural development.
Here one would gradually move through a series of parallel stages gradually come to terms with the true nature - which is potentially unlimited in scope - of this new enhanced scientific approach.
Then at band 6, a certain specialisation in the radial approach would take place. However even here some though a fruitful marriage between both aspects would necessarily exist, some practitioners would be relatively more inclined to the analytic and others to the holistic aspect respectively.
Then finally band 7 would lead to a strengthening of the weaker aspect leading to a greater degree of balance as between the analytical and holistic aspects of understanding.
However I mention the radial purely to provide a greater level of perspective on the overall nature of science.
The purpose of these articles is to attempt to demonstrate the nature of Integral Physics.
So we will conclude now by looking at some of its key features.
Key Features of Integral Physics
1) Rather than being a passive observer of events, a person is now understood as a direct participant - indeed co-creator - of reality. The notion therefore of a separate physical world that can be interpreted in an independent objective manner is no longer tenable.
Reality is now understood as psychophysical i.e. where both physical and psychological aspects necessarily interact.
Whereas conventional science is geared to the quantitative interpretation of physical phenomena, integral science is directly of a qualitative (rather than quantitative) nature.
A key feature of this new understanding is that - when appreciated appropriately in an integral fashion - every physical interpretation of reality has a corresponding complementary psychological meaning with an underlying holistic mathematical structure that is identical. Equally in similar fashion every psychological interpretation can be given a corresponding physical meaning.
Such integral appreciation where physical and psychological interpretations are seen as complementary provides the means for continual transformation of experience through nondual (intuitive) awareness. Growth in such awareness of a directly spiritual nature in turn readily facilitates the ability for further development of complementary understanding with respect to both physical and psychological aspects.
Thus an important goal of integral physics is to establish - from one perspective - a matching complementary psychological partner for every physical notion and from - the equally valid opposite perspective - a matching physical partner for every psychological notion. Indeed it is the ability to holistically appreciate reality in this complementary fashion that constitutes the very basis of the integral approach.
So it will be my intention with respect to the three areas of relativity, quantum mechanics and string theory to demonstrate the manner in which such complementary understanding can lead to a deeper integral appreciation of reality.
2) Conventional physics, which qualitatively is of a one-dimensional nature, is based on linear rational understanding and represents a solely conscious based interpretation of physical events. However the unconscious is also very much involved in qualitative type appreciation. It is especially significant in properly interrelating, in an unreduced fashion, the relationship between whole and part.
So another key feature of integral physics is that two distinct logical systems, which are linear and circular with respect to each other, must be employed in the appropriate interpretation of events.
The linear aspect of this system is provided through either/or unambiguous logic (of clear separation of opposites). The circular aspect is then provided through a both/and paradoxical logic (of the complementarity of opposites).
Thus integral physics employs in qualitative terms a binary rather than unary logical system.
We can perhaps illustrate the potential benefits of this system with reference to the corresponding quantitative representation of numbers in unary and binary format respectively.
In a unary system, 3 would be represented as 3 separate digits 111; 7 would be represented by seven separate digits 1111111 and so on.
Thus to represent 1,000,000 (denary) in unary format, we would need to employ one million separate digits which would be extremely inefficient and cumbersome.
However, as we know, modern computers are largely based on the binary system for representation of numbers where we use 1 and 0.
The zero here is tied up with dimensional transformations.
In the binary system we start with 1.
Then for 2 we introduce the first dimensional transformation by adding one 0 to 1 (i.e. 10). Then 2 * 2 (2-dimensional) is represented by adding a second 0 (i.e. 100). 2 * 2 * 2 is represented by adding a third 0 (i.e. 1000). Thus each dimensional transformation in turn is represented by adding a further 0.
So, firstly the binary allows for (qualitative) dimensional transformations, which are not possible in the unary system.
Secondly even in quantitative terms it allows for a much more efficient representation of numbers.
So instead of requiring one million digits, 1,000,000 in binary requires only 20 i.e. 11110100001001000000.
Now, if we apply such thinking in holistic qualitative terms to a comparison as between the unary (conventional) and binary (integral) scientific models, two key conclusions emerge.
(a) The integral approach allows for proper qualitative (dimensional) distinctions to be made whereas the conventional must necessarily reduce - in any given context - qualitative (whole) to quantitative (part) consideration.
(b) Even in quantitative terms, an integral approach makes for a much more economical and efficient interpretation of relationships. We will see later with respect to string theory, that physicists are experiencing great difficulty in effectively explaining the nature of their findings. I believe - and hope to demonstrative - that an integral approach can greatly help to provide a more enriched framework in which to place certain key findings in proper perspective.
3) Conventional science is based on the quest to know reality (in rational terms) in a manner that remains independent of the self. However the deepest quest is to experience reality as one, through full integration of self with the world in an ineffable manner beyond (and pre-existing) phenomenal recognition. Thus because the ultimate nature of reality in the identity of physical and psychological aspects is nondual and utterly mysterious, it cannot be interpreted consistently in a linear rational fashion.
Thus the current quest to seek a Theory of Everything in dualistic terms is by its very nature untenable.
From an integral perspective therefore, reality can only be properly known through the nondual “Experience of Everything” in mystical oneness with the universe. However indirectly, this ultimate reality can be translated in a coherent symmetrical fashion using three distinctive types of paradoxes with a simple underlying holistic mathematical rationale. These then serve as the proper basis for the more contingent phenomenal interpretations of phenomena.
So, just as analytic science has its special tool in conventional mathematical understanding, likewise integral science has its own special tool in holistic mathematical appreciation.
Thus, in terms of phenomenal interpretation, any theory that is formulated is necessarily of a limited relative nature and cannot therefore operate as a Theory of Everything. So there is a great confusion underlying contemporary physics in its pursuit to unravel a TOE. Ultimately what is known about reality cannot be divorced from the psychological constructs that we use to interpret this reality. So the very quest to understand the world properly centres on the key relationship between the knower and what is known. And this can only finally be resolved when knower and what is known become one in an ineffable spiritual manner.
This article has been of a general nature attempting to place in proper perspective the role of both recognised conventional (analytic) and largely unrecognised integral (holistic) science.
Just as conventional science, on which accepted physical understanding is based, reflects the cognitive structures of the middle band of the spectrum, integral science equally reflects the more refined cognitive structures of the higher bands where intuitive and rational are seen to interpenetrate in a dynamic interactive manner.
A key criticism that I would have with conventional mappings of higher development is that the nature of such refined cognitive structures has not yet been properly articulated, certainly from an accessible Western perspective!
Therefore it has always been a key goal of my own writing to attempt to clarify the nature of such understanding so as to provide the basis for a truly integral approach to both mathematics and science.
And I have never seen such science as limited to vague generalities. Rather I have always had the confidence that it could tackle and indeed unravel key problems that remain unsolved using existing methods.
So in the next article I hope to illustrate this with reference to Einstein's theories of relativity.
1. Even though typically in development, the rational successfully unfolds after the mythic stage, it is important to realise that it is never entirely outgrown. For example powerful mythic influences still underlie conventional religious appreciation in Western society.
2. The aether (ether) - or more fully luminiferous aether - was viewed as a physical medium necessary for the propagation of light. Though experimental attempts to detect the aether proved quite unsuccessful during the 19th century, many scientists stubbornly maintained continued belief in its existence. This even remained in some quarters after Einstein demonstrated that its hypothetical existence was no longer necessary.
3. In relation to the first two sets of polarities there are strong similarities here with Ken Wilber's four quadrants. So the relationship as between internal and external polarities mirrors Wilber's RH (exterior) and interior (LH) quadrants, while the corresponding relationship as between part and whole polarities likewise mirrors Wilber's Upper (individual) and Lower (collective) quadrants.
However my consistent reservation with Wilber's treatment has always related to the lack of any real dynamic interaction as between his quadrants. Therefore I would see his treatment (though valuable in its own regard) as still representative of a 1-dimensional approach that is not adequate for true integral appreciation.
The 3rd set of polarities that I identify (form and emptiness) has no direct correspondent in Wilber's 4-quadrant model.
4. There are various degrees of refinement possible - even within conventional science - as to the nature of this correspondence.
The extreme position would be the pure representation view i.e. that our mental constructs exactly correspond with the (manifest) objective features observed.
A more moderate position, while accepting that the absolute representation view is not strictly valid, would maintain however, that it is not the role of scientists to be directly concerned regarding the precise philosophical relationship as between knower and what is known.
Some scientists would even accept that quantum mechanical findings fatally undermine traditional assumptions regarding the very existence of an “objective” world.
However they still cling in practice to an overall paradigm of science that does not adequately reflect these new findings.
So in effect, conventional science still operates as if reality can indeed be interpreted with respect to its - mere - objective characteristics.
5. For example, 3 raised to the power of 2 entails both a quantitative number (3) and a qualitative dimension (2). However the mathematical value is given in a merely reduced quantitative fashion as 9 (i.e. 9 raised to the default dimensional value of 1).
However if one considered a table top (with each side measured in metres) then 3 (raised to the power of 2) would give a result of 9 square metres. So the result here of 9, expressing an area in 2-dimensional units is clearly qualitatively different from 9 as any measurement in linear terms.
6. In fact momentary reflection on the matter will show that there is a basic asymmetry in our treatment of space and time (with 3 of space and 1 of time).
It is this continuing belief - even at advanced levels of physics - that time is
1-dimensional that above all characterises the linear nature of physical understanding!
7. Though I admire what Ken Wilber has achieved, I have always had a fundamental criticism of his model as an appropriate basis for an integral scientific approach.
Wilber adopts a very Western style treatment of lower and middle stages with considerable emphasis on the various cognitive structures that unfold during these stages.
However he then attempts to graft on a very Eastern style appreciation of higher stages where the emphasis is now largely on the various contemplative states that unfold through successful development. And such refined intuitive awareness - in the growing experience of emptiness - is indeed one very important aspect of such development!
However from a balanced perspective, we must equally outline the nature of the higher structural forms that unfold with these stages and especially in a scientific context the precise cognitive rational forms with which they are consistent. In a dynamic integral fashion both the intuitive states and refined structural forms interpenetrate and are thereby very much interdependent with each other. However we do not find any coherent treatment of the nature of such higher cognitive structures in Ken Wilber's writings, which are not to be confused with vision logic (representing merely the most advanced cognitive structures of the middle levels!)
8. The basic holistic mathematical structure of each dimension (in qualitative terms) is obtained with reference to its conventional roots (from a quantitative perspective).
So for example the 2 roots of 1 are + 1 and – 1 respectively. In corresponding terms, 2-dimensional understanding entails interaction of opposite poles of form (positive and negative) that are complementary with respect to each other.
Thus whereas conventional quantitative understanding of the two roots is based on either/or logic, qualitative 2-dimensional understanding is based on both/and logic.
9. It has to be understood that one born for example as an introvert is still likely to maintain this basic personality characteristic having attained a substantial degree of contemplative union. The key point however is that a level of integration exists with respect to opposite personality characteristics (unique for each individual) which must be reached before such union can be successfully attained.
10. A detailed outline of the relationship as between structures and states for the 4th band can be obtained in Chapter 10 at my site (under specialised higher levels).
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