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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.

A Deeper Significance

Resolving the Riemann Hypothesis

Peter Collins

The key to unlocking the Riemann Hypothesis lies in a qualitative rather than solely quantitative appreciation of mathematical relationships. When viewed in this light it can indeed be resolved whereby it is seen to have the most fundamental implications imaginable for our very understanding of what is meant by Mathematics.


150 years ago in 1859, Darwin's “Origin of Species” was published.

In the same year a brief mathematical paper of breathtaking originality was delivered by Bernhard Riemann relating to the distribution of prime numbers. One observation in that paper mentioned almost as an aside - known since as the Riemann Hypothesis - has assumed such importance that it is now commonly accepted as the outstanding unsolved problem in mathematics.

Throughout its history the Riemann Hypothesis has been the subject of intense investigation by the finest mathematicians. However it has shown itself incredibly resistant to proof with so often, an apparent solution managing to elude final capture in the most tantalising manner. Indeed due to its seemingly impenetrable nature, hints of a more fundamental difficulty can be gleaned through the comments of some of the greatest authorities on the matter.

For example Brian Conrey [1] :

"The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don't understand about the link between addition and multiplication."

And Alain Connes [2] in somewhat similar fashion:

"The Riemann Hypothesis is probably the most basic problem in mathematics, in the sense that it is the intertwining of addition and multiplication. It's a gaping hole in our understanding..."

Finally Hugh Montgomery [3]:

"Sometimes I think that we essentially have a complete proof of the Riemann Hypothesis except for a gap. The problem is, the gap occurs right at the beginning, and so it's hard to fill that gap because you don't see what's on the other side of it."

It is my firm belief that the failure to solve the Riemann Hypothesis relates to a deep philosophical problem that strikes at the very heart of what conventionally is understood as mathematics.

However to put this in context we need to go back 2,500 years in history to a decisive event that subsequently helped to define the very nature of mathematics in Western understanding.

The Pythagorean Connection

The Riemann Hypothesis has profound implications for many problems in number theory.

Perhaps the most famous earlier school associated with the development of number theory is that that of the Pythagoreans and at least two of their contributions - when appropriately interpreted - can be shown to have an intimate bearing on the Riemann Hypothesis.

The first of these - which we will deal with here - is associated with the famed Pythagorean triangle (named after the school).

This momentous discovery related to the fact that in any right angled triangle the square on the hypotenuse (diagonal line) is equal to the squares on the other two sides (opposite and adjacent).

So for example when the opposite and adjacent sides are 3 and 4 respectively, the hypotenuse is 5.

However embodied in this right angled triangle was a further discovery that was to have devastating consequences leading to the eventual breakup in the school and a profound split with respect to the subsequent development of Western Mathematics.

In the simplest version of the right angled triangle both opposite and adjacent sides are equal to 1. However here, the length of the hypotenuse equals the square root of 2, which cannot be expressed as a rational number (i.e. in the form of a fraction). In other words it is irrational.

Now to appreciate the significance of this discovery we must recognise that the Pythagoreans inherently adopted a more integral view of Mathematics where reason and intuition were understood as necessary partners in the pursuit of truth. Indeed from this perspective the ultimate goal of Mathematics was to lead to the highest degree of spiritual contemplation.

So mathematics was seen to have both quantitative and qualitative aspects (that were inherently complementary).

For the Pythagoreans the world was encoded in number. And before their disturbing discovery they believed that all numbers were rational. This then happily coincided in qualitative terms with - what we might refer to as - the rational paradigm in interpreting this world.

Thus the discovery of an irrational number such as the square root of 2 destroyed this complementary balance as they lacked the qualitative philosophical means of explaining why it could arise.

Therefore all mathematical symbols and relationships possess qualitative as well as quantitative aspects. However the subsequent development of Western mathematics has continued in a substantially reduced fashion where effectively the qualitative aspect is ignored.

In conventional mathematical terms concepts are used with little regard for their philosophical (i.e. qualitative) significance. Thus now, besides the rational, we have irrational (algebraic and transcendental), imaginary, complex and transfinite numbers. However these mathematical quantities are all appropriated in reduced fashion within a mathematical paradigm that is solely rational.

So the qualitative understanding that mathematicians have of their concepts is quite inadequate. And this lack of appropriate philosophical understanding, as I will attempt to demonstrate, is central to the failure to properly appreciate what is implied by the Riemann Hypothesis. In fact as we shall see later, the Riemann Hypothesis is essentially a statement regarding the basic mathematical relationship of quantitative to qualitative interpretation. [4]

When we add two numbers a mere quantitative transformation of units (within the same dimension) takes place.

Thus when 1 + 2 = 3, implicitly both numbers are defined in linear terms i.e. with respect to the default 1st dimension, with the result also relating to the same dimension.

However when we multiply two numbers both a quantitative and qualitative (dimensional) transformation in units occurs.

This can easily be represented in the case of 2 X 2 for example where we can illustrate the result geometrically as a figure with an area of 4 square units.

So quantitatively, 2 X 2 = 4. However in qualitative terms the nature of the units involved has now changed from linear (i.e. one-dimensional) to square (two-dimensional).

However from a standard conventional perspective, the qualitative dimensional transformation in the nature of the units is simply ignored with the result interpreted in a reduced quantitative manner.

The line as we know in geometrical terms is one-dimensional. Thus, quite literally, conventional mathematics employs a linear rational approach (where interpretation of number quantities is expressed in reduced one-dimensional terms).

So perhaps we can see right away where this crucial gap in understanding in relation to addition and multiplication lies. To put it simply, multiplication entails a qualitative transformation that requires a truly distinctive mathematical approach for adequate interpretation.

Indeed it was the recognition of this problem that led me some 40 years ago towards the development of this vitally important missing aspect of mathematics (that I term Holistic Mathematics) which requires the use of a more refined circular logic properly suited to qualitative interpretation.

Much of my subsequent work therefore has been geared towards unveiling the true significance of numbers (as qualitative dimensions) and then using this appreciation as a scientific integral means of encoding dynamic transformation processes such as the stages of human development.

So just as 1 and 0 are the binary digits (in quantitative terms) with the power to encode all information, the same two digits (as understood in qualitative holistic terms) have the power to potentially encode all transformation processes.

Fortunately for our purposes, one of the simplest of these qualitative dimensions (i.e. 2) provides the essential ingredients to eventually uncover the mystery of the Riemann Hypothesis.

As we have seen the Pythagorean dilemma related to interpretation of the square root of 2.

Obtaining the square root implies raising a number to the power (i.e. dimension) of 1/2. So if you take out a calculator, input 2, then press the power function and input .5 you will obtain the square root 1.414213562...

The significance of the dimension 1/2 is that it is central to the Riemann Hypothesis.

Thus, through raising the number here to a fraction (1/2) a double transformation is involved. Firstly the quantitative value changes (from 2 to 1.4142…). Secondly the qualitative nature of the number changes from rational (which can be expressed as a fraction) to irrational (which has no such expression).

An irrational number combines in its behaviour aspects that essentially relate to two distinct logical systems. Thus in finite discrete terms the number can be approximated as a fraction. However there is also a continuous aspect to the number in that its decimal expansion is infinite displaying no fixed pattern.

So the two aspects (i.e. discrete and continuous) ultimately correspond to two distinct logical systems. However because conventional mathematics only acknowledges one of these (i.e. linear), irrational numbers are treated in a merely reduced quantitative manner.

However to illustrate more clearly the nature of this problem let us look first at the square root of 1.

In quantitative terms this yields two distinct answers either + 1 or – 1. These can be represented as two equidistant points on the unit circle drawn in the complex plane (i.e. where horizontal and vertical lines are drawn through 0, representing the real x and imaginary y axes respectively).

So in raising the number 1 to a linear fractional dimension, a circular number system is generated (where results lie as equidistant points on the unit circle).

Of course, this is well known in conventional mathematics. However what is gravely missing is the corresponding qualitative appreciation necessary to properly interpret such behaviour.

Two-Dimensional Understanding

The middle stages of development are associated in Western society with significant specialisation of linear understanding which provides the basis for our conventional appreciation of mathematics and science.

Though experience for most people largely plateaus at these levels, some are destined to progress significantly further to more advanced contemplative stages.

Two-dimensional understanding is directly associated with the unfolding of the first of these higher spiritual levels, which I refer to simply as H1. (This can be roughly equated with the psychic/subtle realm!)

The onset of such development often coincides with an existential crisis where one loses interest in conventional pursuits. So as one becomes increasingly disillusioned with conscious phenomena, a more unconscious inward search for meaning steadily gains momentum.

The period is thereby associated with substantial dynamic negation of phenomenal form. This leads in turn to the generation of a purer spiritual intuition that incubates within the unconscious. Then when the time is ripe - often in a dramatic conversion moment - such spiritual intuition is transferred outwards in a brilliant new illumination of conscious experience. Once again rational (linear) understanding is restored but in a more refined manner which incorporates (circular) intuition that is based on the successful fusion of complementary opposites.

One in experiencing nature at this level does so - not as an observer - but rather as an equal co-partner in creation.

For example at the earlier stage a flower, whatever its aesthetic appeal, is interpreted somewhat as “out-there” external to the observer. However now a dynamic interactive process operates when experiencing phenomena as between what is interior and exterior, with both aspects radiating a common meaning.

So, through this shared identity natural phenomena can reveal their true archetypal nature ultimately as Spirit. Likewise the true identity of self is also revealed in a similar manner.

Now the clear implication for mathematics is that when authentic experience of H1 takes place, the nature of its symbols substantially changes so that they likewise become more numinous, ultimately revealing their identity as spiritual mediators.

And in attempting to convey in scientific manner the precise nature of the transformation involved, we find ourselves once again employing the same mathematical symbols where this time their meaning is interpreted in a qualitative rather than quantitative manner.

So the dynamic two-dimensional experience of complementary opposites represents the qualitative counterpart to the quantitative interpretation of the two roots of unity.

In holistic mathematical terms, 1 denotes phenomenal form (strictly oneness as implicit in the recognition of such form) and + signifies positing i.e. making conscious in experience. So the first (linear) dimension relates to the (rational) conscious positing of understanding with respect to phenomenal symbols of form.

However in dynamic holistic terms, – 1 relates to the corresponding dynamic negation of such form which is the very means through which (unconscious) intuition is generated in experience. And when form is fully negated in this manner we have 1 – 1 = 0 as spiritual emptiness (which is the basis for pure intuitive appreciation).

Thus when + 1 and – 1 are used in quantitative terms they are interpreted in accordance with standard (either/or) linear logic as separate opposites.

However when + 1 and – 1 are used in a corresponding qualitative manner they are now interpreted in accordance with a dynamic (both/and) circular logic as complementary opposites.

Properly understood all experience inevitably involves the interaction of both quantitative and qualitative aspects (as equal partners). However Western mathematics - despite its admitted great achievements - has developed in a largely unbalanced fashion (where the qualitative aspect has been all but ignored).

Because from a conventional perspective mathematics is defined in merely one-dimensional terms, this means that only (conscious) reason can be formally recognised in its interpretations.

However in experiential terms, a continual interaction takes place as between conscious and unconscious with both reason and intuition necessarily involved in all mathematical understanding. Indeed for any truly creative work, intuitive inspiration must necessarily be present. However though mathematicians may informally acknowledge the importance of intuitive insight in tackling problems, in formal terms mathematics is presented solely as a rational body of truths.

We can now suggest a solution to the Pythagorean dilemma. At H1, mathematical object phenomena possess a dual identity. While maintaining a rational (discrete) aspect in experience they also radiate a new spiritual (continuous) light.

Likewise the square root of 2, which relates qualitatively to the two-dimensional experience of number, possesses a finite discrete identity whereby it can be approximated to any degree of accuracy. However it also possesses an infinite continuous meaning in that its decimal sequence has no fixed pattern. However both the (discrete) finite and (continuous) infinite aspects relating to rational and intuitive appreciation respectively, correspond to two distinct dimensions of understanding (i.e. 1st and 2nd).

So the transformation in the nature of the number in quantitative terms from rational to irrational, properly requires a corresponding qualitative transformation in understanding to appreciate its nature. However, once again in conventional mathematics, qualitative transformations are simply interpreted in a reduced rational manner. This then leads to basic philosophical reductionism regarding important key notions such as discrete and continuous.

Because of its great importance, it would be appropriate here to explain in a little more detail what is truly meant by two-dimensional circular understanding. Far from being somehow vague, it actually entails using dualistic reason in an increasingly refined manner, whereby it can be made properly compatible with understanding appropriate to holistic (unconscious) meaning.

To do this I will use a simple illustration that I have used on many occasions previously.

If one travels up a straight road left and right directions have a clear unambiguous meaning.

Now if we reverse directions and travel down the road left and right likewise once again have a clear meaning (within this new frame of reference).

So unambiguous linear understanding of direction can take place within each of these frames of reference (when considered independently of each other).

However when we now consider both frames simultaneously as interdependent, paradox immediately arises, for what is “right” in terms of the first frame is now “left” in terms of the second; and what is “left” in terms of the first is “right” in terms of the second.

Linear (1-dimensional) understanding takes place when both frames are viewed independently of each other.

However, circular (2-dimensional) interpretation occurs when both frames are now viewed as interdependent. Though the actual recognition of this interdependence is always directly of an intuitive nature, indirectly it can be given rational expression through the use of complementary pairings of opposites (that seem paradoxical in dualistic terms).

So, circular logic must be properly conceived in a dynamic manner as the interaction of both direct (intuitive) and (indirect) rational aspects. Now the significance of this illustration is that all experience, including mathematical, is necessarily conditioned by polar reference frames such as internal and external.

Thus when we fix the frame of reference as either internal or external respectively, unambiguous understanding can take place in a linear rational manner. However when we now attempt to dynamically relate both poles as interdependent, then all linear distinctions are rendered paradoxical at the circular intuitive level of interpretation.

In general whereas differentiation of phenomena in experience takes place through linear, corresponding integration occurs through circular understanding. 5 Thus in a proper interpretation of mathematical - as indeed all - experience both linear (either/or) and circular (both/and) logic are required.

We have illustrated circular logic here with respect to the simplest case (2-dimensional) which basically relates to appropriate dynamic appreciation of how the two poles internal and external interact in experience.

However it is of vital importance in that it gives rise to the basic question of how best to reconcile linear and circular type interpretation belonging to two distinct logical systems.

And the simple answer which is apparent from the simple diagram showing a circle with its line diameter is that it is the central point which is common to both (at the half-way point on the line).

So both systems are ultimately reconciled with reference to the central point (which is directly spiritual in nature). And this is of vital relevance in ultimately decoding the meaning of the Riemann Hypothesis!

Higher dimensional understanding then requires an increasingly multi-perspectival approach, where partial dualistic recognition with respect to many distinct directions (or poles) can be holistically integrated with each other in a pure intuitive manner. And the unique structural configuration that qualitatively defines each dimensional number is obtained with reference to the root (in quantitative terms) of that same number. The quantitative value of the fractional dimension is thereby the reciprocal of the whole dimension that provides the appropriate structure for corresponding qualitative logical interpretation of that root.

So in resolving the Pythagorean dilemma, the qualitative structure of 2 as holistic dimension, is used to appropriately interpret the quantitative value of ½ i.e. square root corresponding to the partial dimension.

However, because of its considerable importance we must also look briefly at the nature of 4-dimensional understanding which is so essential in dynamically structuring the world in which we live.

If we plot the four roots of unity in the complex plane (as the quantitative counterpart of qualitative understanding) we literally construct a circle divided into four quadrants. We have already dealt with the first two roots + 1 and – 1 respectively. The two new additional roots, + i and – i roots are however imaginary. (The symbol i here relates to the square root of – 1).

I have gone into considerable detail in other places regarding the holistic qualitative significance of the imaginary concept which basically relates to an indirect rational means of conveying holistic meaning that is properly unconscious.

This observation is of the utmost significance for mathematics where imaginary numbers are now widely used in the reduced linear manner (that characterises conventional interpretation).

However what is not at all clearly recognised is that imaginary numbers are but indirect expressions of an alternative logical system.

Thus the proper interpretation of imaginary numbers requires both circular as well as linear logic.

The use of complex numbers (which have both real and imaginary parts) is now an increasingly vital component of conventional mathematical understanding.

However appropriate interpretation of such numbers requires in corresponding qualitative terms, a complex rational approach (which includes both real and imaginary aspects).

Whereas real understanding is provided through conventional mathematical interpretation, Holistic Mathematics is required for the vital imaginary aspect. Here intuitive type understanding pertaining to the unconscious is mediated through appropriate rational use of mathematical symbols. And the combined use of both aspects leads to - what I refer to as - Radial Mathematics. So a major philosophical dilemma - which is central to difficulties in unravelling the Riemann Hypothesis - is that Conventional Mathematics still attempts to interpret complex numbers through a qualitative approach that is solely real.

Roger Penrose in “The Road to Reality” refers continually to the “magic of complex numbers” in recognition of their remarkable holistic properties in many areas of mathematics.

In the early magical stage of infant development we have the overlapping of the holistic unconscious with specific conscious symbols so that these then convey a universal meaning (though in a confused fashion).

In like manner the use of complex numbers (esp. as dimensions) in mathematics frequently gives rise to results with an unexpected holistic capacity. However though these results can often be given a reduced quantitative interpretation, their real significance is continually missed through the lack of genuine philosophical appreciation.

When one reflects on it, i is an obvious example of a circular number (that lies on the circle of unit radius). So not surprisingly it likewise requires a circular qualitative interpretation to appreciate its nature. [6]

The significance of this is that the Riemann Zeta Function (with which the Riemann Hypothesis is intimately linked) employs complex dimensions and generates many results that have no strict meaning in quantitative terms.

However these same mathematical results can be given a truly coherent meaning using holistic mathematical interpretation.

So in a dramatic fashion the Riemann Zeta Function points to the need for incorporation of both systems of logic in the interpretation of results (both quantitatively and qualitatively). And as we shall see, when we do this the Riemann Hypothesis itself dissolves in a remarkably simple explanation.

Prime Numbers

The prime numbers 2, 3, 5, 7,….play a vital role in Number Theory as all (natural) numbers can be expressed as the product of one or more prime numbers.

In this way they are often likened to the atoms in physics as the basic building blocks of matter or alternatively the elements in chemistry which form the basic ingredients for all molecules.

So a prime number has no factors other than itself and 1. In this way its truly linear nature is demonstrated.

However despite their importance, the study of prime numbers has proved a great challenge as they seem in their individual identities to display no obvious pattern. Though Euclid was able to offer an ingenious proof as to why the prime number series is infinite, little further progress was made in understanding their nature for some considerable time.

However there is an equally important aspect to the behaviour of prime numbers, which though recognised to a degree, is not capable of proper appreciation (within the accepted mode of interpretation of Conventional Mathematics).

This relates to the opposite fact that the primes equally are the most circular of all numbers.

This latter aspect can be demonstrated in a variety of ways.

When we take the prime roots of 1 they will appear as unique points on the circle of unit radius (that cannot be derived from any other combination of roots).

Likewise as I mentioned in another article on this site some time ago, when we obtain the reciprocal of a prime number such as 7 its decimal expansion displays fascinating circular properties.

In fact 1/7 is the best known of the full cyclic primes where the number of unique digits (which continually repeat) is 1 less than the prime number involved.

So with 1/7 the digits 142857 continually recur in the decimal sequence.

When one multiplies this six digit number by all numbers from 1 to 6 respectively the same circular sequence of digits will recur (with only the order changing). And this same sequence of numbers serves an important qualitative significance in the Enneagram personality system, providing the path through which integration takes place for six (of the nine) personality types.

I explained earlier in referring to two-dimensional interpretation, how circular (intuitive) understanding is dynamically generated in experience through the negation of the first dimension i.e. – 1 (which is a point on the unit circle). Well, quite literally 1/7 is just another way of expressing 7 (raised to the power of – 1).

So we can see here how quantitative behaviour complements qualitative understanding!

Prime numbers also play a vital role in modular arithmetic where numbers are arranged in circular fashion as on a clock face. This number system now plays a key role in the development of security systems for the Internet.

Our normal clock uses the 12 numbers from 1 - 12 respectively. A modular 12 arithmetic would involve a slight modification in that 12 would be replaced by 0. So 5 X 3 in modular arithmetic = 3 (i.e. 15 = 0 + 3).

However, other product combinations lead to zero e.g. 4 X 3. To avoid this problem, which significantly reduces the value of the system, we need to use a prime number as base. This ensures that any two numbers when multiplied together will yield a positive answer.

Indeed there is a fundamental conundrum regarding our understanding of prime numbers. We tend to view them in linear fashion as the basic building blocks of natural numbers. However further reflection reveals that the primes intimately depend on the natural number system for their precise location.

So there is an inevitable dynamic two-way interaction as between prime and natural numbers with the behaviour of each co-determined through this interaction.

One interesting manifestation is the manner in which primes (with just one factor) are often located as one less than highly composite natural numbers (with many factors).

In fact this is especially relevant with reference to what are called Mersenne primes. So 128 for example is the most composite number possible for its size (with seven factors that are all 2). And when we subtract 1 we get 127, which is prime. This complementary behaviour (where two consecutive numbers have the minimum and maximum amount of factors possible for their size) provides a ready means of searching for new largest primes. So invariably when a new record is established it is another Mersenne prime.

This prime behaviour has also an important correspondent in qualitative psychological terms. Ascetical writers frequently point to the danger of consolation quickly turning to temptation in the spiritual life. So here a strong experience of (composite) integration becomes immediately associated with its opposite in the surfacing of new primitive form.

Indeed we cannot even meaningfully speak about the primes without implicitly involving the natural numbers.

For example if we identify the first prime four numbers as 2, 3, 5 and 7 respectively implicit in this recognition is an ordinal natural number ranking system of 1, 2, 3 and 4. So 2 is the 1st, 3 the 2nd, 5, the 3rd and 7 the 4th prime number respectively.

Also the prime number theorem demonstrates remarkable regularity in the general distribution of the primes.

If you want to know the average gap between prime numbers at any location in the number system simply extract the natural log (ln on calculators) of the number.

For example if you do this for 1,000,000 you obtain 13.8155 (approx). This means that the average gap between primes in that region of the number system (around 1,000,000) is about 14 and the larger the number in question the better is the approximation.

So in the simplest version of the prime number theorem, to find the number of primes less that a given number n, we simply divide n by log n. And the relative accuracy of this measurement has been proven to steadily increase as n becomes larger.

Indeed there are strong similarities as between the behaviour of prime numbers and the quantum behaviour of particles.

Though we know that any quantum event - in isolation - is impossible to predict accurately, yet an amazing statistical regularity exists in the general overall behaviour of particles giving quantum mechanics astonishing predictive capacity. So quantum particles, though apparently acting in an independent unpredictable fashion, actually communicate with each other in a manner designed to preserve an overall synchronistic behaviour.

And we need to look at the behaviour of prime numbers in the same manner.

Indeed this comparison is far from accidental as - properly understood - quantum particles in qualitative terms represent the prime behaviour of matter. Thus what we observe at the macro level of reality are composite natural phenomena entailing the complex organisation of prime constituents that have their roots at the sub-atomic level of matter!

Indeed this realisation of the intimate connection, as between prime numbers and the behaviour of quantum particles, has now come to be recognised in a dramatic fashion through the search for a solution to the Riemann Hypothesis.

Using the language of physics we can say that prime numbers possess both particle and wave aspects (which are complementary).

In the language of psychology we can equally validly say that prime numbers possess both (individual) structural and (overall) state aspects, which are interdependent.

So prime numbers actually entail two complementary aspects (linear and circular) that relate to two distinct logical systems.

Indeed this complementarity is perhaps best expressed in the comments of Don Zagier. [7]

"There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.
The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision."

However to properly appreciate the dynamic complementary relationship of both aspects we need to briefly look at prime numbers qualitatively now in the context of psychological development.

Prime Numbers in Development

In earliest infant experience psychological structures remain entirely undifferentiated; likewise because no differentiation has yet taken place they likewise remain un-integrated.

So here conscious and unconscious remain enmeshed with each other as mere potential for existence. So the two logical systems, linear and circular, that provide the basis for all subsequent differentiation and integration in experience, remain identical with each other (as total confusion).

However once the first differentiation of structures takes place the two aspects start to separate from each other and ultimately can only be united through the pure attainment of spiritual awareness, where again both are reconciled with each other in mature fashion as the identity of form and emptiness.

To grasp what is meant by the prime notion in qualitative terms, we need to probe into the nature of primitive instincts (which especially characterise early infant behaviour).

With such behaviour a basic confusion is involved, whereby what is holistic (properly pertaining to the unconscious) is directly confused with specific conscious phenomena.

Quite literally, the holistic unconscious is responsible for providing the dimensional background (within which distinct phenomena can be identified).

However with primitive events we have a direct confusion of this background with the phenomena that arise.

So space and time thereby immediately collapse in being identified with phenomena. In this way no perspective exists to allow stable experience to continue. So the dimensions in a sense thereby become contained in the forms experienced and so quickly dissolve and pass from memory (in a series of qualitative zeros). [8]

This basically explains why in earliest infancy, appreciation of phenomena is of such an immediate transient nature.

Only when the conscious becomes sufficiently differentiated from the unconscious aspect can the more permanent stable experience of form be sustained.

However though such successful differentiation indeed represents a wonderful psychological achievement, enabling the complex organisation of natural phenomena, it takes place in a somewhat reduced manner.

So the first main phase of development is largely geared to the gradual growth of linear type understanding which then reaches its specialised expression during the middle stages.

Indeed it is fascinating how such reductionism is replicated in mathematical interpretation of the relationship between prime and natural numbers. Here the primes are viewed as the essential building blocks of the natural number system and then directly included as members of the same system.

So in the very listing of natural numbers 1, 2, 3, 4, 5,…. no distinction is made as between prime and composite members (derived from these primes).

In the same way with the linear type interpretation that defines standard mathematics, the unconscious aspect, insofar as it is recognised, is subsumed under conscious type interpretation. In other words Mathematics is formally understood in merely rational linear terms.

However when radical development at the higher spiritual stages unfolds, a substantial change takes place whereby attention shifts away from primes as basic building blocks to their overall general relationship with the natural numbers.

In psychological terms this generally entails a substantial temporary withdrawal from the world of natural phenomena, in an attempt to unearth the roots of natural desire that relate to primitive instincts still repressed through earlier conscious development.

So in the gradual unravelling of this earlier primitive confusion, a harmonious relationship can then eventually be established as between conscious and unconscious.

Thus this second major stage of development is associated with the specialisation of the circular logical system and culminates in the psychological equivalent to the prime number theorem.

In other words through detachment from primitive phenomena, one is thereby enabled to understand their overall holistic interdependence.

In mathematical terms the general nature of primes (in their distribution among the natural numbers) has of course been intensely investigated culminating in the prime number theorem.

However once again this can only be done in a linear manner (geared solely to quantitative interpretation).

Therefore the two types of understanding of primes viz. as basic building blocks of natural numbers and of their general dependency on this number system, lead to two conflicting notions of the primes that are diametrically opposed (and cannot be properly reconciled within a restricted linear framework).

The final main stage of psychological development relates to - what I refer to as - the radial stages. This involves gradual growth in the full interaction of the two systems linear and circular (that have already undergone separate specialised development).

This leads to an interesting problem. At this stage the spiritual contemplative proficient will have achieved substantial general mastery with respect to free control of primitive desires.

However the very process of engaging once again in a more extensive manner with natural phenomena inevitably gives rise to the surfacing of new primitive instincts (as the roots of such phenomena).

Thus for spiritual peace to be maintained in the midst of intense active involvement, a continual dynamic psychological correction process is necessary. This is designed so as to reconcile the unconscious impulses thereby temporarily excited through active involvement, with the deep state of contemplative equilibrium already attained.

Indeed this is central to the whole problem of attachment in the spiritual life. When the primitive roots of desire are not properly recognised, attachment thereby becomes deeply embedded in a rigid experience of form which cannot be successfully resolved at a merely conscious level. Therefore freedom from attachment requires that its roots be properly erased in the unconscious (through the speedy unravelling of primitive desires).

And it is this final stage that directly relates as the psychological equivalent to the “non-trivial zeros” of the Riemann Zeta Function, which have been shown to have amazing properties in the further refinement of the manner in which the primes are distributed.

The Riemann Zeta Function

The Riemann Zeta Function (with which the Riemann Hypothesis is intimately related) starts in a very simple series that again is famously associated with the Pythagoreans.

This series is simply made up of the reciprocals of the natural numbers i.e.

1, 1/2, 1/3, 1/4, 1/5,…….

Now Pythagoras is reputed to have discovered a remarkable musical property of this series. By filling a vessel with water and then striking it with a hammer he produced a musical note. He then removed half of the water and struck the vessel again to find that the note sounded in harmony with the first. Then when more water was removed leaving the vessel 1/3 full, again the next note likewise sounded in harmony and so on with the vessel 1/4, 1/5 full continuing through the various terms of the sequence. So not alone was reality encoded through numbers, but these numbers also vibrated as the music of the spheres.

It is because of this connection that Pythagoras made between the terms of this series and musical notes that it is commonly referred to as the harmonic series.

When we obtain the sum of terms of the series we get

1 + 1/2 + 1/3 + 1/4 + 1/5 +……, which has no finite limit.

Indeed despite its simplicity this series can be shown to have a direct relationship to the primes.

As we know the occurrence of primes thins out considerably as we ascend up the number system.

We have already seen that in the region of 1,000,000 that the average gap between primes is approximately 14. However if we want to see by how much this gap changes as we increase the number by 1 we simply obtain the reciprocal of the number in question.

Thus the average gap between primes, as we move from 1,000,000 to 1,000,001, increases by roughly 1/1,000,000 (which is the corresponding millionth term in the harmonic series). So we have the simplest connection with the primes entailing a natural number n and its reciprocal 1/n. [9]

However one feature which limits the use of the harmonic series is the fact that its sum diverges (exceeding any finite limit).

However if we decide now to square each term (or raise to any other power greater than 1) the series will converge to a finite value.

When we confine ourselves to real powers (greater than 1) we get what is called the Euler Zeta Function, named after Leonhard Euler who carried out remarkable pioneering work in this area. (Zeta in this context is a Greek letter conventionally used to denote the series with respect to whatever power is involved).

Euler demonstrated that there was an important connection between the sum of terms in such series and the product of other terms based on the primes.

For example when the power of each term = 2,

1/1 + 1/4 + 1/9 + 1/16 + … = 4/3 X 9/8 X 25/24 X 49/48 X ….

So the denominators of terms on the left represent the squares of the first four natural numbers 1, 2, 3, 4 while the numerators of terms on the right represent squares of the first four corresponding primes 2, 3, 5, 7 (with both series summed to infinity).

Euler was able to show that the infinite sum of terms (on both sides) was connected with the value of pi (and that this was always the case where an even numbered dimension was involved).

Thus, due to Euler it was established that intimate connections between the prime and natural numbers could be demonstrated through using these series.

However it took the work of Bernhard Riemann to fully establish the truly extraordinary links between both sets of numbers.

When Riemann came to study such series two key further developments had taken place.

Firstly, largely through the work of Gauss, a strong empirical basis had been laid for what was to be known as the Prime Number Theorem. Even though the prime numbers in isolation appeared so irregular, a simple counting function had been discovered (and later improved on) by Gauss that seemingly predicted their overall frequency to a high degree of accuracy.

Using this function one could for example predict well how many primes would occur in the first billion natural numbers (without having to locate any specific primes).

Secondly dramatic developments had taken place in the use of complex numbers. Indeed it was Riemann's genius that he somehow could see that by using complex - rather than confining himself to real - number dimensions in the series, very precise information could be obtained regarding the behaviour of primes.

So the Riemann Zeta Function thereby represents the sum of terms - that first appeared in the harmonic series - when raised to powers (dimensions) that have complex values.

Though the Gaussian prime number counting function had been improved by Riemann, with his new function R(n) enabling a remarkably good approximation of the frequency of primes less than any given natural number, it was not yet capable of giving an exact answer.

However, using his Zeta Function, now geared for calculations using complex powers, Riemann believed that he could eliminate remaining errors from the number function and in principle exactly calculate the no. of primes.

We will say a little about how he proposed to do this later. However what is ultimately more important for our purposes is to show how the use of complex numbers as powers leads to a whole series of important results that cannot be satisfactorily interpreted in conventional mathematical terms.

However these same results can be given a coherent holistic mathematical interpretation. And when we properly interpret such results in a qualitative manner (which is the way they are designed to be decoded) the true nature of prime number behaviour is again revealed as entailing the interaction of two distinct logical systems, linear and circular, which in turn paves the way for ready resolution of the Riemann Hypothesis.

As we have seen Euler showed that his Zeta Function was properly defined for all real values greater than 1. So using these values his function could be shown to converge towards a finite limit (in the standard conventional manner of quantitative interpretation).

However using the enhanced Zeta Function, with s used to denote the complex numbers, a + it, representing the dimensional powers involved, Riemann was seemingly also able to give finite meaning to a whole range of negative values for s.

Indeed he came up with a remarkable transformation formula. Fortunately we do not need to worry how he achieved this. Suffice it to say that he was able to demonstrate that for any value of s through which the function was calculated that an alternative result could be obtained using 1 – s as power.

This meant for example from the value for s = 2, which Euler calculated as (pi squared)/6 that we could thereby calculate the functional value for s = – 1 which the transformation formula gives as – 1/12.

Now there is a famous account of the gifted Indian mathematician Ramanujan who when writing to Hardy at Cambridge regarding his early findings included the seemingly nonsensical result,

1 + 2 + 3 + 4 + ……(to infinity) = – 1/12.

Initially Hardy was inclined to think that he was dealing with a fraud, but on further reflection realized that Ramanujan was in fact describing the Riemann Zeta Function (for s = – 1). He could then appreciate his brilliance as one, who though considerably isolated and without any formal training, had independently covered much of the same ground as Riemann.

However it still begs the question as to what the actual meaning of such a result can be, for in the standard conventional manner of mathematical interpretation, the sum of the series of natural numbers clearly diverges.

The startling fact is that this result - though indirectly expressed in a quantitative manner - actually expresses a qualitative type relationship (pertaining to holistic mathematical interpretation).

We need therefore to understand this point clearly. Conventional mathematical results are based on a (reduced) dimensional interpretation = 1. However whenever a number is defined with respect to any dimension other than 1, an alternative qualitative interpretation can be given. And when quantitative results are thereby obtained (that have no meaning in linear terms) their true significance can only be given in holistic mathematical fashion (in accordance with the qualitative logical structure directly associated with the dimension in question).

Indeed this explains immediately why the Riemann Zeta Function is undefined for s = 1.

When we add up the harmonic series 1 + 1/2 + 1/3 + 1/4 + …. (the function for s = 1) to infinity, it diverges according to standard linear interpretation.

However because in this case, and only this case, 1 is already used as dimension, the alternative qualitative interpretation is necessarily reduced to the quantitative with the series once again diverging in value.

However, to properly illustrate the key insight that is involved here it may be more revealing to start with what are misleadingly referred to as the “trivial zeros”.

So, using Riemann's transformation formula it can be shown that the value of the function = 0 for all negative even numbers i.e. when s = – 2, – 4, – 6, – 8, ……(to infinity)

So if again we were to try and interpret this for s = – 2, in conventional terms it would entail that 1 (squared) + 2 (squared) + 3 (squared) + 4 (squared) + ……(to infinity) = 0 which clearly has no quantitative meaning.

However we have already seen that two-dimensional qualitative understanding entails the identity of complementary opposites, + 1 and – 1.

Now when we look at the famous series which Euler had studied for s = 2, we see that its result can be expressed in terms of the value of pi. Though this result is indeed already meaningful in standard quantitative terms, indirectly the qualitative nature of two-dimensional understanding is present in the form of its expression.

Just as pi directly represents the quantitative relationship between the (circular) circumference and its (line) diameter, two-dimensional understanding likewise expresses the relationship between circular and linear understanding (in qualitative terms).

The philosopher Hegel is one who employed such qualitative two-dimensional understanding extensively in his work. Here thesis (1) and antithesis (– 1), as two linear poles of understanding, are then integrated in a new circular form of integral appreciation (as synthesis) before being differentiated once again as a further thesis.

However in the spiritual traditions it is well recognized how secondary attachment can easily become associated with such experience. Therefore to attain the pure intuitive awareness (of what is expressed to in rational objective terms as the complementarity of opposites) considerable dynamic negation is required with respect to such understanding.

So a state of pure contemplation requires the negation of the second dimension (s = – 2) which, when successful leads to the qualitative state of emptiness (i.e. nothingness).

St. John of the Cross, who in the Western mystical tradition is perhaps the greatest exponent of such negation, literally refers to its purpose as the attainment of nada (i.e. 0 in qualitative terms).

So if we further translate what this interpretation means for the squared series of natural number terms that it contains, then all these are now understood in a purely non-attached manner without rational translation (where the identity of opposite internal and external poles - that dynamically interact in understanding - is intuitively realized in a direct manner).

And when one reflects sufficiently on the matter, our actual experience of number symbols necessarily entails the continual interaction of internal and external aspects.

So contemplative attainment requiring non-attachment, thereby entails a form of understanding of mathematical symbols where dualistic distinctions can be eroded. And one cannot achieve this while remaining confined to mere conventional (one-dimensional) understanding.

Thus there is an appropriate type of mathematical appreciation that is associated with each higher dimension. This requires increasing refinement in the manner in which mathematical symbols are dynamically interpreted - the precise structure of which is given by the corresponding root form of the dimension - thus enabling their inherent numinous quality to be greatly enhanced.

We can quickly deal with all the other “trivial zeros” associated with the negative even dimensions. These can be seen to represent ever greater degrees of spiritual refinement resulting from the more intricate arrangements of complementary opposite terms (as the qualitative counterpart of corresponding roots that are likewise arranged in a complementary manner).

In this way we can perhaps understand advanced contemplative development as the movement to equilibrium dimensional states of increasingly purer levels of spiritual emptiness.

However integration - even in spiritual contemplative terms - is always equally associated with appropriate differentiation of experience. And this is where the odd numbered dimensions come into play. And we have already seen how the first such numbered dimension (i.e. 1) represents a specialisation in differentiated understanding!

One issue that still puzzles mathematicians is the manner in which expressions involving pi for the Euler function, are confined to (positive) even number dimensions.

We can give at least a starting explanation here in that the odd dimensions are associated with differentiation in experience (rather than directly with integration). Therefore we would expect a break in the complementary symmetry that characterises the even integral dimensions. And of course this is replicated in quantitative terms by the fact that odd numbered roots are never arranged in a fully complementary (i.e. circular) manner!

However it is the odd negative values for s that are especially fascinating.

Once more as we have seen in the natural number series for s = – 1, the value that arises for the function, – 1/12 has no meaning in conventional quantitative terms.

It is revealing however that this answer (and indeed all answers that arise when the power is a negative odd value) is rational.

Now again the clue to what is involved in terms of qualitative interpretation can be given through reference to St. John of the Cross.

St. John distinguishes clearly in his writings as between active and passive purgation. Passive purgation relates directly to holistic type understanding (where an unwanted indirect conscious attachment remains) and implies withdrawal from reality in a deeper type of contemplative awareness.

However for development to remain healthy an appropriate form of differentiation must also take place. So between each higher state of integration (the even dimensions) a higher state of differentiation in an appropriate form of active involvement is also required (the odd dimensions). And here selfish attachment is removed through the loss of any sense of intuitive light. Thus one strives to carry out activities in faith, with no supporting intuitive guidance. In fact this is how the use of reason is sharpened for one must learn to greatly economise on conscious effort as one tends only to what is truly essential.

One can see that – 1/12 is of small magnitude (as a rational number). This already suggests a limited use of reason (which befits the growth in contemplative type awareness).

Interestingly, further values become even smaller up to s = – 5 suggesting even greater economy in the use of reason before eventually increasing - slowly at first - and then more rapidly. This would too be in keeping with the onset of the radial life at the higher dimensions which is associated with a great increase in active involvement.

Also the values keep switching sign which again suggests that the focus of attention keeps likewise alternating during the differentiation stages as between external and internal cleansing of the rational faculties.

So intuitive reasoning is purified through the removal of associated dualistic rational elements (which befits deeper contemplative integration); in reverse manner reason is purified through the removal of associated intuitive support as is suited to active involvement (appropriate to purer spiritual engagement with reality). And both of these aspects are given expression through the negative integer values of s in the Riemann Zeta Function, which as we have seen pertain directly to holistic (qualitative) rather than conventional (quantitative) interpretation. [10]

So to sum up here we have been at pains to show how these results that arise from the use of complex numbers as dimensions in the Riemann Zeta Function lead to numerical results - that to be meaningful - must incorporate both quantitative and qualitative interpretations.

Also the transformation formula shows that the quantitative results (for s greater than 1) and corresponding qualitative results for s less than 0) are indissolubly linked. Though we cannot see this in conventional rational terms, there is an inherent unity as between both quantitative and qualitative aspects and it is this key feature that above all captures the true inherent nature of the primes.

Now once again the Riemann Hypothesis is directly associated with the dimension which has real part = 1/2.

Revealingly when we use the transformation formula for s = 1/2 (i.e. .5) both sides of the equation are equal.

Bearing in mind what we have concluded regarding the intimate links as between quantitative and qualitative type appreciation, we can thereby see that it is only at this value that the equality of both types of interpretation can be reconciled.

Though we will have a little more to say regarding the famed non-trivial zeros, this in a nutshell is what the Riemann Hypothesis is really all about.

In other words the Riemann Hypothesis is necessary to ensure that both quantitative and qualitative aspects (which are inherent in the very nature of prime numbers) can both be successfully reconciled with each other.

Non-Trivial Zeros

When we allow the value for s (i.e. the dimension to which each term in the Zeta Function is raised) be complex nos., the value of the function will equal zero for an infinite set of these numbers.

This critical set of complex numbers (for which the function is zero) can be referred to as the roots of the function and are widely celebrated as the non-trivial zeros. 11

Indeed this represents unfortunate terminology reflecting clearly the bias of conventional mathematics. Because the trivial zeros can be more easily calculated and have - in direct terms - a qualitative rather than quantitative significance, they are dismissed as somewhat irrelevant.

The non-trivial zeros - by contrast - have a direct quantitative (as well as unrecognised qualitative) importance and Riemann in a truly ingenious manner was able to show how these values could be used to correct the predictions that his improved counting function was giving him regarding the general distribution of primes.

Indeed in principle he believed that he had acquired the means to exactly predict the number of primes below any natural number one cares to suggest.

It was no easy job in Riemann's time to calculate the complex values that would give the roots of his function. It wasn't discovered till much later that he had actually devised a new method (more efficient than was previously available) for doing this, using it to calculate the first three roots to a fair degree of accuracy.

Now Riemann noticed that in each case the real part of these complex values was equal to .5 and though not able to prove that this would always occur, suggested that it would very probably be true for all roots. And so the Riemann Hypothesis was born!

Thus Riemann did not succeed in proving his famous Hypothesis. Likewise, though he did provide an exact formula for counting the primes, it took another 37 years for the prime number theorem to be finally proven (based on his pioneering ideas)!

If we assume the Riemann Hypothesis is true and that the real part of the complex values (giving the zeros) is always .5, then we need only concern ourselves with the imaginary part. So all the non-trivial zeros then lie on the vertical imaginary line drawn through .5 which - not surprisingly - is referred to as the critical line.

The first two zeros occur at 14.1347… and 20.0220… on the critical line and as we move up the line, the zeros appear with greater frequency. Riemann was able to suggest how these imaginary values could be used to create a whole series of wave patterns that would cumulatively correct the initial approximation of primes provided by his general function.

However in contrasting fashion to the primes, which gradually thin out as we move to larger natural numbers, the zeros become ever more dense requiring a rapidly increasing number of corrections to exactly predict the primes as we move up the scale of the natural numbers. [12]

The wave patterns that these imaginary values exhibit of course give rise to musical comparisons. So in a sense we have come full circle. The Riemann Zeta Function has its roots in the harmonic series of the Pythagoreans (with its strong musical connections). And then in the sophisticated use of this function to predict the frequency of primes the musical comparisons are once again apparent.

Also we saw earlier that the problem created by the square root of 2, related to a critical distinction as between the discrete and continuous notion of number. And again in like manner the wave patterns created from the non-trivial zeros are used to bridge the continuous smooth curve of the general counting function used to approximate the primes and the actual occurrence of individual primes, which is of a discrete nature.

One of the most exciting breakthroughs in recent years has been the discovery that the non-trivial zeros are seemingly very closely related to energy values in certain quantum chaotic systems. So they appear to have a direct relevance to the sub-atomic physical world.

However what is not at all clearly recognised is that the non-trivial zeros equally have a direct relevance in complementary super qualitative terms to psycho-spiritual reality. And once again, correctly understood, both physical and psychological aspects are indissolubly linked.

Indeed mirroring the language of physics these zeros equally correspond to a high energy qualtum chaotic system which characterises - what I refer to as - the specialised unfolding of radial development.

Perhaps we get the greatest evidence of what this means in the lives of celebrated mystic activists i.e. people who have reached a deep and constant state of contemplative depth and yet who successfully combine this with extensive committed involvement in worldly affairs.

The Riemann Hypothesis would suggest that the real part = ½. What this simply means in psychological terms is that to avoid dualistic attachment, a golden mean must be maintained as between opposite conscious polarities (such as exterior and interior). If you envisage a circle with its line diameter the midpoint of the unit line is equally the midpoint of the circle (which is literally ½)!

In other words it is at this point, which represents the central value, that the line and circle are reconciled in quantitative physical terms (and also linear and circular understanding in a corresponding qualitative spiritual manner). And, because this very reconciliation constitutes the inherent nature of primes (both in terms of quantitative behaviour and qualitative understanding) the Riemann Hypothesis is thereby axiomatically true.

So, successful adherence to the golden mean between polarities enables refined non-attached engagement with natural phenomena (which is compatible with continual contemplative absorption in Spirit).

However even here, chaotic type experience (i.e. new activity that is necessarily unpredictable) continually gives rise to primitive imaginary projections. However in the case of the zeros these always occur in pairs. Indeed there is a remarkable connection here with the nature of virtual particles, which have an extremely transient existence through being immediately cancelled out by the opposite pole.

Though of course it is an ideal that can only be loosely approximated, with the purely spiritual person engaged in such active pursuits, imaginary projections are likewise speedily erased as soon as they arise, thus ensuring that little primitive attachment can take hold. Such a person then has the capacity to act as a superconductor of Spirit in a life of service that can bear amazing fruit.

And when little attachment remains at a primitive unconscious level, likewise very little can become embedded at the macro level of conscious phenomena, thus facilitating maintenance of the golden mean.

So, when appropriately interpreted in holistic mathematical terms, the Riemann Zeta Function (and Hypothesis) has a remarkable relevance for the precise qualitative understanding of certain psycho-spiritual dynamics at the most advanced levels of human development (which complements corresponding understanding of quantum physical interactions at the sub-atomic levels of matter).

And once again, because in the very dynamics of understanding these two aspects (quantitative and qualitative) are indissolubly linked, the Riemann Hypothesis - when correctly interpreted - necessarily relates to both aspects. Strictly speaking, in dynamic experiential terms, it is appropriate psycho-spiritual qualitative appreciation of the zeros that enables full quantitative interpretation of these same zeros that is at once rationally based yet intuitively meaningful. So again, Conventional Mathematics, in ignoring the qualitative dimension, at best can hope to obtain a limited partial understanding!

Expressing it more accurately the Riemann Hypothesis relates to the fundamental relationship between both quantitative and qualitative aspects (where both can be successfully reconciled as identical). 12

And so in this statement likewise is encoded the secret inherent in the nature of the primes, the ultimate understanding of which can only be attained through an ineffable spiritual experience, where any remaining division as between quantitative and qualitative aspects is finally dissolved.

Deeper Significance of the Riemann Hypothesis

By substantially widening the frame of reference to include qualitative as well as quantitative interpretation of mathematical symbols, the Riemann Hypothesis resolves itself in a surprisingly simple manner.

However it might be worthwhile here to comment just a little further on the nature of this resolution which is certainly not a proof in the conventional sense.

Indeed one might ask if the quest for such a proof (i.e. in accepted linear terms) is still attainable!

Well perhaps surprisingly, this resolution of the Hypothesis answers that question.

For, once we accept that prime numbers necessarily combine in their nature two distinct logical systems (that are linear and circular with respect to each other) and that the Riemann Hypothesis represents a fundamental statement regarding the ultimate identity of these two aspects, then it is not possible to prove this essential identity with reference to just one aspect (i.e. linear).

So assuming that the axioms of conventional mathematics are sufficiently watertight to exclude use of the alternative circular logical system, no acceptable proof can be found.

Indeed this really signals the need for a decisive mind change among mathematicians regarding the very possibility of proof (in the conventional sense) for a wide range of fundamental prime number problems. [14]

It is now accepted in physics that quantum mechanical behaviour is subject to the uncertainty principle where merely probable statements can be made regarding the behaviour of particles. When understood appropriately it is similar with prime numbers (which from one valid perspective represent the numerical language of quantum interactions). And when one further considers that quantum particles can holistically communicate with each other literally in a prime (i.e. primitive instinctive) manner, then clearly the implication is that a comprehensive interpretation of prime numbers requires both conscious (quantitative) and unconscious (qualitative) aspects of understanding, which this article has sought to demonstrate.

In his assessment that the Riemann Hypothesis was very probably true, Riemann - perhaps unwittingly - had already correctly pointed to its proper status (when viewed from the conventional perspective). So, though the proposition may be demonstrated to be true with an extremely high degree of probability, absolute proof is not possible.

And this indeed is where the true significance of the Riemann Hypothesis lies for it really points to a much deeper problem. For far too long, a tragic split as between the quantitative and qualitative aspects of mathematics has existed. Though this has allowed the considerable specialisation of one aspect (quantitative) with admittedly remarkable achievements, it has also led to a dangerous imbalance in our overall scientific approach to reality which is extremely unhealthy from an integral viewpoint.

One example of such imbalance is the present threat to our global environment which if not speedily rectified will inevitably lead to a whole series of major ecological disasters this century. And as Conventional Mathematics serves as the indispensable tool of science, correcting this imbalance requires rectifying the problem at source in the fundamental way that we view mathematical relationships.

So to conclude I will briefly propose what I see as an appropriate framework for a more comprehensive mathematical approach. It comprises three major strands.

  1. Conventional Mathematics: this represents the standard quantitative approach based - as we have seen - on linear rational logic. Realistically it will continue to dominate for some time with tremendous further advances to be made using its established methods. However some problems - e.g. in relation to primes - will continue to prove especially difficult to unravel (due, as we have seen, to the fact that primes cannot be successfully viewed in a merely quantitative manner).

  2. Holistic Mathematics: this represents the alternative qualitative approach based on appropriate multidimensional philosophical understanding of mathematical symbols (which like a vast unexplored continent remains almost entirely undeveloped). In principle every symbol, relationship, hypothesis with an already accepted standard interpretation in conventional terms can be given a corresponding qualitative meaning (with fascinating alternative applications). Holistic Mathematics thus has enormous potential as the indispensable tool for a truly integral scientific approach as in the study of human development.

  3. Radial Mathematics: Conventional and Holistic Mathematics are like two (separate) blades of a scissors. However to obtain full benefit from these blades they must be used in conjunction with each other thus leading to mathematical activity that is truly radial.

Radial Mathematics while maintaining an appropriate correspondence as between both quantitative and qualitative aspects thereby has the capacity to greatly enhance both (as they are inherently complementary).

Indeed I would distinguish three main types of Radial Mathematics.

Radial (Type A): here though a certain mature balance is maintained as between both aspects the emphasis is mainly on the holistic (qualitative) side as an aid to better philosophical appreciation of the nature of mathematical relationships. Indeed I would describe my own recent research as a - very - preliminary form of Radial (Type A) Mathematics. So for example in this article, the unravelling of the Riemann Hypothesis owes more to qualitative - rather than quantitative - appreciation. However, because so little use is presently made of such appreciation, it has enormous scope to literally throw revealing new light on many outstanding mathematical problems.

Radial (Type B): again this starts from a mature appreciation of the necessary interdependence of both quantitative and qualitative aspects with however greater emphasis here on using creative insight to inspire analytical type work of a truly original kind. Implicitly, many great mathematicians such as Ramanujan and Riemann correspond to this type (without however formally acknowledging the qualitative aspect).

Radial (Type C): this will offer by far the greatest possibilities for mathematical work that can be both extraordinarily creative yet immensely productive. For example in such an approach a satisfactory proof of a proposition requires that both aspects (quantitative and qualitative) be satisfied.

The Pythagoreans for example were able to prove in quantitative terms that the square root of 2 is irrational. However they were unable to provide the necessary qualitative explanation as to why it is irrational. And this lack of radial proof subsequently led to the break up of the school.

So we now have the appropriate context in which to place the Riemann Hypothesis, serving as a particular striking illustration of what is the basic axiom in radial mathematics i.e. the ultimate identity of both the linear (1) and circular (0) logical systems.

Properly understood, mathematics is necessarily experiential in nature and radial understanding allows for the fullest, most passionate dynamic realisation of such experience. Here one is never just satisfied to know but rather seeks to become continually transformed through such knowledge so as to attain fullest realisation in Spirit (which is the true ultimate goal of mathematics).

It is here that the mystery of the primes - which is the same mystery as life itself - is at last realised in that ineffable experience where (rational) form and (intuitive) emptiness merge and become inseparable from each other. And in this spiritual moment always present, with pure desire, you can find this same mystery ever more wondrously revealed again and again and again. 15

Sweet music indeed! And if you listen closely, you may hear the sounds of Pythagoras still ringing in the distance.


1. ref. Sabbagh “Dr. Riemann's Zeros” (Atlantic, 2002), p.160

2. ref. Sabbagh “Dr. Riemann's Zeros” (Atlantic, 2002), p.208

3. ref. Sabbagh “Dr. Riemann's Zeros” (Atlantic, 2002), p.227

4. Building on Jungian insights, Marie-Louise von Franz is one authority who has spoken persuasively regarding this Western cultural split in Mathematics as between its quantitative and qualitative aspects.

5. One of my consistent criticisms of Integral Studies is that the nature of integration is rarely properly distinguished in qualitative terms from corresponding differentiation. Even when translated from a vision-logic stance, a reduced notion of integration thereby results (consistent with the linear asymmetric method of relating developmental variables employed).

6. As we have seen when we raise 1 to a linear number (e.g. a rational fraction such as 1/2) the resultant value is circular (i.e. appears on the circle of unit radius). In fascinating reverse manner when we raise 1 to a circular number such as i (which already lies on the circle of unit radius) the resultant value is linear. So 1 (raised to the power of i) is .001867….. This clearly demonstrates how numbers used as dimensions are qualitatively distinct from numbers representing base quantities corresponding to two distinct logical systems.

7. Don Zavier “Bonn University inaugural lecture”

8. There are very close structural comparisons here with the nature of strings. So we can validly look at strings as the primitive instinctive nature of matter (where mathematical prime notions have a key role to play in both quantitative and qualitative terms).

9. n is raised to the linear dimension 1 and its reciprocal 1/n, to the circular dimension – 1 (with – 1 a point on the unit circle!) Thus both linear and circular aspects can be seen here to intimately affect prime behaviour. Strictly, as we move from 1,000,000 to 1,000,001, the increase in the average prime gap should be taken as the reciprocal of the average of the two numbers i.e. 1/1,000,000.5. Another way of expressing this is that the variation in the probability of a number being prime is intimately related to the reciprocal of the number in question.

The simplest prime counting function (conceptually) can be attained through adding successive terms in the harmonic series up to n (representing a given natural number) and then dividing by the resultant sum of terms.
For example the first million terms of the harmonic series add up to 14.384 (approx). (The excess here over our earlier estimate, 13.8155.. using the log n formula is explained by Euler's constant which = .5772 approx.)
Then when we divide 1,000,000 by 14.384 we get 69,522 (with actual number of primes less than 1,000,000 = 78,498). This is not yet a good estimate. However when n increases, the approximation, provided by using the harmonic series, steadily improves in relative accuracy.

Fascinatingly, the sum of the first 8 terms of the harmonic series = 2.7178571…, which is a surprisingly good estimate for e = 2.7182818…. In my own account of development the psychological equivalent of the prime counting log function (which is directly based on e) is attained with 8-dimensional understanding!

In 2002, Jeffrey Lagarias discovered an elementary problem, based on use of the harmonic series which is equivalent to the Riemann Hypothesis!

10. We have demonstrated here a qualitative psychological explanation for the meaning of the “trivial zeros”. However because of the necessary complementarity of both physical and psychological aspects this entails that a complementary qualitative physical explanation must also exist. This would suggest that there are various dimensional configurations at the quantum levels in which pure physical energy can arise (corresponding to the negative even numbers for s). Also various dimensional forms exist through which particles and anti-particles can be generated (corresponding to negative odd numbers). Such particle types could then perhaps be identified through different (rational) quantum numbers.

A striking feature with respect to zeta values for even integers of s can be mentioned here, all of which entail pi expressed to the power of s multiplied by a rational fraction. The denominators of these fractions are built from prime numbers that are arranged in a remarkably coherent order. So, if s is a power of 2, then all prime numbers - and only these - from 2 to s + 1 will be included as factors. In all other cases for s, primes from 3 to s + 1 - and again only these - will be included.

For example when s = 16, which is a power of 2, the denominator for the zeta value = 325641566250 = 2 X 3 (to power of 7) X 5(to power of 4) X 7 (to power of 2) X 11 X 13 X 17. So here all primes from 2 to 17 are used (and only these primes).
When we consider that the negative even values for s represent the integration of what is differentiated (at the positive even values), then we can perhaps appreciate the quantitative importance of the trivial zeros. Just as the non-trivial zeros are related to obtaining the exact general frequency of the primes, in this manner the trivial zeros are related (through corresponding integration in experience) to the exact particular primes from 2 to s + 1. So in this sense both sets of zeros serve a complementary function.

11. A root of the equation for which the value of the Zeta Function = 0, corresponds to s (as power or dimension) = ½ + it. Therefore because, for any value of s, the transformation formula gives the corresponding value for 1 – s, when s = ½ + it is a root, 1 – s = ½ – it must thereby also be a root. As I have demonstrated in the article, Right Hand and Left Hand sides of the transformation formula correspond to interpretations that are quantitative and qualitative with respect to each other.
Thus if we take s = ½ + it as a quantitative solution to the equation, then 1 – s = ½ – it, thereby represents a corresponding qualitative solution.

However we can equally start by taking s = ½ – it as a quantitative solution whereby 1 – s = ½ + it now represents the corresponding qualitative solution. So in fact the two roots ½ + it and ½ – it can be given quantitative and qualitative interpretations (which are indissolubly linked). So, clearly both must be equally emphasised (as the Riemann Hypothesis is essentially a statement regarding the relationship of both aspects).

As we know a complex number combines a real and an imaginary part. Thus in this context, ½ in the real part is a linear number (i.e. lies on the real number line). By contrast i, in the imaginary part, is a circular number (i.e. lies on the unit circle).

Fascinatingly what is true in quantitative terms is replicated through corresponding interpretation.

Thus the identity of the real aspect of both sides of the transformation formula as ½, corresponding to quantitative and qualitative meanings, is here translated in a rational linear manner by breaking the unit diameter line into two equal parts.
However the corresponding identity of the imaginary aspect on both sides, + it and – it, arises through circular logic (i.e. as the complementarity of opposites). So once again both in quantitative and qualitative terms, two systems of logic (linear and circular) are involved.
It is not possible therefore to properly interpret the Riemann Hypothesis (using solely quantitative notions translated through linear logic).

12. Gauss showed the chance that a number t is prime can be given by 1/log t. A simple formula exists for counting the frequency of non-trivial zeros along the critical line in the vicinity of the value t, which is 2pi/log(t/2pi) This would suggest for example an average gap of 2.27 (approx) between zeros in the region of 100! Now 2pi represents the circumference of the circle of unit radius. So if we replace the circle (i.e. the circumference 2pi) with its linear radius (1) in the formula we get 1/log t, i.e. the same as the chance that the number t is prime. So seen in this way there are close connections between both results (that are linear and circular with respect to each other).

13. The Riemann Hypothesis represents a special case of a more general class of zeta functions (L-functions) sharing similar characteristics. All these - in the General Riemann Hypothesis - have critical lines going through ½ on the real axis. Indeed if anything this should only strengthen belief that a truly fundamental explanation - such as demonstrated in this article - must underpin all.

14. Two such - apparently simple - problems that immediately come to mind in this context would be (i) the Goldbach Conjecture that every even number > 2 can be expressed as the sum of two prime numbers and (ii) the Twin Prime Hypothesis that the number of occurrences of consecutive primes such as 11 and 13 in the prime number system, has no finite limit. 15. Though Conventional Mathematics cannot directly enshrine holistic unconscious notions in its approach, indirectly they keep shining true in the accounts of the greatest mathematicians. Especially in speaking of the mystery of the primes (and the Riemann Hypothesis in particular), it is as if they know they are really speaking of a deeper meaning that transcends all mathematical speculation.

Indeed I recently read a quote of Hilbert - who became to a degree obsessed about the significance of the Riemann Hypothesis - that not alone was this the most important problem in mathematics but absolutely the most important for humanity!

And in a sense Hilbert was right! For properly appreciated, the Riemann Hypothesis amounts essentially to a mathematical expression of the fundamental dynamic nature of life itself. Indeed in mathematical terms it is expressing exactly what is implied by the famous Buddhist sutra:

“Form is not other than emptiness; emptiness is not other than form.”

So - especially in the appropriate understanding of prime numbers - let me offer this adapted sutra as a future motto for mathematicians to follow!

“Quantitative (interpretation) is not other than qualitative; qualitative (interpretation) is not other than quantitative”. Mathematical interpretation therefore ultimately needs to be radial (i.e. employing conventional and holistic aspects in complementary fashion).


Edwards, Harold M.: Riemann's Zeta Function (Paperback): Dover Publications, Dover Ed. Edition June, 2001

Sabbagh, Karl: Dr. Riemann's Zeros: Atlantic Books, September 2003

Du Sautoy, Marcus: The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (Paperback): Harper Perennial, April, 2004

Derbyshire, John: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Paperback): Plume, May, 2004

Conrey, J. Brian: The Riemann Hypothesis: Notices of the AMS, 2003

Riemann, Bernhard: On the Number of Prime Numbers less than a Given Quantity (November, 1859 Translated by David R. Wilkins).

Devlin, Keith: How Euler discovered the zeta function:

Klarreich, Erica: Prime Time: New Scientist November, 2000

Watkins, Matthew: The Riemann Hypothesis:

Collins, Peter: The Riemann Hypothesis - a new approach:

Hegel, G.W.F.: Phenomenology of Spirit (Paperback). Oxford University Press, 1979 (Translated by A.V. Miller)

Kavanaugh, K & Rodriguez, O: The Dark Night of the Soul in the Collected Works of St. John of the Cross: ICS Publications, 1991

Von Franz, Marie-Louise: Number and Time; Reflections Leading Toward a Unification of Depth Psychology and Physics: (Studies in Jungian thought) (Hardcover). Northwestern Univ Pr, 1974

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

Judge, Anthony: “Laetus in Praesans”: Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks:

A special thanks to:

Mike Wirth, who at various times accompanied and supported me on this journey:

Anthony Judge, for directing me to his valuable work on the psychosocial significance of other important mathematical concepts.

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