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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 I | Part II | Part III

Dynamic Nature of
the Number System

Part III: The Bigger Picture

Peter Collins


My basic position is that understanding of the number system must be placed in the context of development itself.

Indeed properly understood in a dynamic interactive manner, the number system is seen as inseparable from such development.

So right away, we have to tackle the huge fallacy that somehow the number system can have an (external) meaning independent of (internal) interpretation. Even the (mistaken) belief in the purely objective nature of this system, reflects a corresponding (mistaken) interpretation i.e. that both aspects can be absolutely separated from each other in experience.

Indeed to put what follows in perspective, it might be helpful to refer to the same notion of “a spectrum” as used by Ken Wilber in his first published book “The Spectrum of Consciousness”.

As we know natural light comprises just one small band on the overall electromagnetic spectrum.

In a similar manner, the common-sense type of psycho spiritual light (i.e. intuition) that informs conventional mathematical interpretation likewise forms just one small band with respect to the overall spectrum of potential intuitive development.

Indeed the very nature of such conventional intuition as we have seen is to quickly reduce it in terms of rational type interpretation that is merely quantitative in nature.

However, as we shall see, there are many further bands of understanding on the spectrum which can unfold, each associated with a unique increasingly refined type of intuition (with corresponding refined rational structures of a circular paradoxical nature). These bands enable both quantitative and qualitative type appreciation of mathematical symbols to unfold in an increasingly dynamic interactive manner.

Our experiential understanding of the number system thereby changes with each band on the spectrum. Thus the relationship of the (external) objective aspect of this system to our (internal) understanding of its nature is unique for each band.

Now what happens with Conventional Mathematics is that interpretation consistent with just one band, which is Band 2 on my map - where truth is of an absolute nature - is then imposed on all other bands.

In fact, as such Mathematics is completely lacking a psychological developmental context, the possible relevance of other bands is not even considered.

In my limited discussions with practitioners on the nature of Mathematics, I have consistently come up against this problem - even among the most open minded - of a marked inability to recognise that other bands on the spectrum of development are associated with uniquely distinct mathematical systems.

In this respect, it is similar to one who, used to defining electromagnetic energy in terms of natural light, is thereby unable to accept the existence of other bands (not visible to the ordinary eye).

Map of Spectrum of Development

It may help to place matters further in perspective by discussing my map of the spectrum of development.

In its most recent form, I outline 7 bands (representing the most general grouping of stages) on the spectrum.

Each band then comprises 3 main levels and then within each level are numerous other stages and sub-stages.

Now, as one might expect from what has been already said, I am especially anxious to portray development throughout the spectrum in a dynamic interactive manner.

Indeed I find the traditional asymmetrical approach to development very misleading.

From my perspective, everyone - by definition - already has a certain dynamic access to all stages on the spectrum.

Of course rudimentary access is distinct from the mature specialised understanding of each stage! So certainly there is the need for “ascent” of the spectrum in attaining more mature development of “higher” level stages.

However, what is not at all properly emphasised in traditional accounts is the equal need for continual “descent” of the spectrum. In other words, healthy integration from a vertical perspective must be both “top down” (where “lower” stages are integrated from the perspective of “higher”) and likewise “bottom-up” (where “higher” stages are integrated from the perspective of the “lower”). And this is a continuing process throughout development. In fact, as we shall see, this distinction is vitally important for proper appreciation of the nature of the Zeta 2 and Zeta 1 zeros respectively.

Now, it would be very misleading at this early point in our human evolution to suggest that complete development of the spectrum is remotely possible. Even the linear understanding that characterises current mathematical and scientific development is only highly specialised in a relative sense (when compared to less developed bands). Thus the potential for further progress, even with respect to this recognised band, is truly unlimited.

Having said this, I do believe we have reached a stage where very rapid development - at least in some important respects - of further bands, hitherto largely neglected, can take place in the near future. Indeed, a better future for our species on this planet, may well depend on such rapid development!

The Analytic Bands

So my own map of development is offered as a template of what I believe is possible at this early stage in human evolution, in relation to the various bands (with a special emphasis on the structures relating to scientific type understanding).

Overall, I would see 3 main phases in terms of the full developmental ideal. With respect to Mathematics, the first two bands relate to analytic understanding of a (conscious) rational nature; the next two relate to corresponding holistic appreciation of an (unconscious) intuitive nature, while the final three bands relate to the dynamic two-way interplay, in a mature balanced fashion, of both analytic and holistic aspects.

Initially - as well recognised in developmental psychology – in early childhood, both conscious and unconscious are still greatly entangled in experience with each other.

Thus, the first task of development is devoted to the gradual differentiation of conscious type experience (thereby reducing the confused influence of the unconscious).

So Band 1 on the Spectrum is devoted to this task and I would be in broad agreement with existing accounts of development in this regard. From a scientific perspective, this would entail the gradual emergence of linear type understanding of reality.

Band 2 then relates to the specialised development of such conscious understanding, which in a mathematical and scientific context represents linear (1-dimensional) interpretation.

However what is extremely unhealthy in our culture is the undue domination of this mode of appreciation. Therefore the unique nature of further bands on the spectrum is thereby reduced in terms of this mode (especially in mathematical and scientific terms).

When it comes to the number system, contemporary mathematical interpretation reflects an extreme version of the understanding of Band 2.

Just as the first two bands are devoted to the unfolding of linear type understanding, the next two bands are devoted to the unfolding of refined intuitive type awareness, which is consistent with circular (paradoxical) type interpretation from a rational perspective.

These bands have been typically associated in the past with the various mystical traditions in the pursuit of authentic contemplation.

However my point - which I now believe our culture urgently needs to embrace - is that these bands are equally important for Mathematics and Science, with a unique distinctive holistic (qualitative) interpretation thereby associated.

In the last article I referred to the zeta zeros (Zeta 1 and Zeta 2). Even though we may not yet realise this truth, properly understood, these two sets of numbers are of similar importance in our number system, as the analytic aspects (cardinal and ordinal). In fact quite simply they represent the holistic counterpart that - when appropriately understood - fully complements the analytic aspect of the number system in a dynamic interactive manner.

Now, just as understanding of the analytic number system is a product of human development, likewise understanding of the holistic number system (with its Zeta 1 and Zeta 2 aspects) is likewise a product of such development.

The Swiss psychologist Piaget demonstrated in his studies, how in early development a degree of unconscious confusion is inevitably associated with an infant's appreciation of number.

However, through the growing differentiation of consciousness, quantitative type recognition is gradually freed from such confusion leading to the consolidation of linear analytic ability.

Due to the narrow specialised emphasis in Mathematics on linear type understanding, a highly abstract appreciation of the number system has now emerged. Not alone have we real and complex numbers but further extension has occurred with the inclusion of increasingly abstract notions such as p-adic number systems and adele rings.

So the development of Mathematics - though in some respects wonderful to behold - has taken place almost entirely within the specialised linear understanding associated with Band 2.

However, what most mathematicians have not yet grasped is the fact that an entirely distinctive holistic qualitative type appreciation of all its symbols potentially emerges over the next two bands. And when it comes to clarification of the nature of the Zeta 1 and Zeta 2 zeros, these higher bands are especially relevant.

The Holistic Bands and the Zeta Zeros

Bands 1 and 2 are largely geared in psychological terms to unfolding of the differentiated aspect of (conscious) experience. The next two Bands are geared to unfolding of the corresponding integrated aspect of (unconscious) type experience.

Then, just as understanding of the analytic number system unfolds through Band 1 (with specialised interpretation taking place at Band 2), corresponding understanding of the holistic nature of this number system, i.e. the zeta zeros is associated with Band 3 (with specialised appreciation unfolding at Band 4).

Indeed the key issues with respect to these zeros are especially relevant in a psychological developmental context.

So once more, using psychological terminology, the two great tasks of development relate to differentiation and integration respectively.

Again as we have seen, the first two bands are associated largely with successful differentiation (in an analytic type manner).

However differentiation without integration would be highly imbalanced. So the next two bands are geared to the corresponding task of mature integration (in a holistic fashion).

Now as I have mentioned from a vertical perspective, integration can be viewed from two complementary directions i.e. “top-down” (where one attempts to integrate the “lower” from the “higher” levels) and “bottom-up” (where one attempts to integrate the “higher” from the “lower” levels).

The Zeta 2 and Zeta 1 zeros are intimately associated with this task in development.

So, just as both “top-down” and “bottom-up” represent here two complementary aspects with respect to the integration of psychological experience, equally the Zeta 2 and Zeta 1 zeros represent two complementary aspects with respect to the integrated (i.e. holistic) appreciation of the number system.

When “higher” spiritual contemplative type development starts to unfold (Band 3, Level 1) initially - perhaps more in the case of males - it tends to take a strong transcendent focus.

One thereby operates on the principle that reason is superior to the senses. Thus one uses reason e.g. through a range of ascetic disciplines, to control the “lower” physical impulses of the unconscious in the pursuit of “higher” spiritual growth.

When authentic progress is the made in this regard, thereby enabling experience at a “higher” contemplative level of awareness, typically one then attempts through a process of “top-down” integration to integrate the “lower” stages from the perspective of the “highest” now attained.

Though this process initially serves a valid function, it can lead unwittingly to imbalance through a false form of hierarchy i.e. where “higher” stages are considered superior to ”lower”. Therefore, because one now identifies progress with these “higher” stages, one can experience a false sense of elitism, as part of a spiritual minority (operating at an advanced contemplative level).

However the very process of attempting to maintain rational discipline - even at a spiritually refined level - can lead to the gradual repression of “lower” primitive impulses, which are deemed incompatible with pure spiritual awareness.

Indeed I have written in other places that a significant part of the depression that can arise during the “dark night of the soul” is likely to be due to the accumulated repression of “lower” level physical instincts.

So, too much focus on transcendence is unbalanced, as it leads to an undue emphasis on the “top-down” aspect of integration.

The Zeta 2 zeros are (initially) directly associated with this transcendent aspect of psycho spiritual development.

In this way, through the growth of spiritual intuitive awareness, the fundamental polarities of experience (such as external and internal) increasingly interact.

Indeed, my own realisation of the nature of 2-dimensional understanding sprung directly from such development.

Also, from an early stage, I was fascinated with the relationship of this new appreciation to conventional experience (based on 1-dimensional interpretation).

So it was this very process of “top-down” understanding i.e. in attempting to relate the qualitative 2-dimensional notion (of the interdependence of opposites) in a reduced (quantitative) 1-dimensional context, that led to my appreciation of the first of the Zeta 2 zeros.

And this knowledge of the 1st of these zeros then served as the basis for investigation of the nature of the other zeros (associated with higher dimensional numbers).

One may only realise the limits of the - mere - transcendent approach to development (with its unbalanced emphasis on rational discipline) through a severe psychological crisis. Resolution of this crisis can then lead one to gradually change course in the opposite direction, thereby enabling one to give due expression to the corresponding immanent aspect of spiritual development.

From this perspective, rather than attempting to control - and indeed censor - lower primitive instincts, one gradually lets them speak for themselves, through being projected into conscious awareness.

Now initially - because of remnants of undue super-ego influence - these will tend to be projected in a somewhat involuntary manner, whereby they become attached to, and thereby confused with conscious type phenomena.

So a long process follows, where ever deeper primitive impulses are slowly brought into consciousness, initially projected in an involuntary fashion.

Thus the task here is to gradually integrate, through a growing holistic affective awareness, such physical promptings in a seamless manner.

So this aspect is thereby geared ultimately to the full differentiation of primitive impulses - whereby they are effectively brought to conscious light - which equally entails their full holistic integration at an unconscious level.

Therefore by definition, the full (conscious) differentiation of prime i.e. primitive impulses (relating to repressed unconscious material) requires their corresponding full (unconscious) integration in a collective manner.

And this is the way in which “bottom-up” integration is obtained, where “higher” are integrated from the perspective of the “lower” stages. So again this process entails (a) the individual differentiation in conscious terms of - former - unrecognised prime instincts and (b) their holistic collective integration in an affective manner, leading to a - now - purified unconscious awareness.

What is truly remarkable is that this exactly parallels the role of the Zeta 1 (Riemann) zeros with respect to the number system.

Therefore in a dynamic interactive context, the very existence of the prime numbers (as distinct individual quantitative members of the number system) is only possible through their corresponding collective holistic integration in qualitative terms (which occurs through the Zeta 1 zeros).

So the very process of solving the “primitive problem” i.e. integration of conscious and unconscious, in an internal psychological manner, is exactly complementary with the corresponding solution of the prime number problem (which entails reconciling the individual quantitative nature of each prime with their collective qualitative nature).

This provides valuable insight into the nature of the Riemann Hypothesis.

All the (non-trivial zeros are postulated to lie on the line through 1/2. As we have seen this implies maintaining a perfect balance as between (analytic) differentiation and (holistic) integration.

The linear nature of these zeros simply entails in psychological terms the differentiation of unconsciously emitted impulses so that they are made fully conscious and thereby fully compatible with linear type understanding in experience. The initial involuntary nature of projections also reflects the fact that they have not yet been properly appropriated in a conscious manner!

So again it is only when these projections are fully differentiated (in conscious terms) that they can equally be fully integrated (from an unconscious perspective) with their involuntary nature thereby ceasing.

The imaginary nature of the line simply means that these projections refer directly to the unconscious - rather than conscious - aspect of experience and in holistic terms, the imaginary line provides the indirect conscious means of expressing their inherent unconscious nature.

Thus the Zeta 1 zeros are therefore vital in enabling such seamless psychological “bottom-up” integration to take place.

This equally entails - though we are as yet unaccustomed to look at matters this way - that the very resolution of primitive instinctive behaviour leads to the creation in experience of qualitative dimensions (which are exactly represented by the Zeta 1 zeros).

Now, these qualitative dimensions can be understood as pure (intuitive) energy states in psycho spiritual terms.

Through appropriate development, these then become increasingly immanent and grounded in everyday experience, thereby facilitating the seamless integration of both conscious and unconscious. So involuntary instinctive impulses - that earlier arose due to confusion of unconscious desire with conscious phenomena - through successful differentiation, are now gradually associated with these refined intuitive states.

These then act to harmonise the differentiated conscious expression of such prime instincts with the underlying holistic nature of unconscious desire (to which they properly relate).

Put more simply this ultimately requires the full identity of both conscious and unconscious aspects of experience.

So from the internal (psychological) perspective, solving the primitive problem ultimately requires the identity of both conscious and unconscious aspects of experience.

Equally, from the complementary external (physical) perspective, solving the prime number problem - in understanding the relationship of the zeta zeros to the primes - likewise requires the full identity of both analytic and holistic interpretation.

The mystical traditions provide one way of distinguishing the Zeta 1 and Zeta 2 zeros from a psychological perspective.

Once again we must distinguish the two complementary aspects of immanence and transcendence respectively.

Transcendent realisation relates to the spiritual seeing of the (quantitative) parts as collectively included in a (qualitative) whole, which is however not confused with the parts.

Immanent realisation relates to the corresponding spiritual seeing of the (qualitative) whole uniquely contained in each (quantitative) part, again without confusion of either aspect.

For example Blake's famous line “To see a world in a grain of sand” relates to this latter immanent type of realisation.

Now, the Zeta 1 zeros can be properly viewed as “grains of sand” in this context, through which the holistic nature of the number system is reflected, in a highly refined spiritually immanent manner. So from this perspective, they represent pure psycho spiritual energy states that express, in a highly dynamic manner, the complete harmony throughout the number system of its quantitative and qualitative aspects. They thereby enable a perfect tension in interpretation to be maintained as between the individual and collective nature of the primes with respect to the natural number system.

And this behaviour with respect to the primes is perfectly replicated with respect to the corresponding psychological behaviour outlined.

So this internal psychological development provides the appropriate holistic means of interpretation with respect to the physical nature of the zeta zeros in external terms.

Physical Connections

In dynamic terms, both the physical and psychological aspects of the number system are complementary.

In this respect, it is interesting to note that a striking correspondence has been observed by investigators as between the Zeta 1 (Riemann) zeros and the physical energy states of quantum systems.

This finding did not surprise me when I first read about it, as coming from the holistic perspective, I already realised that the zeta zeros should indeed relate to quantum states at the sub-atomic level of matter.

But why in this regard, the now recognised connection as between quantum mechanics and chaos theory?

I will just point to one interesting complementary feature of both theories.

Though in qualitative philosophical terms, the understanding of quantum mechanical behaviour appears quite paradoxical (non-linear) its actual (quantitative) equations are expressed in a linear manner.

It is somewhat the reverse with respect to the formulation of chaos (in any relevant system). Here its (quantitative) equations are of a non-linear nature. However in terms of qualitative understanding, its behaviour belongs to the classical realm (based on linear notions of interpretation).

So it would not be surprising, from this perspective, that the ultimate two-way relationship as between quantitative and qualitative aspects could thereby entail quantum chaology. [1]

Some physicists postulate that a physical or quasi-physical system (of a quantum chaological nature), with identically matching zeta zeros, will eventually be discovered, thus enabling proof of the Riemann Hypothesis.

However this, I believe, represents confused thinking! This ultimate dynamic system, for which the physicists are searching, corresponds simply to the number system. However, though necessarily inherent in physical reality, it cannot be directly identified with the phenomenal aspects of such reality.

One could validly say that as we approach its phenomenal limits (where dynamic interaction reaches an extraordinary rapid level) that the behaviour of physical reality would then approximate closer to the inherent dynamic nature of the zeta zeros.

In a sense, the semi classical system of chaos, underling quantum reality, provides a means of bridging the physical world and the world of number. Therefore the zeta zeros (as the dynamic extreme of the number system), would then dynamically manifest themselves through physical (and related psychological) phenomena, that display quantum chaological features.

This thinking would also be very relevant in terms of the origins of the physical Universe.

The Big Bang where the ineffable “nothing” gives birth to “something”, primarily relates to the birth of the number system itself (in a dynamic framework of space and time).

Though this in no way removes the great mystery of creation, it does however extend the limits of investigation beyond physical phenomena into the origin of number itself.

And here we are faced with the equally enormous problem of how the number system, by its very existence, has already innately solved this incredible mystery of enabling the consistent relationship of the quantitative and qualitative aspects of phenomena throughout the Universe.

In this sense the mystery of the Universe is the same mystery inherent in the number system.

When one begins to truly appreciate the nature of the number system, it becomes the most exciting possible area of study, filling one with great wonder that is utterly boundless in nature.

It is indeed a great pity that Conventional Mathematics still seeks to fruitlessly solve this mystery through a reduced form of analytic interpretation (in a merely quantitative manner).

Rightly understood, the number system is like the most marvellous jewel imaginable, ultimately surpassing all rational thought and thereby worthy of endless contemplation.

Indeed, I have no doubt that in future eras, the zeta zeros will play a special role in advanced spiritual practice, as the most appropriate way of entering ever more fully into the great mystery of “who we are” and our place in the Universe.

Now when we fuse energy and form we get dynamic transformation.

So, again quite remarkably our number system - which we misleadingly view in an abstract unchanging manner - ultimately represents the most dynamic transformation process possible, compatible with phenomenal meaning.

Approaching ultimate spiritual realisation requires moving ever closer towards ineffable reality.

Likewise ultimate realisation of the number system likewise entails continual movement from the knowledge of number as (mere) form towards an extremely dynamic appreciation of number as energy states before ultimately dissolving in complete mystery.

In fact the number system serves as the finest bridge connecting both the phenomenal and ineffable worlds and in that resides its enormous importance as the fundamental mediator of all evolution.

There is a further surprise!

Normally we associate number interpretation with rational understanding of an analytic kind. And I extended this to show that qualitative appreciation requires intuitive awareness that is holistic in nature.

However the Zeta 1 zeros (initially) are directly associated with affective - rather than cognitive - understanding.

This again implies that ultimately the proper understanding of the number system (and by extension Mathematics and the Sciences) requires cognitive (rational), spiritual (intuitive) and affective (emotional) understanding.

This is just another way of expressing once again that ultimately the number system is inseparable from the full spectrum of psychological development.

Now, in truth both Zeta 1 and Zeta 2 zeros are ultimately fully interdependent. So the most advanced level of Band 3 (i.e. Level 3) would be concerned with two-way integration with respect to both sets of zeros.

Just as Band 2 is related to the specialisation of (linear) quantitative understanding, Band 4 is related to the specialisation of (circular) qualitative type appreciation.

Advanced Bands and the Number System

I have three further bands on the Spectrum (Band 4, Band 5 and Band 6).

Once again the first two bands (Band 1 and Band 2) relate to the differentiation of (conscious) experience.

The next two bands (Bands 3 and Band 4) then relate directly to integration with respect to (unconscious) type awareness.

Thus the final major task in development is the mature marriage of both the differentiated and integrated aspects of development.

Initially, specialisation with respect to both of these aspects takes place in a relatively independent fashion.

So the final task is the two-way integration of both types of appreciation in an increasingly interactive fashion (whereby they mutually enhance each other in experience). I have on many occasions referred to this as radial understanding, which combines both the notion of a circle (as for example represented by a wheel) with various lines drawn from the centre to meet this circle.

Radial thereby entails the mature marriage of both conscious (linear) and unconscious (circular) understanding, through reference to a central point of being (i.e. the will). So ultimately it represents the mature integration and thereby seamless activity of cognitive, affective and volitional faculties.

Band 5 would represent the preparatory stages of such development.

The understanding associated with this band is vital I believe for obtaining a fundamental grasp of the true nature of the number system.

We have identified these two aspects as analytic and holistic with respect to each other.

Earlier in development, the fixed absolute type appreciation of the analytic nature of the number system unfolds (with respect to its cardinal and ordinal aspects) in a quantitative manner.

Then later, the dynamic relative type awareness - in appreciation of the holistic nature of this system with respect to the interaction of its cardinal and ordinal aspects - potentially unfolds.

This culminates in the knowledge of the zeta zeros, directly representing such holistic understanding, as pure psycho spiritual energy states (ultimately inseparable from pure contemplative awareness). Equally from the complementary physical perspective, it culminates in the knowledge of the zeta zeros as representing pure physical energy states at a sub-atomic level.

So the proper interpretation of the zeros as representing energy states at the “low” physical level requires an appropriate degree of intuitive awareness (i.e. spiritual energy) at the corresponding “high” psychological level. Thus in dynamic interactive terms, we must increasingly see physical and psychological aspects as interdependent (in both horizontal and vertical terms).

However, the huge remaining task is to somehow combine both of these extreme aspects in a harmonious fashion. So this involves the attempted marriage of fixed rational understanding in analytic terms, with highly dynamic intuitive understanding of the holistic aspects of the number system (corresponding to the Zeta 1 and Zeta 2 zeros respectively).

In other words the task is to successfully marry the analytic (quantitative) with the holistic (qualitative) appreciation, so that both can retain their unique distinctive identities, while each also being greatly enhanced through the mutual interpenetration of both aspects.

And this as I see is a task very much belonging to Band 5, which in a preliminary manner is what I have been attempting to illustrate in these articles.

However more than any single point made, what I really wish to get across is an appreciation of how Mathematics, when appropriately understood, is truly an all-spectrum affair. And when it comes to the fundamental nature of the number system (with the need to coherently combine both its analytic and holistic aspects) then this presumes that both analytic and holistic appreciation have already achieved appropriate mature development.

So once again from this perspective, it is simply not possible to properly address this task in Band 2 terms (which defines Conventional Mathematics). This implies an attempted marriage, where the presence of one of the two partners seeking union (i.e. holistic) is not even recognised.

Band 6 would then represent the specialised development of radial development. In mathematical terms, this would be compatible with both analytic and holistic abilities combined with each other in a harmonious fashion (to the nth degree).

So it would thereby represent mathematical activity of an immensely productive yet highly creative nature that would be seamlessly integrated with both artistic and spiritual type awareness.

Band 7 on my map relates to the need for a possible final correction in development.

Even at Band 6 an imbalance may remain whereby one key aspect of development is over-emphasised.

For example some are naturally by personality designed to be “activists” while others are “contemplatives”.

So, in the final Band (Band 7) one would seek to operate against type with an activist choosing a more contemplative existence (and a contemplative a more active one) though I recognise that there may be considerable limits with respect to this final adaptation later in one's life.

From a mathematical perspective, this would imply that the more analytic inclined practitioner - we are referring here to those already substantially operating at the mature radial level - would deliberately emphasise the relatively weaker holistic aspect in understanding (and vice versa).

However in this context, we must keep reminding ourselves that the holistic aspect is not even recognised yet in formal mathematical terms!

So in this regard there certainly is a great need now for the Mathematics profession to collectively address this vitally important - though completely neglected - key issue.

Number System and Evolution

True appreciation of the number system, in a dynamic interactive manner, has vast implications for our understanding of just about everything.

No longer is number seen as enjoying some abstract existence, independent of human experience. Even the mistaken belief in such an abstract existence is necessarily a product of human experience!

Rather, number is now seen as deeply inherent in all phenomenal processes (both with respect to physical and psychological aspects) as the most fundamental means of their encoding.

We are accustomed now in biological terms to recognise the importance of DNA (through which all the varied characteristics that manifest themselves in living species are genetically encoded).

However over the years, I have grown to realise that the most fundamental encoding of all phenomena is with respect to number.

In Physics, String Theory is considered by many as best candidate for “The Theory of Everything”. Here it is postulated that matter is ultimately composed of incredibly tiny “strings”, with unique vibrations of these strings giving rise to the manifest phenomenal particles.

However this physical theory is built on the same quantitative reductionism that defines present interpretation of the number system, with strings portrayed as being devoid of qualitative features. So the very notion of such “strings” as the homogeneous “stuff” of the universe is indefensible from any coherent philosophical perspective.

It is however very interesting how deeper investigation of the minute nature of sub-atomic particles does indeed lend itself increasingly to notions of a purely mathematical nature.

What seems to me a much more tenable proposition is that all manifest phenomena ultimately represent the dynamic interaction of both the quantitative and qualitative aspects of the number system.

So when both aspects are identical the notion of number is ineffable and ever present in a timeless (and spaceless) reality.

When however there is some separation of both of these aspects, number itself is then born in a dynamic manner, as the inherent encoding of phenomena (with respect to both their physical and psychological aspects).

In this sense, created phenomena do not so much serve as reality itself but rather as appearances which veil a deeper existence (representing dynamic number interactions that ultimately are ineffable).

Thus, if we could trace these phenomena fully back to their original nature as pure number, by definition we would reach an ineffable state, where quantitative would no longer be separated from qualitative aspects in space and time.

So for number to be meaningful in a phenomenal context, it must unfold in a dynamic relative manner in space and time.

When one begins to understand the ultimately mysterious synchronous relationship of the primes and the natural numbers (and natural number and the primes) it becomes increasingly difficult to avoid the conclusion that the ability to communicate is an inherent inescapable aspect of the number system. In this sense the number system represents a true living system!

Now, initially this might seem a ridiculous suggestion. However, when one accepts number as the most inherent basis of all phenomena, and that communication is subsequently possible between phenomena, this implies that such communication ultimately reflects the activity of number (with respect to both its quantitative and qualitative aspects).

And remember that it has already been physically established at the sub-atomic level that communication between particles such as light can indeed take place (even at a great distance)!

So, I would go much further by saying that this - what can only be conceived as - telepathic capacity at the sub-atomic level of matter, has its deeper roots in the even richer inherent telepathic nature of the number system. Here the individual and collective nature of the primes is perfectly synchronised throughout the system (from both the quantitative and qualitative perspectives).

Thus this holistic capacity, enabling collective synchronicity, is thereby an in-built feature of the number system from its very inception, which then becomes inherent in all processes (physical and psychological) throughout nature.

Such understanding will however require the most fundamental change in our relationship to the physical world.

Due to the specialisation of analytic interpretation within science, especially since the time of Newton, holistic appreciation of nature has all but been eradicated.

It constantly amazes me for example, that non-local activity at a quantum level is not readily accepted for what it truly represents i.e. clear evidence of the holistic communication ability of sub-atomic particles. And if such holistic ability exists at the sub-atomic level, then it is a feature of all nature.

However having - falsely - tried to convince ourselves for several centuries of the impersonal nature of matter, we are still psychologically unprepared for this massive scientific transition. This requires acceptance that all nature is, in truth, holistically connected in an integral manner, which cannot possibly be understood through the conventional paradigm.

And the deeper roots of this unrecognised holistic dimension are to be found within the number system itself (in the two-way relationship of the primes and the natural numbers).

We will once again recall that Hilbert considered the problem of the zeta zeros, which is central to this working of the number system, in the relationship of the primes and natural numbers, as not only the most important in Mathematics but absolutely the most important!

When one comes to the startling realisation that the most fundamental requirement everywhere throughout evolution is the consistent relationship of both its quantitative and qualitative aspects, and that this truth is originally enshrined in the number system (through the two-way relationship of the primes and natural numbers that is mediated through the zeta zeros), then perhaps one can appreciate the aptness of Hilbert's claim.

Thus the number system is inherently ineffably encoded in everything in creation as its original potential DNC (Dynamic Number Configuration). And the subsequent decoding of number then manifests itself in space and time through all the distinctive phenomenal attributes of created reality (both quantitative and qualitative).

Mapping of Mathematics and Science

I will finish this section with a simple illustration of the potential power of this new dynamic manner of appreciating number.

When reading again Bill Gates' book “The Road Ahead” I realised once more the great significance of the present digital revolution, with exciting new information technologies developing with increasing rapidity.

Ultimately digital information is based on the power of the binary system to potentially encode all information in terms of the two digits 1 and 0 (understood in a quantitative manner). So this represents the Type 1 aspect of the binary system.

However what is not yet recognised is that a complementary binary system exists that can potentially encode all transformation (processes) again in terms of the two digits 1 and 0 (understood in a qualitative manner). Now this qualitative binary understanding entails the balanced relationship of both linear (1) and circular (0) type interaction i.e. all processes involving both differentiation and integration. This represents the Type 2 aspect of the binary system.

Then the most comprehensive understanding of the binary system involves the harmonious integration of both the Type 1 and Type 2 aspects. So the binary digits now potentially can encode all information and transformation processes. This represents the Type 3 aspect of the binary system.

In fact, ultimately we can view the whole Universe as the manifestation of the Type 3 aspect of the binary digital system, potentially relating to all its information and transformation processes. And this is simply based on the numbers 1 and 0 (viewed in terms of both their quantitative and qualitative aspects).

It is my belief that the rapid change that is being brought about by the explosion of information (on so many levels) in our present culture will soon require a counter revolution, entailing a fundamental transformation in our very manner of relating to reality.

Such transformation cannot - by definition - take place through further advances in information technology. Rather it will need a completely new kind of holistic understanding (which is not yet even recognised by the mathematical community).

There is a huge urgency about this!

Though admittedly enormous advances have been made in scientific understanding in the past few centuries (thereby enabling remarkable technological advances for mankind), it is now in danger of becoming hugely unbalanced.

Essentially the present paradigm (with its roots in Mathematics) is based on analytic type appreciation of a mere quantitative nature.

This indeed is admirably suited for attaining the multi-differentiation of knowledge in a vast array of ever more specialised disciplines.

However, it is not at all suited for the corresponding requirement for proper integration of these disciplines in a coherent manner. So despite more knowledge than ever (as information) being available, we are facing the growing danger of considerable fragmentation in so many aspects of life.

Indeed our world is facing increasing threats in economic, political, social and environmental terms. [2]

A radical change in perspective is therefore now required before a significant proportion of the world's population becomes swamped by tsunamis of change (which can no longer be properly understood or controlled).

I will conclude this article with a brief outline again of the 3 Types of Mathematics that need to be recognised.

Type 1, represents the traditional analytic approach to Mathematics based on mere quantitative interpretation. As we have seen this is directly associated with the specialised linear (i.e. 1-dimensional) interpretation associated with Band 2 on the Spectrum. So Conventional Mathematics simply represents this Type 1 aspect!

Type 2, represents the (unrecognised) holistic approach to Mathematics based on qualitative interpretation, or more correctly in dynamic terms the intersection of quantitative and qualitative meaning. Though directly of a highly intuitive nature, indirectly it is associated with circular (paradoxical) rational interpretation. In its mature form, it would be associated with Band 4 on the Spectrum.

In a balanced approach to Mathematics, the Type 2 aspect would be considered equally important to the Type 1. The fact that this statement is unlikely to resonate strongly in the context of strongly established quantitative notions, only indicates how unbalanced current interpretation has become.

Type 3 represents the most comprehensive approach to Mathematics. Whereas Type 1 initially develops in absolute isolation from the other Types (which are not even recognised), Type 2 initially unfolds in a relatively separate manner, where it is not yet integrated with Type 1 understanding.

However a big change with Type 3 appreciation is that the Type 1 and Type 2 aspects are now understood in a relatively complementary manner whereby both increasingly can interact with each other. It thereby represents the fullest expression of mathematical understanding, whereby quantitative and qualitative aspects can maintain a relatively distinct identity, while equally capable of integration with each other (in a manner that greatly enhances both aspects).

So from the perspective of Type 3, the limitations of Type 1 and Type (in isolation) can be more clearly seen.

One of the great problems with Conventional (Type 1) Mathematics - even with respect to the analytic quantitative type problems within its range - is that it can become greatly lacking in the creative inspiration that is necessary to fuel relevant work (especially of an original nature).

Equally while Type 2 Mathematics by its nature gives rise to a plentiful supply of refined intuition, a key problem is that such intuition may never be properly grounded in quantitative type appreciation.

So the equal recognition of both aspects, in an increasingly dynamic interactive manner, therefore has the ready potential to lead to mathematical work that is immensely productive (in quantitative terms) yet highly creative (from a qualitative perspective).

In its specialised developed form it is associated with Band 6 on the Spectrum.

At some stage in the future, all Mathematics will simply be recognised in Type 3 terms. However at this stage in our human evolution, we are still a long way from such a realisation.

Equally of course all the Sciences can be defined in Type 1, Type 2 and Type 3 terms.

So for example with respect to Physics, Type 1 is associated with the (reduced) merely quantitative approach to physical reality (which has now reached its limits in String Theory).

Type 2 is then associated with an altogether distinctive holistic qualitative approach, which will be especially important, from a integral perspective, in providing an overall framework to physical reality that is philosophically meaningful (in an intuitive manner).

In particular I see it leading to a totally new holistic appreciation of the nature of physical dimensions. One of the big problems at present is that physicists are trying to pose ultimate questions as to the nature of reality through a Type 1 lens of enquiry (which is completely inappropriate for the task). So this is where the Type 2 approach should be of special value.

Type 3 again is associated with the most comprehensive understanding, where again the Type 1 and Type 2 aspects are integrated with each other in an increasingly dynamic interactive manner. The most important area where this change in perspective is now required is in Mathematics itself (as our scientific notions are so deeply rooted in mathematical understanding).

And the most important area in Mathematics that needs to be urgently addressed is with respect to the number system. As we have seen, at present it is based on highly reduced - and thereby gravely distorted - assumptions.

As I have said there is now a great urgency about this! If we fail to adapt quickly enough, we are likely to become increasingly estranged from our natural and social environment with massive problems rapidly ensuing.

Though the domination of a particular mathematical (and scientific) worldview we have become accustomed to viewing reality in a greatly unbalanced manner.

We now need to address this chronic imbalance rapidly.

Once again, the future of our species on Earth may ultimately depend on it!


1. It is sometimes said that the Universe is born out of chaos. Perhaps it would be more accurate to suggest that in fact it is born immediately into chaos, with an extremely high unpredictability as to the likely course of its development at the earliest stages.

So physicists are attempting to identify those very special conditions required to ensure stable evolution after a Big Bang, ultimately enabling biological life to emerge.

Of course this birth into chaos would then merge with the formation of the first particles (at a quantum level).

Now, when we view the number system in dynamic terms as that most fundamental special condition of all, in enabling phenomena possessing both quantitative and qualitative aspects to coherently emerge, then it would not be surprising that it could represent itself as the pure limit of a dynamic system that is quantum chaological in nature.

2. I was drawn to the fact in reading Bill Bryson's book “A Short History of Nearly Everything” of how the human species is unique in the sense that it has been responsible in its short history on the planet for the total extinction of a great variety of species. It made me realise that our very future on the planet is far from secure and that there is a real danger that we could ultimately bring about the extinction of our own species.

Indeed, though is necessarily of a merely speculative nature, it is possible that intelligent life - assuming that it exists throughout the Universe - may regularly face the dilemma of unwittingly making itself extinct at a comparatively early stage of its evolution.


The Spectrum of Consciousness: Ken Wilber, Quest Books (2nd Edition, 1993).

Updates on Classification of Stages: though written more than 5 years ago this provides a brief account of the key features of my 7-band map of development referred to in this article.

The Spectrum of Development: another more recent on-going personal blog designed to give a more detailed experiential account of the various levels and stages on each band of the spectrum.

The Road Ahead: Bill Gates, Viking (First Edition edition, 1995).

A Short History of Nearly Everything: Bill Bryson, Broadway Books, (1st edition, 2004)

Tapestry of the Number System: I refer in the article above to the fact that the zeta zeros (Zeta 1) cannot be mathematically understood in the absence of an affective dimension. So this blog entry, from my “Spectrum of Mathematics” blog, attempts to relate the holistic nature of the zeros to the wonderful lyrics of the Don McLean song “Tapestry”.

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