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As an economics’ student in Dublin in the late 1960’s, Peter Collins underwent a significant “scientific conversion”. Since then he has devoted considerable attention to the implications of a full spectrum developmental approach for radical new interpretations of mathematics and its related sciences. Though potentially of growing relevance for better understanding of our present problems, so far, he believes, these have been greatly overlooked by both the scientific and integral communities.

A New Scientific Vision

Part 3: The Integration of Science and Spirituality

Peter Collins

"In the beginning, mountains are mountains and rivers are rivers; later on, mountains are not mountains and rivers are not rivers; and still later, mountains are mountains and rivers are rivers." Zen saying

Radial Development

With respect to science, “n the beginning, mountains are mountains and rivers are rivers” refers to its analytic aspect, where object phenomena appear to have an independent unambiguous identity. And as we have seen, the scientific approach to evolution is built on this narrow rational perspective.

“later on, mountains are not mountains and rivers are not rivers” relates to the holistic aspect that I dealt with in the last article. So rather than independent phenomena in a finite quantitative manner, one attempts to appreciate their holistic qualitative interdependence as truly infinite. Though directly of a spiritual nature, this can be indirectly translated in a circular rational manner through the complementarity of opposites. And this represents the contemplative vision of science.

“and still later, mountains are mountains and rivers are rivers.” This now relates to the radial view of science. Here the independent nature of distinct physical phenomena is restored in a free unrestricted manner, while equally being combined with the holistic interdependence of all phenomena. It is only at this stage that material and spiritual aspects of reality can be seamlessly integrated. Put another way this represents the true marriage of science and religion.

So the third band of radial science entails a double perspective, where both the analytic and holistic aspects of understanding coherently interpenetrate in two-way fashion with each other.

Radial science cannot be divorced from authentic development of the radial stages.

In a certain sense, because one has continual access, everyone already enjoys a certain experience of the radial stages. However it usually remains somewhat attenuated due to both a lack of sufficient differentiation of stage structures and perhaps to an even greater degree, lack of sufficient integration of these same structures. Also considerable imbalance with respect to the primary modes, affective (emotion) cognitive (reason) and volitional (will) may exist. [1] For example, the cognitive mode can undergo specialised development with however the affective lagging far behind. Then the volitional aspect designed to harmonise both, inevitably becomes hampered in this integral task leading to a lack of authentic freedom in terms of experience. Therefore mature radial development requires that a substantial balance be achieved as between reason and emotion. This entails a considerable amount of shadow work in both lessening the dominance of the superior, while equally recovering the inferior mode from the unconscious depths of personality. There are characteristic differences between Western and Eastern mystical traditions. Greater emphasis on form in the West typically leads to a more active mystical expression at the radial stages. This entails that as higher development unfolds, a substantial amount of backward integration with the middle band takes place.

In this way, activity associated with the middle can become greatly enhanced through a genuine spiritual dimension. However, because such experience remains strongly rooted in the world of form, it is unlikely that the highest contemplative levels such as causal and nondual would emerge here in their fullest expression, though lesser access is indeed possible.

There is a second passive form of mysticism more associated with Eastern traditions, where a much greater emphasis on vertical development of spiritual contemplative stages takes place. In many ways this resembles the climbing of a steep mountain. Rather than turning back to base before completion, one continues the climb to even greater heights. However as one inhabits a more rarefied spiritual air, there is a great danger that one thereby loses contact with the ground below.

In fact there is a notable imbalance in many accounts with undue emphasis on the ascent. I can speake with some feeling on this, as for many years I was greatly influenced by St. John of the Cross, who is probably the best known advocate in the Western tradition of this second type of mysticism. His most comprehensive book on the topic is “The Ascent of Mount Carmel”. So, for St. John the spiritual journey is indeed like a steep mountain climb with very few destined to reach the summit at the top.

And there is a crucial distinction as between the notion of the ascent as used here and by Ken Wilber.

For Wilber, the ascent relates to the differentiation of distinctive stages. Again there is a certain discontinuity in his treatment here in that up to the centaur level, stages are identified largely in terms of phenomenal structures, whereas the higher stages are treated more in terms of spiritual states. And his understanding of stage development is of an asymmetrical nature proceeding from prepersonal to personal to transpersonal.

However when St. John of the Cross mentions the ascent, he is referring directly to the notion of integration. And as we have seen, as integration relates to continual negation of what has been consciously posited in development, thereby leading to a refined spiritual form of intuitive awareness, this in effect leads to a profound regression in terms of differentiated notions of progress. So firstly there is an active night of sense followed by an active night of spirit. The former relates to sense attachments (of both a positive and negative kind) whereas the latter relates to the more deeply rooted conceptual and volitional aspects of personality. And in the active nights, one exercises a degree of conscious control in attempting to undo these attachments.

Later on in the journey, one encounters the passive nights again of both sense and spirit. Here a much deeper unconscious cleansing of attachments takes place with the passive night of spirit especially going to the very roots of personality. So at this stage the profound erosion of all the phenomenal apparatus that constituted one's former conscious identity occurs.

One can perhaps criticise his approach in that he lays too much emphasis on the via negativa. The positive side of spiritual progress relates to the illumination of new differentiated stages in personality. Then the negative side—in the dynamic negation of conscious understanding—entails purgation, in what directly relates to the corresponding integration of these structures.

This is brought out very well in Evelyn Underhill's classic “Mysticism” where she shows a marked capacity to recognise the psychological dynamics of authentic mystical experience, with both illumination in relation to differentiated stages and purgation in the corresponding deep integration of these stages taking place. Now Ken Wilber makes the misleading claim that Underhill's treatment is broadly in line with his own formulation of Eastern mystical states. However the two approaches could hardly be more different as Wilber to my mind does not remotely capture the true existential nature of what is portrayed in Western mysticism.

A major weakness that I have found both in Eastern and Western approaches is an undue emphasis on transcendence. Again one can look at the physical analogy of climbing Mount Everest. Though one may be indeed elated at reaching the summit, in many ways the most difficult part of the journey remains in being able to negotiate the descent successfully. Most accidents occur on the descent as a great deal of energy may already have been expended in reaching the summit. However, many contemplative accounts misleadingly give the impression that the spiritual journey has been largely completed on attaining the summit in the transcendent experience of God.

However this is somewhat inaccurate as the corresponding spiritual descent, which is necessary to successfully ground the spiritual progress already attained with everyday mundane reality, is not then properly addressed.

I would see this descent as potentially the most dangerous part of the journey. It takes a considerable amount of dedicated psychological energy to reach the highest transcendent stages of spiritual development. However, unwittingly this can also be associated with considerable repression of the primitive instinctive aspects of personality.

Therefore, if one relaxes as it were at the summit, thinking that the job has been completed, one's long repressed shadow side is then likely to be released in a somewhat naked fashion.

And this is what happens with so many charismatic gurus, who undoubtedly have attained authentic experience of higher spiritual stages. However rather than then seeking to continue with the descent to the lowest levels in the humble confrontation with their primitive instincts, they fall off the right path and give full rein to the still unreformed shadow self through greed, lust and the abuse of power.

And again this is where I would be strongly at variance with the account of psychological development given by Ken Wilber, which places an undue emphasis on the ascent to higher levels, with integration then viewed in a largely top-down manner. However when one has achieved the transcendent summit the even greater task of properly revisiting the lower levels remains. Only then can the spiritual light shine with sufficient intensity to reveal those blockages and instinctive desires that remained inaccessible since earliest childhood. So one can now gradually lessen superego control and let these instincts and feelings speak for themselves. And this is a very necessary preparation so as to enable the immanent aspect of spirit to properly unfold. Through social pressures, we are typically forced to create a persona that can considerably mask our vulnerability and weakness. So probably the best measure of spiritual attainment is the extent to which one can properly let go of this mask and accept one's true humanity. In the Christian literature this is referred to as humility. The very word humility is derived from the Latin word humilis which can be translated as grounded or from the Earth. It also implies a lowly state in line with the return to the lowest stages of development.

Therefore the important point is that spiritual attainment should be properly grounded in the physical world, where spirit and matter can at last be experienced as truly complementary with each other.

Therefore, I prefer to look at the radial stages as representing an enhanced form of normality, where recognition of the human potential for true greatness is properly reconciled with corresponding knowledge of its limitless weakness. In this sense, the radial stages represent but a simple clarity of understanding free of false pretensions.

Another Approach to Number

I have drawn attention before to the fundamental importance of the number system.

Conventional science is built on the notion of quantitative measurement (which directly relates to number).

And as I have been pointing out, there is a corresponding holistic qualitative aspect to number which has remained largely unexplored. In the last entry, I explained how this holistic aspect gives rise to a number spectrum providing a precise scientific ordering for all possible stages of development.

And in this context the natural numbers are especially important.[2]

There are two ways however of constructing the natural number system.

The first is based on addition.

So we start with 1 and by progressively adding 1 we generate this number system.

Thus 1 + 1 = 2; 2 + 1 = 3; 3 + 1 = 4 and so on.

This is deceptively very simple but there is an unexpected problem. In mathematics the units are treated as independent and absolute. However, this begs the question as to how these units can then be combined (which implies interdependence) to obtain the new whole numbers.

Quite surprisingly, in a podcast exchange also involving Richard Dawkins, Sam Harris raises this very issue. So in discussing 2 + 2 = 4, he quite accurately implies that we cannot understand the relationship without intuition.

Therefore an additional holistic aspect is properly required to understand how the natural number system operates.

So let us now address the simplest case i.e. 1 + 1 = 2.

Here the units possess in analytic terms an independent identity (as separate quantities) and in holistic terms an interdependent identity (as possessing the shared quality of oneness). And this potential intuitive awareness of the shared quality of units is required so that the new number quantity of 2 can be recognised. Indeed there is a great irony here as mathematics refers to the natural numbers as integers or whole numbers. However in conventional terms, interpretation completely excludes the integral holistic aspect of understanding!

In the last article, I drew attention to the considerable importance of this holistic notion of number.

Again conventional science is 1-dimensional. However, we then have differing relative interpretations with respect to the other natural numbers with 2, 4 and 8 being especially important from an integral perspective.

And in earlier work on this forum, my criticism of Ken Wilber was largely based on this holistic mathematical interpretation of number as the most appropriate scientific means of interpretation of qualitative type processes such as human development. Just as the quantitative notion of number underlies the analytic approach to science, equally the qualitative notion underlies this holistic approach in the context of development.

However there is an alternative manner of constructing the natural number system which is based on the prime numbers.

The primes are those numbers which have no factors (other than 1 and the number in question). The first five for example are 2, 3, 5, 7 and 11. It has been known since the time of Euclid that an infinite amount of these primes exist. Also—and very important—in what is known as the unique factorisation theorem, every natural number can be represented as a special combination of prime factors. For example, 30 = 2 * 3 * 5. And this combination is unique with no other configuration of primes giving the same number.

As no doubt most are already aware, the individual primes occur in a random fashion. Therefore there is no simple way of calculating the next prime in a sequence from knowledge of the previous primes. And this random nature of primes is an important observation as genetic mutations with respect to natural selection also supposedly occur in a random fashion.

However, there is a great deal of difficulty in defining what randomness in a mathematical context—or indeed in any other context—properly means.

For example if one spins an unbiased coin, the occurrence of a head on each individual toss is random. So therefore one cannot predict whether the next outcome will be a head or a tail from knowledge of previous outcomes. But if one tosses a coin repeatedly, a far greater degree of order becomes apparent with respect to the collective behaviour of these trials. Thus common sense suggests that roughly 50 of—say—a hundred tosses should be heads, though it would be surprising perhaps to get exactly 50. And as we increase the number of trials, the collective outcome can be predicted with a progressively greater degree of accuracy. Therefore with a sufficiently large number of trials, the total of H's obtained approximates ever closer to 50%.

So randomness at the individual gives way to a high level of order at the collective level of investigation. It is somewhat similar with respect to the random nature of primes. Each prime is not random in exactly the same manner as the individual outcome on tossing a coin. However, the same collective pattern is in evidence, so that one can predict with an ever greater degree of percentage accuracy, the frequency of primes up to a given number.

For example, one simple formula is given by n/ln n (where the natural logarithm is used). So for example when n = 100, we get 100/4.605… = 22 (when rounded to the nearest integer). There are in fact 25 primes up to 100. So the estimated answer is already reasonably accurate. And the percentage accuracy increases as n becomes larger.

So again we can see a distinct pattern emerging whereby what is random at an individual level of investigation gives way to an increasing level of order (i.e. non-randomness) at the collective level. Thus effectively, in moving from the individual to the collective we are switching as between the two notions of the number system based on multiplication and addition respectively.

In fact properly interpreted, randomness and order are complementary notions, which can only be understood with reference to each other.

Though regarding primes, new formulae were being used by the mid 19th century that could predict their frequency up to a given number with a high relative degree of accuracy, a residual problem still remained in that it was not known how to absolutely predict the correct number that would arise. In other words, no one could fully reconcile the two approaches to number based on addition and multiplication respectively. However in what constitutes one of the most original mathematical contributions of all time, this was soon to change.

Darwin and Riemann

As is well-known, Darwin's great work “On the Origin of Species” was published in November, 1859. Remarkably, in that very same month, a short mathematical paper on prime numbers was published by the German mathematician Bernhard Riemann, the long-term implications of which I believe are likely to be greater than Darwin's work.

Though a shy and socially awkward individual, Riemann possessed a hugely original mathematical mind. Indeed five years before his publication on primes he had unveiled a new geometry of curved surfaces that came to be known as Riemannian Geometry. This was then to prove of inestimable value to Einstein in formulating his General Theory of Relativity.

However his stunning original work with respect to the nature of prime numbers was his greatest achievement.

Riemann dealt with what is known as the zeta function, which represents a way of connecting the primes and natural numbers.

The following is a simple example of the zeta function that had been investigated by Euler (before Riemann).

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + … = 2^2/(2^2—1) * 3^2//(3^2—1) * 5^2//(5^2—1) * …

So on the left side we have an additive series involving all of the natural numbers and on the right a multiplication series involving all of the primes. In this case the dimensional power (of both the natural numbers and primes) is 2. And it is remarkable that we are able to connect both sides of the expression for any real value of the power (greater than 1).

Riemann's genius was to examine what would happen if instead of a real number as power, a complex number (i.e. containing both real and imaginary parts) was used instead. And he found that for special values of this complex number, both sides would be zero.

So these solutions are referred to as the zeta zeros and there are an unlimited number of them with several billion having already been calculated.[3]

Well what is the significance of all this?

I have quoted before how when David Hilbert—possibly the best known practitioner in the early part of the 20th century—was asked what was the most important problem in mathematics he replied:

“the problem of the zeros of the zeta function, not only in mathematics but absolutely the most important of all problems”.

I know it must be difficult to appreciate this statement relating to what initially appears a somewhat abstruse technical issue. However it has truly the utmost significance not only for mathematics but for all the sciences and indeed human behaviour generally.

A child can quickly learn to count 1, 2, 3, 4, … in sequence. At this stage of our evolution, the counting i.e. natural numbers are hardwired into our brains as the archetype of order.

However underlying the counting numbers, which we take so much for granted is an intricate wave like system i.e. the zeta zeros (composed of complex numbers). This forms the substructure of our everyday system. So without this substructure the natural numbers could not exist. Put in more psychological terms, there is a deep unconscious basis to our everyday notion of natural numbers. And we have not yet remotely begun to plumb the depths of this great mystery.

This parallels—though much more fundamental—what was discovered in the early 20th century with quantum mechanics, when a new world of particles that also manifested as waves was found to comprise the substructure of matter. And the behaviour here was of a totally unexpected nature not in keeping with conventional views.

However due to the great dominance of the quantitative approach in Mathematics, the enormous holistic implications of Riemann's discovery have never been taken on board.

It is not only physics that requires revision. At a much deeper level of understanding, this fundamentally applies to mathematics.

Remember mathematics is built on the number system! So, if the number system is not what we imagine, then this is equally true of all mathematics.

And because mathematics provides their necessary foundation, this likewise applies to all the sciences.

The most important unsolved problem in Mathematics is generally recognised as the Riemann Hypothesis, which intimately relates to the nature of these zeros. Riemann calculated the first few zeros (which was no mean feat before the age of computers) and found that they lay on an imaginary straight line known as the critical line.

He then postulated that all of the zeta zeros lay on the same line. This is the famous Riemann Hypothesis, the proof of which has become the holy grail of mathematics. Just as he had postulated, many billions of these zeros have been calculated and all found to lie on the same straight line.

If the Riemann Hypothesis is true, this means in effect that the primes are distributed in an optimal manner.

One valid way of expressing this is by saying that the individual primes then occur as randomly as possible that is consistent with the greatest degree of their collective order with respect to the natural number system. This equally implies that the two approaches to number (addition and multiplication) can then be perfectly reconciled with each other. This is what mathematicians often refer to as the music of the primes, where like a vast orchestra, each prime contributes its own unique note to the natural number system, yet is perfectly synchronised with all other prime notes.

The very metaphor “the music of the primes” suggests a qualitative holistic aspect to prime number behaviour. However, again because mathematics is defined solely with respect to the analytic quantitative paradigm, this aspect cannot be explored through present understanding.

However, I had long been developing a new approach to mathematics and had already come to understand the holistic nature of the natural numbers (through the additive system), which I illustrated to a degree in my last article. So, some 15 years ago, I became deeply involved in the corresponding attempt to unravel the holistic significance of this new set of zeta zeros.

For some considerable time, this was for me an all-consuming problem. And eventually it did yield its secrets (at least some of them anyway).

And as Jung had proved of inestimable value some years previously, once again by placing the problem in a Jungian perspective, I was better able to see its significance.

In a very true sense, when the primes are viewed in the customary analytic manner as real number quantities, then in this relative context, the zeta zeros represent the holistic qualitative counterpart, originating from deep within the unconscious regions of personality, of these numbers.

You may remember in the last essay that I gave the example of how conventional interpretation offers but a reduced explanation of multiplication. So once again, when we multiply 2 * 3 besides the quantitative transformation to 6, a qualitative transformation in the nature of the units takes place (from 1-dimensional to 2-dimensional).

So I had been consistently wondering what enables this reduced treatment of multiplication in mathematics. So again given that the nature of the units has changed, are we then justified in treating multiplication in a reduced linear fashion?

And now at last I was able to see clearly the answer to this problem. Again from one valid perspective, the zeta zeros represent the holistic counterpart to the primes. So when we look at these zeros, it is as if we are thereby enabled to extract the qualitative aspect of prime number behaviour and treat it in a relatively independent manner.

Thus the assumption that all the zeta zeros lie on a 1-dimensional imaginary line is the qualitative counterpart to the assumption that all natural number quantities lie on a corresponding 1-dimensional real line.

So I now had the answer to that tantalising question. We are therefore entitled to treat the multiplication of primes in a reduced quantitative manner as natural numbers on the real line, if in fact the zeta zeros (as their qualitative counterpart) lie on an imaginary line. Once again the imaginary is the way in which the holistic qualitative notion expresses itself indirectly in a quantitative manner!

However an even deeper implication of this finding goes back to a question I posed in the first of these three articles.

Given that reason and intuition relate to two distinct forms of understanding as analytic (quantitative) and holistic (qualitative) respectively, how can we be even confident that they are compatible with each other?

And the answer here once again depends on the holistic truth of the Riemann Hypothesis.

If the hypothesis is true then we can indeed combine both reason and intuition and by extension the analytic and holistic aspects of science, confident in the knowledge that they are—though distinct in nature—fully compatible with each other.

However the Riemann Hypothesis has not been proven to be true. In fact given the truly dynamic way that the number system operates with respect to its wave (zeta zeros) and particle (primes) aspects, it appears obvious to me that there cannot be a proof of the Hypothesis within the established confines of conventional mathematics.

Put another way, properly understood, the very nature of the Riemann Hypothesis transcends rational thinking, while also being necessarily immanent in the very assumptions required for mathematics to operate.

And chief among these assumptions is that all the natural numbers lie on the real line! Again this is justified if all the zeta zeros lie on the imaginary line. However in attempting to prove this, we have to assume that the natural numbers do indeed lie on the real line. So there is a total circularity involved here which cannot be resolved in a linear rational manner.

This leads to the remarkable realisation that a dynamic synchronicity characterises the true nature of the number system at its very roots, where in order to proceed, quantitative and qualitative aspects must be assumed to be inherently reconciled. However the very nature of this synchronicity is a deep mystery! So we are light years away here from accepted conventional understanding.

Just one other aspect at this stage which has a big repercussion for psychological development!

I mentioned earlier that comparatively little emphasis is placed on the descent in spiritual contemplative terms.

So there is a tendency to believe that spiritual union can be achieved solely with respect to its transcendent aspect.

Inevitably however, the attempt to reach the spiritual summit represses to a significant degree the instinctive aspect of personality. So during the ascent, the senses and emotions of the “lower” self are typically under the control of reason, even if a considerably refined form at the higher stages of development.

Therefore to bring a proper balance, one must return—perhaps for a prolonged period of time—to the lower stages. Here one must attempt to listen sensitively in affective terms to what is revealed in unacknowledged desires, feelings, trauma and various forms of disassociation that can date back to earliest childhood (and even time in the womb).

So only then can one become freely in touch with the spontaneous instinctive aspect of personality.

And it had already struck me that there are strong connections here with the very notion of prime numbers.

Primitive instincts tend to occur in an involuntary random fashion. Thus, maturity in development entails that one can properly deal with such instincts, without their intrusion into consciousness in an unwanted manner.

In childhood, the solution to this problem entails a slow learning process, where in gradually adapting to social norms, one finds that certain behaviour is no longer acceptable. As rational control in one's life increases, this then inevitably tends to repress a large area of instinctive response.

So again the question arises as to whether one's primitive instincts can properly be integrated into behaviour without the domination of superego rational control.

And this is where perhaps surprisingly the holistic nature of the zeta zeros can potentially play a huge psychological role.

Primitive impulses, though of unconscious origin, become immediately embedded in experience with conscious phenomena.

So there is a basic confusion of meaning where holistic unconscious desire is directly identified with conscious symbols.

In mathematics, the zeta zeros play the holistic role of keeping the primes, as it were, in place, so that they can then assume optimal positions consistent with maintenance of the overall order of the number system. It is similar in an unconscious manner. So the psychological appreciation of the zeta zeros, thereby entails coming to terms with the hidden nature of primitive instincts, which when brought to conscious light, enables one to properly harmonise in affective terms, holistic (unconscious) desire with manifest (conscious) behaviour.

In future, I am very confident that when their holistic nature is properly appreciated, the zeta zeros will come to play an enormous psycho-spiritual role in a special type of meditation that is geared to mastery of instinctive behaviour without undue rational interference. Alternatively such meditation represents mastery of rational control free from involuntary projections, with both aspects now seen as fully complementary with each other.

And because of present undue emphasis on the transcendent aspect in the attaining of higher spiritual stages, I would say that this represents in practice a most important aspect of authentic spiritual development, enabling one to be become fully grounded in a genuine form of humility that lies at the other extreme of behaviour to what so-called gurus often represent.

Or to put it in simpler terms, the holistic understanding of the zeta zeros fully enables reconciliation, through the will, of the cognitive and affective aspects of personality.

And as cognitive and affective represent in effect the human means of control and response regarding reality, we can generalise this principle therefore likewise in physical terms.

So the zeta zeros represent the manner in which all complex control and response patterns in nature can be co-ordinated leading to various dynamic configurations entailing randomness and order respectively. However this requires both quantitative and qualitative appreciation and cannot therefore be properly encapsulated within analytic science.

And radial development from this new perspective appears as a return to normal life. However, it is now an enhanced type of normality where the spiritual is seamlessly integrated with everyday phenomena. And this also serves as the blueprint for radial science to then unfold.

Principle of Universality

My own prolonged investigation of the Riemann Hypothesis was an early excursion into—what I term—radial mathematics. This represents a dynamic form of understanding, where both the analytic and holistic interpretations of mathematical symbols are understood to coherently interact with each other.

And what I discovered was that the quantitative notion of randomness in the behaviour of the individual primes has no meaning distinct from the corresponding qualitative nature of their overall interdependence with the natural number system. Once again this cannot be properly understood in a reduced quantitative manner. And I have repeatedly made this point that the greatest revolution yet in our intellectual history will emerge when the true nature of mathematics with respect to both the analytic and holistic nature of its symbols is eventually accepted.

And this then in turn becomes a starting axiom or principle for radial understanding in all scientific contexts.

So once again from the radial perspective, the key issue is how to relate both the quantitative (analytic) and qualitative (holistic) aspects of scientific interpretation in a truly coherent fashion.

Again this issue does not even arise in the accepted scientific understanding of evolution. Here, the qualitative aspect relating to the whole is simply reduced to part understanding that can then be understood in a quantitative manner.

However as we have seen in the first article, while science does not explicitly recognise the role of intuition to which the holistic aspect properly relates, implicitly it must use this holistic aspect whenever the recognition of wholes in any context is required.

So analytic interpretation represents a gross misrepresentation of the true radial situation, where both analytic and holistic aspects are explicitly recognised.

Equally holistic understanding in its focus on the qualitative interdependence of reality does not sufficiently recognise the quantitative parts.

Therefore in this sense it can also be said to misrepresent the true nature of reality (where both quantitative and qualitative aspects are involved). So it is only with radial understanding that both analytic and holistic interpretation, formerly treated in a somewhat separate fashion, can be seamlessly combined with each other as two sides as it were of the same coin.

One of the earliest recognitions of the wider implications of the zeta zeros was in relation to atomic physics.

I have stated before in previous articles that Riemann's work on prime numbers was in fact deeply suggestive of the underlying nature of quantum physics. Though quantum physics did not exist in 1859, Riemann himself was attempting to make such physical connections through perturbation theory.[4]

There is an interesting anecdote told by Jon Keating, who is one of the foremost practitioners in the quantum physical implications of Riemann's zeta function. Though a lot of Riemann's notes were destroyed, a famous notebook called the Nachlass survives. When on vacation, Keating and a friend stopped off at the library in Göttingen holding the Nachlass, one requested to see his physics work on perturbation theory and the other his mathematical work on the calculation of the value of the zeros. To their great surprise, they were handed the one set of notes showing that Riemann was simultaneously working on both problems.

However because of the growing separation of mathematics from physics, when the quantum mechanics revolution unfolded in the early part of the 20th century, mathematicians were unable to recognise its deep connections with Riemann's work.

Then a famous chance encounter in the 1970's dramatically changed the situation. Hugh Montgomery, a mathematician who had just formulated an interesting hypothesis regarding the spacing between the zeta zeros was visiting the Institute for Advanced Study in Princeton and in a brief discussion mentioned it to Freeman Dyson, who immediately replied that the same result had been found in physics with respect to the energy levels of atomic nuclei.

We will avoid the technicalities here regarding the role of random matrices in establishing such physical results. However the key point is that an intimate relationship seemed to exist as between a result from pure mathematics and one in the experimental field of quantum mechanics. Further investigation led chiefly by Michael Berry and Jon Keating has now established a remarkably close connection that is no longer disputed as between the zeta zeros and what is known in physics as a quantum chaotic system. However again because of the rigid analytic approach that mathematicians still cling to, they are unable to explain why such a connection should exist.

Interestingly, because of my own development of holistic mathematics (and more recently radial mathematics), I had already long assumed that some connection of the zeros with quantum mechanics necessarily existed.

As I mentioned in the previous article, the very nature of quantum mechanical behaviour is that analytic notions of independence and holistic notions of interdependence can no longer be clearly separated from each other. Then in relation to the primes, I had come to understand their behaviour in a similar fashion, where the quantitative could not be understood in the absence of a distinct qualitative holistic aspect.

So what mathematicians have yet to realise is that the true nature of number is inherently dynamic and inseparable from the corresponding behaviour of sub-atomic particles at the quantum level. Such physical behaviour simply manifests an example of number activity, where analytic and holistic aspects mutually interact. However it will require, as I have said, a total revolution of the traditional mathematical approach before this can be understood clearly.

The characteristic spacing of the zeros on a straight line represents a phenomenon that is now being recognised in many branches of science as the principle of universality. Thus in varied complex systems, it is being found that behaviour corresponds to the same relationship that characterises the zeta zeros of the Riemann zeta function.[5]

In dynamic interactive terms, universality can be expressed as a random-ordered system. In mathematics the natural numbers comprise the best example of an ordered system. So the points representing the numbers on a straight line are all a uniform distance from each other. When we plot randomly distributed numbers on a straight line a different type of characteristic arrangement is in evidence where some points will be very close to each other and others very far apart. The red pattern exhibits a precise balance of randomness and regularity known as “has been observed in the spectra of many complex, correlated systems. In this spectrum, a mathematical formula called the “correlation function” gives the exact probability of fin two lines spaced a given distance apart. However the zeta zeros which provide the archetypal behaviour with respect to universality seem to fall somewhere in between the ordered and random sequences.

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The red pattern exhibits a precise balance of randomness and regularity known as “universality”, which has been observed in the spectra of many complex correlated systems. In this spectrum a mathematical function called the correlation function—which is what Hugh Montgomery showed to Freeman Dyson at their chance meeting—gives the exact probability of finding two lines spaced a given distance apart.

All complex correlated systems appear to be governed by the same system that underlies the behaviour of the zeta zeros.

And remember again that from a mathematical perspective, a complex number comprises both real and imaginary parts. So there are real and imaginary aspects to all such complex systems, which simply imply both analytic and holistic aspects of behaviour, with the analytic related to the independent aspect that can be measured in a quantitative fashion and the holistic to the corresponding interdependent aspect that is understood in a qualitative manner.

Universality arises therefore when a system is very complex, consisting of many parts that strongly interact with each other to generate a spectrum. And associated with all of these systems, the same distinctive pattern emerges representing a precise balance as between randomness and regularity (as with the zeros of the zeta function).

One example relates to bus timetabling in Cuernavaca Mexico in 1999, where a Czech physicist Petr Šeba noticed young men handing drivers slips of paper for cash. Thus, each driver was paying “a spy” to record when the bus ahead departed from the stop.

If this happened recently the driver would slow down, letting passengers accumulate at the next stop. However if it happened long ago, he would speed up to prevent other buses passing him. In this way the driver would best maximise his profits. Šeba eventually managed to get information from the “spies” regarding bus departures and when he plotted the data, it resembled the familiar spacing that characterises the zeta zeros and quantum chaotic energy systems. So this was just a simple example of economics behaviour representing the balance between randomness and order that is referred to as universality and which according to Van Vu a mathematician at Yale University seems to be a law of nature.

Also it appears that the more complex a system the more robust its universality features appear.

Universality is also being used in social terms to study the structure and evolution of the Internet and in an environmental context to formulate climate change models.

It has also been found to apply in biological terms. In a fascinating experiment on the cone cells of the eyes of chickens, the mix representing a balance between order and randomness was found to exist, this time however in two dimensions.[6] It has also been found to apply to fluids and materials and a variety of other disparate situations.

With a radial interpretation, evolution is understood to be based on principles corresponding in the most fundamental manner to control and response patterns with respect to matter. So we have the masculine principle of control (associated with natural selection and order) and the feminine principle of response (associated with beauty). Thus the two principles of evolution, natural selection and beauty (control and response) really represent the masculine and feminine principles respectively. Though distinct, they mutually interact with each other in a complementary manner. And these match the manner in which both reason and emotion interact in human experience.

And the deeper root of both these principles is to be found in universality, which in turn is ultimately based on the dynamic behaviour of the number system. However, to be truly meaningful, the principle of universality itself must be widened to include holistic as well as analytic appreciation. It cannot be properly grasped through the reduced approach of analytic science.

The huge problem at present is that scientific discussion takes place from just one band of the spectrum. So the particular reduced interpretation that is given in the modern theory of evolution conforms well to the application of this mindset. The conclusions reached therefore correlate closely with the limited assumptions used for their derivation.

However each level of the spectrum yields a different understanding. And this represents not just a different psychological interpretation but equally a different physical reality.

So the higher band leading to holistic science enables us to see how interdependence with respect to every possible relationship is qualitatively distinct from corresponding analysis of the parts on which conventional understanding of evolution is based.

Then radial understanding shows how both holistic and analytic aspects can be seamlessly integrated in a much richer interpretation, which again corresponds to a much richer reality. And here the personal dimension of interaction with the material world is restored; here the spiritual intuitive dimension relating to qualitative appreciation is likewise restored; finally the teleological dimension in appreciation of the sheer mystery of existence is restored.

And at this radial level, natural selection and beauty are seen as two complementary aspects of evolution that are expressive of the principle of universality, where randomness and order are both understood as complementary.

The full mystical expression of this principle relates to the ability to give each random physical phenomenon, regardless of how seemingly minute and insignificant, its unique identity (as an immanent reflection of spirit) yet likewise be able to see the interdependence of all the ordered diversity of life as equally the expression of spirit (in a transcendent manner).

In this worldview nothing is dispensable; nothing is merely impersonal, nothing lacks meaning, as everything is now seen to radiate spirit as the source and purpose of all that exists. And here the spiritual and material views of reality can be seamlessly merged.

Though of course I support this spiritual view of evolution, I would question an undue emphasis on the ascent.

Just as with human development, the ultimate task is to see higher and lower stages of evolution as merely relative. So in the true spiritual, which now complements the material view, both evolution and involution, as mirror copies of each other, radiate from a central point in the present moment, with forward movement in one representing backward movement in the other and vice versa. Thus though it may often indeed seem that evolution has an unambiguous forward trajectory leading to the emergence of higher order wholes, this reflects the misguided attempt to view objective reality as being somehow independent of the observer, when it has no strict meaning apart from such personal interpretation.

The principle of universality in fact provides the unconscious basis for appropriate linear scientific interpretation.

I mentioned in my first article how the holistic integral aspect of understanding is effectively blotted out through the accepted rational scientific approach. Then in the second article I used the holistic interpretation of the 1st version of the natural number system (the additive approach) to show how this integral dimension can be scientifically expressed.

However the 2nd version of the number system (the multiplication of primes approach) leads to the realisation that the zeta zeros have a corresponding holistic relevance for the successful differentiation of phenomena.

Many contributions to the Reading Room have implied that because analytic science has proven so successful in providing a detailed knowledge of the mechanism of evolution, that the spiritual view is thereby rendered superfluous. I have already argued however that this view is unacceptable because it ignores the holistic qualitative aspect of integration! However it equally ignores the holistic quantitative aspect of differentiation.

Just as there is an unrecognised unconscious basis for successful integration to take place, equally there is an unrecognised unconscious basis for successful differentiation of phenomena, as in analytic science, to occur. And the holistic understanding of the zeta zeros shows how both of these coincide in a truly complementary manner.

Though I am not questioning the benefits of its recognised procedures (in their rightful place) the idea that conventional science can operate in a truly objective manner is somewhat naïve. Scientists are often unconsciously motivated by attachment to a particular theory. This can then lead to a rejection of research findings not in support of this viewpoint. Also personal status may be involved in scientific disputes, which can again greatly colour objective preferences. Likewise as in all human endeavours, petty rivalries, jealousies and blind spots can exercise an undue influence on viewpoints held. And a considerable resistance to new thinking may be in evidence even when existing theories no longer provide satisfactory explanations.

So when advocating a new economics vision at the height of the Great Depression, John Maynard Keynes stated in his preface to “The General Theory of Employment, Interest and Money ”.[7]

“The ideas which are here expressed so laboriously are extremely simple and should be obvious. The difficulty lies, not in the new ideas, but in escaping from the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds.”

And despite exercising great personal authority and influence, his ideas were not truly accepted until rapid increases in military spending during the 2nd World War, provided a striking confirmation of his approach.

One of the benefits of holistic unconscious development is that in forcing one to confront the shadow at ever deeper levels of being, it can to a considerable extent free one from the blind projections that can impede conventional scientific understanding.

In holistic terms therefore, the zeta zeros can be validly interpreted as providing the unconscious basis for appropriate linear understanding, where it can operate in a free and unrestricted manner. Only then can one hope to study reality in an analytic fashion without the many prejudices that can greatly limit conventional scientific practice. There is yet another way of appreciating what this is all about.

What seems random at an individual yet can reveal a deep sense of order at a more collective universal level. And then in turn from another related perspective, what is random at an individual can be shown to reveal a true internal sense of order, while what was ordered at the external collective level can now internally be seen to be random.[8]

So again the principle of universality shows at a fundamental level that randomness and order are intimately connected in a two-way fashion. And the key to truly understanding this principle is the realisation that randomness and order operate in a complementary fashion as quantitative to qualitative (and qualitative as to quantitative).

The most important—and most beautiful—objects that exist in Mathematics are called L-functions. The Riemann zeta function is the most famous L-function, allowing for the extreme archetypal connection as between randomness and order.

However, an unlimited number of other L-functions exist, with over 20 million documented in some detail at the LMFDB (L-functions and Modular Forms Database) web-site.[9]

And all of these can be validly seen as representing the dynamic relationship as between varying configurations of randomness and order respectively (where once again both quantitative and qualitative aspects interact).

However I must emphasise, from the perspective of the radial vision, how limited is the analytic scientific approach. The present information age is based on use of the binary digits 1 and 0. Imagine now how inefficient it would be to operate with a unary system based on just 1. Therefore, for example, to write out 1 billion in this system, it would take 1,000,000,000 digits (all 1). However to represent this same information in binary terms would take just 30 digits.

From a holistic mathematical perspective, the present scientific approach is indeed a unary system, where the linear analytic aspect is solely recognised. This is therefore based on just one holistic digit i.e. 1 relating to the quantitative aspect of reality. However the radial approach is based on the two holistic digits 1 and 0 (relating to both quantitative and qualitative aspects respectively). So, just as a unary system is hugely inefficient in analytic terms as a means of encoding information, equally a unary system in holistic terms is hugely inefficient as a means of encoding transformation processes. And once again conventional science represents such a holistic unary system!

In particular, I see a big problem with evolutionary science in its attempt to interpret reality using this reduced unary system.

It has seemingly reached the mistaken conclusion that we can now dispense with God. However what it is really demonstrating is that we need to move to a different notion of God free of all the mythic resonances of the past. So we are now being faced with the responsibility of realising our own identity as God. And there is nothing arrogant about this as it equally requires recognising the identity of all other beings (human and otherwise) likewise as God with respect to their essential spiritual nature. And it is in this merging of both individual and collective aspects (in quantitative and qualitative terms) that the true mystery of existence is thereby continually realised. In truth, reality is created from moment to moment. And we and everything else that exist are co-creators of our world and therefore inseparable from the continual process of evolution, wherever it occurs.

Science as Monological, Dialogical and Translogical

I will now briefly return to the three bands of science that I have identified i.e. analytic, holistic and radial indicating how monological, dialogical and translogical aspects are involved at each band.

As we have seen in the first essay in this trilogy, in explicit terms, analytic science is monological. It represents the study of material form (which excludes spiritual emptiness); it is carried out in a quantitative manner, where the whole in any context is reduced to its constituent parts and it assumes an objective (impersonal) reality independent of one's (personal) interaction with it.

However, implicitly other elements are necessarily required often to a significant degree. So for example in the manner in which intuition and imagination are involved it is dialogical and then in the sense of wonder and mystery that drives scientific endeavour it can be translogical.

However even here there can be a distinction as between scientists who properly recognise the existence of these other dimensions and those who remain somewhat blind as to their nature. So Einstein while remaining consistent in his quest for an objective interpretation of reality was strongly aware of the nature of intuition and mystery in science. Richard Dawkins, though again considerably motivated by the same instincts seems much less consciously aware of their distinctive contribution.

Then holistic science in explicit terms is of a dialogical nature. So firstly the interaction between the external world and internal observer is properly understood; then the interaction as between the qualitative nature of the whole and quantitative nature of parts is recognised; finally and most fundamentally, the most refined interaction as between the world of material form and spiritual emptiness is realised. Though it implicitly retains its monological aspect, it is no longer emphasised at this stage. It is also translogical, initially in a strong transcendent spiritual manner, as scientific understanding is seen to go well beyond what can be understood at the material level.

Finally, it is only with radial science that the full expression of monological, dialogical and translogical aspects can be attained. So here science is monological in a renewed ability to engage in analytic enquiry, considerably free of the unconscious projections that limit normal engagement. It is also dialogical in the manner that the holistic aspect of science can now be fully integrated with the analytic with a greatly enhanced intuitive ability evident. Finally it is now fully translogical, with the immanent aspect of spiritual understanding within nature now combined in a balanced manner with the transcendent, where the spiritual aspect of reality goes beyond the material.

However, even with respect to radial science, I would identify three relatively distinct types.

The first more active approach can be referred to as radial 1. Though requiring authentic holistic type appreciation, it is geared mainly to analytic enquiry. This represents in fact a much enhanced version of conventional analytic appreciation. So interpretation can take place now in a greatly enlarged framework, which is seen as fully compatible with spiritual awareness. Also, it would be considerably free of the projections and prejudices that can greatly limit conventional scientific understanding. Finally, it would allow for the much more creative use of intuition, greatly facilitating for example within the fields of evolutionary biology, new interconnections between data and fruitful novel hypotheses for investigation.

The second more passive approach can be referred to as radial 2. Though properly grounded in analytic type appreciation, it is mainly suited to holistic enquiry as between disciplines in the search for new patterns and linkages that would likely be completely overlooked by those working within established fields. Though not excluding analytic investigation, this type of science is therefore most suited to the employment of fundamental holistic notions that can successfully integrate various fields of study. So I would classify my own recent attempts in clarification of the enormous potential importance of the zeta zeros as a very preliminary version of such radial 2 science.

The third, which can be referred to as radial 3 represents in effect a balanced mix of both radial 1 and radial 2. And because these aspects are truly complementary, this therefore allows by far the most comprehensive scientific approach that can be both amazingly productive in analytic terms, yet equally extremely creative from a holistic perspective. However we are still a long way from this type which at the present moment must remain an ideal of what perhaps science can become in the future.

Ken Wilber did engage with the issue of the integration of science with religion chiefly in his “Marriage of Sense and Soul”.[10]

There were many aspects of this book that I found commendable e.g. a clear outlining of the nature of the problem and also his emphasis on the need for a more universally acceptable notion of the spiritual not specifically tied to the mythical representations of conventional religious traditions.

However the main reservation I had was his inability to see how narrow science can be given distinctive expressions at the more advanced stages of development.

So for example there is not just one valid interpretation of physics but a number of increasingly refined interpretations in accordance with the full spectrum. And again for convenience, I divide these interpretations into three main classes i.e. analytic, holistic and radial. It is true in practice that (narrow) science is certainly given a monological interpretation in explicit terms. However Wilber did not sufficiently emphasise how such science implicitly can operate in a very different manner with dialogical and even translogical elements involved.

Then in terms of higher level science, he offered no true integral approach. He suggested the analytic model as a means of measuring brain states associated with advanced spiritual development. However, this does not represent a proper science of the higher levels. Rather it indicates precisely the problem that I mentioned in the last article that Wilber tends to identify higher stages with their internal psychological characteristics and then unduly in terms of the characteristic states of these stages. However, there is also a need to define their integral structures with respect to both (internal) psychological and (external) physical aspects of reality. Only then can a science of the higher levels be properly developed.

And because he failed to provide any true holistic integral mode of scientific enquiry, this entailed the corresponding lack in his approach of the more comprehensive radial vision of science based on the interaction of both analytic and holistic aspects.

I have only had the opportunity to sketch out the barest vision of what radial science might entail in this article. By linking the problem of the zeta zeros to their deep unconscious roots in a psychological context, I have demonstrated how it is inseparable from the need to properly harmonise the affective, cognitive and volitional modes of personality. This then serves as the starting basis for an enlightened form of scientific enquiry that is intimately connected with true artistic and spiritual appreciation.

If at the very least these last three articles encourage participants of Integral World to question anew the dominant position of analytic science in our culture they will have served some purpose.

Notes

1. These modes bear comparison with the notion of “lines of development” in Ken Wilber's work. However in terms of integration it is very important to distinguish as between primary and secondary modes. One must pay special attention therefore to the primary modes i.e. affective, cognitive and volitional in terms of integration. However this does not necessarily apply to a secondary mode such as musical ability. Wilber's “lines of development” refer largely to secondary modes in my terminology. Also as regards the integration of modes, the notion of “lines” is inappropriate, as this refers to their differentiated rather than integrated expression.

2. Leopold Kronecker, a German mathematician, famously said “God created the natural numbers. All else is the work of man”. Though we might more correctly say today “All else is the work of humans”.

3. The zeta zeros (Riemannian zeros) or more correctly the non-trivial zeta zeros occur in pairs of the form ½ + it and ½—it respectively (where i = square root of—1). There are an unlimited number of these zeros which become ever more plentiful as we ascend (and descend) the imaginary number scale. The first zero occurs at t = 14.134725… There are also trivial zeros that are intimately related to the holistic notions that I dealt with in the last article. For example the first trivial zero occurs when the dimensional power of the zeta function is—2. So whereas 2 relates to the simplest form of holistic interdependence that rationally is expressed as the complementarity of real opposites,—2 relates to the direct intuitive recognition of such interdependence (which literally entails negation of its rational expression).

The true significance of the (non-trivial) zeta zeros is that they provide the means of reconciling both the additive and multiplication approaches to the number system. Using these zeros, Riemann was able to derive a formula that could absolutely predict the number of primes to a given number. In deeper psychological terms, the zeta zeros entail the integration of both the “high-level” cognitive and “low-level” affective approaches to development both of which are ultimately understood in a purely relative manner. And such integration is a prerequisite for the radial stages to unfold. So the scientific approach at the radial levels involves recognition of affective and cognitive aspects with respect to both the differentiation and integration of phenomena. And these are continually reconciled through the central volitional mode. Though the scientific still remains relatively distinct it enables the artistic mode (through direct emotional experience) to be likewise involved with again the will mediating as between both. So the full reconciliation of scientific (cognitive) artistic (affective) and religious (volitional) modes are now possible. Remarkably, when appropriately understood in both an analytic and holistic manner, the number system, like a hidden software code, is seen to underlie all created phenomena. In fact these manifest phenomena properly represent but the dynamic interaction of number with respect to both its quantitative and qualitative aspects. In this sense, number represents the fundamental intermediary as between (material) form and (spiritual) emptiness.

4. Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by working from simpler exact solutions to related problems. Though it did not exist in Riemann's time, subsequently perturbation theory came to play a huge role in quantum mechanics.

5. See video What is Universality from Quanta Magazine.

6. A Bird's Eye View of Nature's Hidden Order from Quanta Magazine.

7. The General Theory of Employment, Interest and Money; John Maynard Keynes

8. This is intimately true of the number system. In quantitative terms each individual prime e.g. 5, is of a random nature. However when one looks qualitatively at the potential relationship between its ordinal members, i.e. 1st, 2nd, 3rd, 4th and 5th a highly ordered pattern is in evidence. Then each natural number in quantitative terms is part of an ordered arrangement. However when one looks in corresponding qualitative terms at the prime factors of a natural number, a highly random picture is in evidence. It has been proven in what is known as the Erdős-Kac Theorem, that the prime factors of the natural numbers are in fact distributed in a normal manner (similar to the frequency of H's with repeated tosses of a coin). So what is random from a quantitative perspective is ordered from a qualitative; and what is ordered from a qualitative is random from a quantitative perspective. This applies to the relationship as between the prime and natural numbers and by extension to everything in nature in what represents the complementarity of randomness and order. However this principle of universality cannot be properly understood as at present in mere analytic terms (in a solely quantitative manner).

9. See The L-functions and Modular Forms Database

10. The Marriage of Sense and Soul: Integrating Science and Religion; Ken Wilber; Broadway Books, Paperback ed. 1st May 2000.

Other Reference

See Prime Evolution by Matthew Watkins and Revisiting Prime Evolution by Peter Collins