8. Summary and Conclusions (1)

8 - Summary and Conclusions (1)

The following represents both a summary of some of the main points covered in previous Chapters and also an update with respect to several issues in what is an ongoing process of reflection on the radial mathematical significance of the Riemann Hypothesis.

Reduced Nature of Conventional Mathematics

Conventional Mathematics is based on a reduced rational mode of linear understanding (relating - literally - to a one dimensional logical form of interpretation).

This can be most easily appreciated with respect to number.

When a number is raised to any dimension (other than 1), properly speaking, a qualitative - as well as quantitative - transformation is involved. Likewise when numbers are multiplied together, a similar qualitative transformation is involved.

So again for example 2² (representing square units) is qualitatively distinct from 4¹ (representing linear units).

So the expression of the value of 2² as 4 (i.e. 4¹) represents a reduced interpretation (in merely quantitative terms).

The eminent mathematician Brian Conrey has said

"the Riemann zeta function represents a connection between addition and multiplication that we don't yet understand."

The fundamental reason for this problem of understanding is due to the reduced nature of conventional mathematical interpretation.

Addition of numbers entails a mere quantitative transformation (in linear terms).

However, multiplication entails both a qualitative and quantitative transformation (entailing the interaction of two distinct logical systems).

So if we are to unravel the secrets of the Riemann Zeta Function, we must incorporate two distinct logical systems in interpretation that are linear and circular with respect to each other.

So it is to this latter system that we now turn.

Holistic Mathematics

When we look directly at the qualitative - as opposed to quantitative - nature of mathematical symbols, an entirely distinct holistic form of understanding emerges that is philosophical in nature.

As we will shortly demonstrate numbers as (quantitative) objects and numbers as (qualitative) dimensions conform to two distinct modes of logical interpretation that are linear (1) and circular (0) with respect to each other.

So if we denote the natural number system

1, 2, 3, 4,…..,

This conforms to the linear system as (quantitative) objects.

Implicitly, all of these numbers as (quantitative) objects are defined with respect to a default (qualitative) dimension of 1.

So this number system could be more fully represented as

1¹, 2¹, 3¹, 4¹,…..

When any number in this system is initially expressed with respect to another power (or dimension) its reduced quantitative value is given in terms of the 1^st dimension.

 

This number system serves as the basis for Conventional Mathematics.

However we can define an alternative number system with respect to a default  (quantitative) number object of 1, where the emphasis is now on the natural numbers as changing (qualitative) dimensions. 

This alternative system can be represented as

1¹, 1², 1³, 1⁴,…..

From a standard quantitative perspective such a number system seems somewhat trivial (as the reduced quantitative value in each case = 1).

However its significance begins to appear when we obtain the corresponding natural number roots of unity (where a fascinating circular system emerges).

For example when the dimension, n = 4, the corresponding four roots of 1 will appear geometrically as four equidistant points on the unit circle (i.e. of radius 1) in the complex plane.

Now the existence of such a system is of course already recognised from a quantitative perspective in Conventional Mathematics.

However the true significance of this system is in fact of a qualitative nature.

So corresponding - for example - to each quantitative root n^th root (that can be written as a fractional power) is a corresponding integer qualitative dimension which provides the coherent mathematical structure for definition of a unique logical mode of interpretation (with respect to this dimension).

When properly grasped the significance of this statement is truly enormous for it means that there is not just one valid logical means of interpretation with respect to mathematical symbols but potentially an infinite set.

So every number - and this can be extended from the natural numbers to all other numbers, rational, irrational, imaginary and complex etc. - serves equally as both a quantitative object (as understood in the conventional sense) and as also as a qualitative dimension (where it serves, in holistic terms, as a unique means of logical interpretation of all mathematical symbols).

Now this is of vital significance when we come to look at the Riemann Zeta Function. Here numbers generated from the Function correspond to both number categories (quantitative and qualitative). Also the ingenious functional equation provides a ready means of converting a value which has meaning in terms of one system (e.g. linear and quantitative) into a corresponding value with respect to the alternative system (circular and qualitative).

For example

ζ(2) = 1/1² + 1/2² + 1/3² + 1/4² +……   = p²/6,

corresponds to the standard quantitative linear mode of interpretation.

Through the functional equation we are equally able to calculate the value of

ζ(-1) = 1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1 +……1 + 2 + 3 + 4 +….. = - 1/12.

Now clearly this result has no meaning in terms of the standard quantitative system of interpretation. However it does have an alternative valid meaning in terms of the alternative qualitative holistic system!

So when properly understood the Riemann Zeta Function and its consequent Riemann Hypothesis entail the interaction of two distinct modes of interpretation that are linear (analytic) and circular (holistic) with respect to each other.

The recognition of the existence of Holistic Mathematics as an alternative qualitative mode of interpreting mathematical symbols goes back some 40 years to the late 60’s when I was a University student in Dublin. Its subsequent development has been in relation to the attempted provision of a more scientific means of structuring the various stages of psychological growth that unfold on the full Spectrum of potential development.

We can identify various Bands (as general groups of stages) on this Spectrum.  So in terms of this Spectrum, the specialised rational linear understanding - that typifies the conventional mathematical approach - corresponds to the stages of the Middle Band.

However more advanced stages unfold in what - I term - the Higher and Radial Bands, where both refined intuition and reason are combined in varying configurations. Quite remarkably these Bands can be precisely structured in terms of appropriate mathematical interpretation of the corresponding “higher” holistic dimensions that unfold.

And this represents but just one - though admittedly important - possible application of Holistic Mathematics.

Ultimately it has to be grasped that every mathematical symbol, relationship, formula, hypothesis (with an already established mathematical interpretation in conventional quantitative terms) has an equal coherent philosophical qualitative interpretation (from a holistic mathematical perspective). 

However because some holistic dimensions are especially important in terms of forming a deeper appreciation of the mystery contained in the Riemann Hypothesis we will explain these now in a fuller manner.

Holistic Interpretation of 2^nd Dimension (- 1)

In holistic terms, dimensions are defined in terms of the invariant unit quantity (1), just as in reverse fashion (reduced) number quantities are likewise defined in terms of the invariant unit qualitative dimension (1).

The 1^st dimension can be written as + 1. In holistic terms 1 has the connotation of (unitary) form and the 1^st dimension (which represents the holistic interpretation of the corresponding 1^st root) = + 1.

So linear (i.e. one-dimensional) understanding is based on the interpretation of phenomena of form that are consciously posited in experience.

Because here there is only one direction i.e. positive, understanding tends to be of an unambiguous nature.

Thus in Conventional Mathematics what is true, is clearly distinguished from what is not true, leading to the goal of absolute proof with respect to its various propositions.

So from a linear perspective mathematical understanding is formally expressed in a merely rational fashion based on consciously understood phenomena that can be clearly posited i.e. made objective in experience.

Now when we look at 2 dimensions (rather than 1) we now find - with respect to corresponding roots – that two now exist (which are + 1 and – 1 respectively).

Once again the corresponding holistic qualitative interpretation relates - in this context - to the 1^st and 2^nd dimensions.

We have already encountered the first as rational posited linear understanding of form (which once again characterises conventional understanding).  

However the 2^nd here (i.e. – 1) holistically relates to the negation of such form.

Psychologically this implies that the generation of intuition (which is spiritually of an empty formless nature) arises from the successful dynamic negation of such form.

Though such intuition is present implicitly in all experience - and to a marked extent with truly creative work - formally, its role is not recognised in conventional mathematical terms (based on linear reason).

So perhaps we can begin to appreciate here the reduced nature of such understanding.

In other words, from a dynamic perspective all mathematical interpretation involves a dynamic interaction of both reason and intuition. However in conventional understanding this is reduced in merely rational terms. And as we have seen this is directly due to the one-dimensional nature of such understanding (which cannot properly allow for both quantitative and qualitative meaning).

As I have said, though intuition is necessarily present (at least implicitly) in all mathematical understanding, it is quite rare for its development to explicitly take place to any marked degree.

Again in dynamic psychological terms, all interpretation entails the interaction of both conscious and unconscious elements.

And quite simply the development of the unconscious entails the successful dynamic negation of consciously posited form!

Thorough his process one is thereby enabled to move from the specific, and necessarily limited, understanding of finite phenomena of form to a truly holistic, ultimately infinite, nature of the underlying pattern to such form (which is of a spiritually empty nature).

In the past, the radical development of this purely intuitive type of integral understanding was especially identified with the religious contemplative traditions.

One striking Western expression that readily lends itself to this holistic mathematical treatment is provided by St. John of the Cross.

So the “dark nights” which the spiritual disciple must encounter on the way to pure spiritual union entail the radical negation (i.e. emptying) of attachment to all posited form.

However it is vital to grasp here that this 2^nd dimension (of dynamic negation of form) entails an entirely distinctive type of understanding (than operates with linear interpretation).

Thus whereas one-dimensional (linear) understanding is based on the clear separation of polar opposites (e.g. external and internal) two-dimensional understanding by contrast is based on the inherent complementarity (and ultimate identity) of such opposites.

So the second dimension - in dynamically entailing the negation (of what has already been posited as form - can be thereby holistically expressed as + 1 – 1 (where these opposites are complementary).

Just as in static analytic terms, + 1 – 1 = 0, likewise in dynamic holistic terms, + 1 – 1 = 0. However in this context nothingness (or emptiness) merely implies the absence of specific form, which is compatible with purely intuitive spiritual experience.

Though this might seem - at first impression - to be far removed from the Riemann Zeta Function (and Riemann Hypothesis) it is truly vital in terms of attaining any clear comprehension of what is involved.

Once again successful interpretation of the full Riemann Zeta Function entails using both (quantitative) rational and (qualitative) type understanding.

In particular whereas it is possible to provide a coherent (quantitative) interpretation of the positive values of the Zeta Function (> 1) in standard linear terms, a corresponding coherent interpretation of the negative values (< 0) requires holistic understanding of a qualitative intuitive nature (based on the complementarity of opposites).  

Holistic Interpretation of 4^th Dimension (i)

In a sense the first two dimensions are the most important as they relate to the corresponding two distinctive types of understanding (analytic and holistic) involved in all understanding.

Whereas the first represents standard linear (either/or) logic based on the clear separation of polar opposites (e.g. truth and falsehood), the second represents a more subtle – inherently paradoxical – circular (both/and) logic based on the corresponding complementarity of polar opposites. Thus to admit that prepositions can be both true and false (simultaneously) really points to the relative nature of truth (which is always arbitrarily defined in a limited context).

I will briefly illustrate this important point once more as it is central to all our discussion.

If we wish to define turns on a straight road this can be successfully achieved (in a linear fashion) once we define the direction of movement along that road (which is necessarily arbitrary).

So If I point in one direction and define that as “up” then left and right turns have an unambiguous meaning; likewise if I now move in the opposite direction “down” the road, left and right also have unambiguous meanings.

However if I now attempt a holistic explanation - simultaneously - combining both directions, then left and right have an arbitrary meaning depending on the polar reference context. 

So in this holistic context what is right is also left; and what is left is all right.

Now the great importance of this observation is that all understanding is likewise necessarily conditioned by polar opposites (that are arbitrary in nature).

Thus the absolute dualistic distinctions - that we make for example in Mathematics -at the linear (one-dimensional) level of rational understanding, are rendered paradoxical in terms of the corresponding circular (two-dimensional) level of intuitive awareness.

And these two modes of understanding are the very means by which one achieves both differentiation and integration respectively in experience.

Thus once again the attempt to reduce mathematical understanding to the merely linear form of interpretation (suited for differentiation of meaning) creates significant difficulties in terms of corresponding holistic interpretation (suited for the overall integration of meaning).

Thus there is an alarming deficit in current mathematical understanding with respect to any adequate philosophical appreciation of the concepts that it uses.

Even worse, this lack of philosophical curiosity - which indeed is a considerable barrier in terms of true appreciation of the nature of the Riemann Hypothesis - is by many practitioners considered a virtue (rather than the great deficit which it truly represents!)  

Indeed I would go considerably further. Many propositions require the interaction of both systems for satisfactory proof. In particular this is true of the Riemann Hypothesis which correctly understood in holistic terms, points to the intimate interconnection of both systems.

Thus to sum up, the 1^st and 2^nd dimensions are the most important in holistic terms as they provide the basis for both linear (1) and circular (0) interpretation.

Higher dimensions can be fruitfully understood as representing ever more subtle configurations of the two systems (which in psychological terms relates to varying dynamic interactions of conscious and unconscious respectively).    

So just as we can represent a potentially infinite set of numbers (as data information) through use of the binary digits (1 and 0), Likewise in qualitative terms we can represent a potentially infinite set of numbers (as dynamic psychological transformation processes) through the same two digits (1 and 0) now given their qualitative holistic interpretation.

Thus dimensions > 2 holistically relate to ever more refined interactions of reason and intuition (where literally rational interpretation is given an increasing number of directions) that leads to a multi-perspectival approach.

Of particular importance here is the 4^th dimension which leads to the generation of the important imaginary no. i (=√– 1).

Every mathematical symbol with a quantitative intervention in conventional linear terms, can equally be given a qualitative interpretation in holistic terms (where the new direction is seen in a circular context).

As we have seen in holistic terms – 1 as the 2^nd dimension, directly relates to formless intuitive understanding. This unconscious type understanding results from the dynamic negation of what is posited (as conscious phenomenal form) and represents the complementarity of opposites (i.e. + 1 – 1 = 0).

However the question then arises as to how give such understanding expression in terms of the conventional linear mode of experience.

We are thereby required to express what is inherently two-dimensional and circular (relating to the complementarity of opposites) in reduced one–dimensional (linear) terms (based on the corresponding separation of such opposites).

In effect this requires us therefore to obtain the square root (of the two-dimensional expression).

It thereby results in √-1. (So again this represents the reduced expression in standard linear terms of what is inherently of an intuitive unconscious nature. And such intuition is based on the interaction of two dimensions of understanding whereby what is initially posited is then dynamically negated in experience!

So in holistic terms the mathematical notion of the imaginary relates to the manner by which understanding – pertaining directly to the unconscious – is accommodated to rational conscious expression.

Essentially it relates to the manner by which intuition interacts with rational experience.

Thus once we recognise the necessity of this interaction (reason and intuition) with respect to all mathematical processes, then in holistic terms we must accept the need for a complex rational approach.

So just as there are complex quantities (in conventional mathematical terms) with real and imaginary aspects, likewise in holistic mathematical terms, we must employ a corresponding complex qualitative approach with corresponding real and imaginary aspects.

And this imaginary aspect - in qualitative terms - relates to the manner by which the unconscious (through intuition) is accommodated to conscious rational understanding (i.e. that is real).

Now again the great problem with conventional mathematical understanding is that it attempts to deal with understanding based on the interaction of conscious and unconscious, in a solely real conscious manner.

We can also use imaginary numbers to further demonstrate how numbers (as quantitative objects) and numbers (as qualitative dimensions) are linear and circular with respect to each other.

We have seen that when we raise 1 to a rational fraction such as ½ the result lies on the circle (of unit radius in the complex plane).  

Thus, though 1 and ½ are both linear nos. when considered independently of each other in this context, where a number quantity is raised to a number dimension, the result is circular in nature.

Fascinatingly, in reverse fashion, we can now raise 1 to an imaginary dimension (that is already a circular no.) the result is now a number quantity (that is linear).

Thus 1ⁱ = (e^2pi)ⁱ = e^-2p = .00186744.

This also explains quite simply the philosophical reason as to why iⁱ results in a real number quantity (that is linear).

For iⁱ  = (1)^i/4 = .207879..

All dimensions (> 2) entail varying configurations of real and imaginary aspects.

In holistic mathematical fashion this entails that a unique structure with respect to the holistic binary system exists, relating to the distinctive manner in which refined reason and intuition operate, for these dimensions.

The great significance of all this for appreciation of the Riemann Hypothesis is the fact that Riemann Zeta Function is quantitatively defined in terms of complex number dimensions (entailing number behaviour according to two distinct types of numbers). This entails that corresponding qualitative appreciation of the Function should also be complex entailing understanding that conforms to two distinct logical interpretations.

Indeed as we shall further demonstrate later, coherent interpretation of number behaviour with respect to the full Riemann Zeta Function cannot be made in the absence of two distinct logical systems.

Prime Numbers: Holistic Interpretation

The holistic mathematical approach is vital in terms of the provision of an adequate philosophical explanation of the nature of prime numbers.

Indeed, intimate connections can be made as between the fundamental nature of prime numbers and primitive instinctive behaviour in psychological terms.  

The very basis of primitive instinct is that both conscious and unconscious are entangled with each other in an undifferentiated manner. Thus the two logical systems - that underline all experience - are here directly confused with each other.

Putting it in holistic mathematical terms, an infant - for example - that is still subject to purely instinctive response, is unable to distinguish qualitative dimensions providing the holistic framework for events to occur, from the phenomenal objects involved in such events.

Thereby, qualitative dimensions of experience are directly confused with the quantitative object phenomena that arise. Because no background perspective is available (through stable dimensions) in which to properly locate objects in space and time, experience is of an immediate fleeting nature that quickly passes out of memory.

Because psychological and physical aspects are complementary, the parallel in material terms relates to processes well below the macro level.

For example it is striking how strings have traditionally been expressed in terms of one-dimensional objects (which at the same time in a sense contain the dimensions in which they are located). So in fact at the string level of reality both the linear aspect of existence which gives objects independence and the circular aspect of dimensions (which enables their interdependence) are largely undifferentiated from each other. Thus as we cannot yet separate objects from dimensions, we cannot observe their existence (which entails the existence of separate space and time). 

So in holistic mathematical terms, string reality directly represents the prime basis of material existence. In corresponding psychological terms, earliest infant experience represents the prime basis of human understanding.

The importance of all this for the understanding of prime numbers is the clear recognition that with prime numbers two logical systems are inevitably involved which ultimately are inseparable from each other.

This obviously raises a big problem for conventional mathematical interpretation, where attempted understanding of prime numbers takes place within one system only.

Thus when the linear system is solely used, undue emphasis is placed on the independence of prime numbers (as the basic building blocks of the natural numbers).

However there is an equally important circular system where the interdependence of prime numbers becomes evident.

For example when we take the prime number roots of unity, this is directly revealed through the circular geometrical arrangement of points that arises. Though this is recognised in conventional mathematical terms, understanding still takes place in linear terms. Thus the holistic significance of their corresponding prime dimensions (as unique logical systems of interpretation) is not appreciated.

Likewise when we express a prime number with respect to the negative – rather than the positive - linear dimension (i.e. – 1) again a distinct circular pattern is revealed.

So for example 7^-1 = 1/7 = .142857 (recurring). This is fact is the best known of the cyclic primes (which exhibit amazing circular properties).

The circular aspect also arises in terms of the general distribution of the primes (where a remarkable regularity with the natural numbers is evident).

So from this perspective, far from being independent of the natural numbers (as basis building blocks), the prime numbers are in fact fully interdependent in general terms with the natural number system.

Thus two opposite perspectives are true with respect to the prime numbers

(1)  they are the most independent of numbers (which corresponds directly to their linear interpretation).

(2)  They are equally the most interdependent of all numbers and ultimately indistinguishable from the natural (which corresponds to their circular interpretation).

 Indeed this point has been well made by Don Zagier.

"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 the great problem remains that conventional mathematics attempts to deal with this conundrum through use of just one logical system (linear), when - by the very nature of prime numbers - two are involved (linear and circular). 

Holistic Nature of e

As one final illustration of the holistic mathematical approach we will look at the holistic nature of e (which plays a crucial role with respect to understanding of the primes).
Now e is one of the two best known of the transcendental numbers (the other being p).
Indeed as p plays an equally important role, let us briefly look at it first.
In conventional mathematical terms, p is defined as the ratio of the circumference of a circle to its line diameter.
In corresponding holistic terms, this relates to interpretation which is based on the relationship between the two binary aspects of understanding i.e. circular and linear respectively.
So from a holistic mathematical perspective, transcendental understanding is based – neither on linear or circular understanding as considered separately – but rather on the interaction between both.
Furthermore such understanding is expressed in conscious (real) terms representing one of the most subtle forms of reason possible (that necessarily entails a considerable intuitive aspect)!
It is somewhat similar with e in holistic terms – as like all transcendental numbers – it represents this interaction as between (circular) intuition and (linear) reason. 
Indeed we can draw here on another fascinating parallel as between conventional and holistic mathematical understanding.
In conventional terms when we differentiate e^x we get the same result i.e. e^x.
Then in reverse fashion when we integrate e^x, we once again obtain e^x.
In corresponding holistic terms, differentiation and integration are fundamental to all psychological (and physical) processes of growth.
So from this holistic perspective, e represents a state of organic growth that is so refined that the processes of differentiation (with respect to object phenomena) and corresponding integration (with respect to a qualitative dimensional framework) are indistinguishable.
And this is generally identified with a pure contemplative state.
So the reason why e is so important with respect to prime numbers (as with the prime number theorem) is because it is here that the two distinctive logical systems – that underline prime number behaviour – are identical.

The prime number theorem (n/log_en) relates to the general distribution of primes in the number system.
The corresponding holistic equivalent relates to the refined contemplative state where perfect harmony is maintained as between primitive instinctive and natural (conscious) understanding. 
And both relate to a fundamental unity of the two logical systems (where neither can be separately identified).
As we will see, this simple point provides the basis for the solution to the Riemann Hypothesis.

Radial Mathematics

We have already distinguished two types of Mathematics

1) Conventional i.e. quantitative - representing a real approach based on the linear

    logical system of interpretation.

2) Holistic i.e. qualitative – representing an imaginary approach based on the

    circular system of interpretation.

However the most comprehensive system combines both the quantitative and qualitative aspects in a manner whereby both can greatly enhance understanding of each other. So the third form, which I call radial, can be defined as follows.

3) Radial i.e. both quantitative and qualitative representing a complex approach

   based on the balanced interaction of the two binary systems (linear and circular).

Indeed to place my own approach in context, I will define three variants on the radial approach. 

·        Radial 1 - which is based largely on specialisation of the quantitative aspect, that is indirectly informed with a background qualitative appreciation of symbols. This is - relatively - more analytic than holistic in orientation.

·        Radial 2 – based on the specialisation of the qualitative aspect, that is indirectly informed with a background quantitative appreciation of symbols. This - relatively - is more holistic than analytic in orientation.

·        Radial 3 – based on balanced specialisation of both aspects leading to the greatest possibilities for truly creative - yet highly productive - endeavour.

In this context, I would describe my own approach to the Riemann Zeta Function as a preliminary Radial 2 approach that is geared largely - though by no means exclusively - to appreciation of its qualitative philosophical significance (in holistic mathematical terms).
When one grasps the basic significance of the nature of a prime number (as combining the intimate identity of two distinct logical systems), then one can perhaps appreciate that solution of the Riemann Hypothesis must be conducted in a radial mathematical manner.
From a radial perspective the solution to the Riemann Hypothesis appears quite obvious as no more than a basic axiom of this system.
Indeed I would see the true significance of the Riemann Hypothesis as establishing the need for this more comprehensive approach to mathematical understanding (i.e. that is radial).