A New Mathematics

When one properly recognises how the qualitative aspect of understanding (pertaining to interdependent relationships) is necessarily involved in all mathematical understanding and that furthermore this aspect cannot be appropriately understood from a quantitative perspective (which entails gross reductionism) then the very nature of Mathematics is transformed in a radically different manner.


For from this new perspective, Mathematics is understood inherently in a dynamic manner (through the interaction of both quantitative and qualitative aspects).

Now every mathematical symbol, theorem, relationship etc with an established quantitative interpretation can equally be given a distinctive qualitative interpretation (of a holistic nature).

Though I have made this point repeatedly over the years, I have the strong impression that very few have grasped the significance of what it means.

For we are not talking here at all about an extension of existing Mathematics (as it is currently understood) but rather an entirely new type of Mathematics which will have many important applications that cannot yet be even imagined.


Conventional Mathematics is based on a very strong specialisation in the linear rational approach.

Though intuitive capacities may informally be recognised as also necessary for such Mathematics (especially where creative work is involved) it is given no formal role in interpretation. Quite simply therefore the very nature of intuitive understanding is grossly reduced in a misleading rational manner (which ultimately distorts its true nature).

So the qualitative aspect of Mathematics is based directly on specialisation of the intuitive nature of understanding (which indirectly can then be interpreted in a circular logical fashion).


Thus with respect to Mathematics we have two extremes 1) the specialised quantitative approach based on linear reason and 2) the specialised qualitative (contemplative) approach based directly on intuition (that indirectly can be given rational expression in a circular logical fashion).

And in between these two extremes we have the inevitable interaction of both quantitative and qualitative aspects in a dynamic manner. And this is where mathematical activity properly takes place.


Most of my own work has been with respect to the qualitative interpretation of number and I spent many years in attempting to precisely clarify the holistic qualitative meaning of all the major number types i.e. the original numbers (1 and 0), the primes, natural numbers and integers, the rational and the irrational numbers (algebraic and transcendental) the transfinite and most of all imaginary and complex numbers.


For example it strikes me as amazing that mathematicians make widespread use of imaginary numbers (without having any clear philosophical notion of what this entails).

The understanding of the deepest problems in Mathematics and elsewhere ultimately requires a strong philosophical dimension and this is especially true with respect to the fundamental nature of prime numbers. No amount of mathematical techniques, regardless of how sophisticated, can substitute for example for a proper philosophical understanding of the true nature of the Riemann Hypothesis.

Indeed I would maintain that such understanding would quickly lead one to the realisation that this proposition cannot be proven (or disproven) in conventional mathematical terms! In fact in the best sense of the word it transcends the very nature of current mathematical interpretation!


A dynamic approach to Mathematics immediately establishes a direct relationship with the physical world where - by definition - it is necessarily seen as an encoding of what in some sense already phenomenally exists (in a physical manner).

Once again the Riemann zeros lead inevitably to the view that the prime numbers do not in fact abstractly exist (which itself arises from the merely quantitative perspective) but in fact are already inherent in physical matter - literally - at its most primordial (i.e. prime) level.

When one follows this to its logical conclusion, as we probe ever closer to the original nature of matter that Physics and Mathematics become inseparable. So in this sense the ultimate nature of the physical universe is inseparable from the ultimate nature of the prime numbers.


Now admittedly there is some growing recognition of this fact with the Riemann zeros being likened to the vibration of some physical system. I would put this more emphatically. The prime numbers (in their relation to the natural numbers) in fact correspond to the vibration of an original physical system that subsequently governs the nature of physical evolution!

However Mathematics, from the dynamic perspective, not alone has a direct relationship to physical phenomena (through which they are encoded) but also psychological reality.

Just as external (objective) and internal (subjective) dynamically interact with respect to experience, this means that Mathematics - when seen from this dynamic perspective directly relates to all psychological life.

Now again speaking in the context of the Riemann zeros, this is an aspect that has been totally missed with respect to their nature.

For not alone do these zeros have immense significance in physical terms, they equally have immense importance from a psychological perspective.

The prime numbers in qualitative terms bear a close relationship with primitive instincts where basically conscious is directly confused with unconscious interpretation. Indeed in this sense we could refer accurately to the beginning of physical creation as a totally primitive state! Now attaining the highest contemplative state requires a profound mastery of such instincts!


I have little doubt that some future stage in our evolution, qualitative mastery of the nature of the Riemann zeros will enable many to reach extraordinary levels of spiritual realisation.

So the quantitative problem of understanding the nature of the primes (lying at the beginning of physical evolution) ultimately coincides with the complementary qualitative problem of fully mastering the nature of primitive instinctive behaviour (lying at the end of spiritual evolution).

Indeed both these tasks are identical as appreciation of the ultimate nature of the prime numbers in pure mystery (from a quantitative perspective) can only be fully achieved through the corresponding realisation of an ineffable spiritual state (from a qualitative perspective).

The Central Issue for Mathematics

The implications of the Riemann Hypothesis can be stated in a number of different ways.

Ultimately however it relates to the fundamental nature of the number system - which strange as it it might appear to many - is grossly misrepresented in conventional mathematical terms.

The critical basic issue relates to the relationship as between independence and interdependence. Now this applies to all relationships (including of course mathematical).


If we are to understand this relationship in a meaningful fashion then the two aspects independence and interdependence must be understood in a dynamic relative fashion.

This therefore implies that the number system itself must be understood in a dynamic interactive fashion where numbers - from one valid perspective - possees a relative independent existence; yet - from an equally valid perspective - they can be understood as - relatively - interdependent with other numbers.


Expressed in an equivalent manner, numbers thereby possess both quantitative (independent) and qualitative (interdependent) aspects; alternatively we can say that numbers possess both a cardinal and ordinal identity.


Now putting it quite simply the prevailing intellectual paradigm that governs Mathematics (as conventionally understood) is fundamentally unsuited for a coherent dynamic treatment of the number system.


As I have continually stated this paradigm is characterised by its linear logical approach.

Now it may be of value to clarify once again what this precisely means!

All phenomenal experience is governed by the interaction of fundamental poles (that operate as opposites).

At the most basic level these can be reduced to two key sets.

On the one hand we have, in any experiential context, external (objective) and internal (subjective) aspects; also we have whole (collective) and part (individual) aspects.

So for example we cannot have knowledge of a number "object" such as "2" without a corresponding internal (mental) perception of this "object". So properly speaking, what we always have in mathematical experience is the dynamic interaction of two aspects of understanding (external and internal) that are - relatively - positive and negative with respect to each other.


Likewise we cannot have knowledge of a specific number - again such as "2" - without a corresponding collective number concept (that holistically contains this number).
Thus again, mathematical experience properly entails the interaction of both part and whole notions that are positive and negative with respect to each other.


Now, like the four key directions on a compass we could illustrate these four aspects as four equidistant points on the circle of unit radius (in the complex plane). So we position external and internal as opposites on the real axis and whole and part as corresponding opposites on the imaginary axis.

When one stops for a moment to reflect on this, the very manner in which the fundamental poles (dynamically underlying all experience) interact, gives rise to a circular number system that corresponds (from a qualitative perspective) with the roots of 1 (in quantitative terms).


So the four roots of 1 are 1, - 1, i and - i (understood as separate co-ordinates). Likewise in the context of 4, the four dimensions qualitatively are understood in terms of the same set of co-ordinates (this time with opposites interpreted in a complementary manner). Thus the relationship of external to internal can be modelled in terms of + 1 and - 1 (as real complementary opposites). Likewise the relationship of whole to part can be modelled in terms of i and - i (as imaginary complementary opposites).


In conventional terms the relationship as between whole and part is understood in a highly reduced - merely quantitative - fashion. Here objects are understood as holons (part-wholes) with every whole part of a larger whole (in a quantitative manner).

However the unreduced) nature of wholes and parts requires appreciating them in true holistic fashion as archetypes of what is universal. So in this refined intuitive fashion, from one perspective, parts are qualitatively contained in the collective whole (that preserves a qualitative distinction from the parts). Equally from the reverse perspective, the whole is uniquely reflected in each part (again in a manner that preserves the qualitative distinction of each part from the whole).


So in the context of numbers, each individual number (as a quantitative part of the number system) is contained in the collective number concept (that is understood as qualitatively distinct from the specific number); likewise the collective number concept is uniquely reflected through each specific number that now attains a distinctive qualitative nature (which is not confused with the quantitative).


And this is the very meaning of the notion of the imaginary in a qualitative mathematical sense i.e. where a symbol such as number reflects a holistic qualitative meaning (as distinct from its real quantitative interpretation)!


Therefore to properly understand the number system, not alone do we need to allow for real and imaginary aspects (as quantitatively understood). Equally, we need to allow for real and imaginary interpretation (as qualitatively appreciated).

So the relatively independent aspect (as quantitative) comes from the real aspect of understanding (relating to reason). The relatively interdependent aspect (as qualitative) comes from the imaginary aspect (relating to holistic intuition).


However, interpretation corresponding to such (imaginary) intuition, emanating from the unconscious, can indirectly be expressed in a rational manner through the circular use of logic.

Now the simplest form of such circular logic relates to the complementarity of just two opposite poles i.e. external and internal, which constitutes 2-dimensional understanding.

Here, mathematical interpretation is inherently of a dynamic interactive nature.
For example with respect to the understanding of the number "2", we recognise that this number has an external (objective) status as relatively independent in understanding; equally we recognise that it has an internal (subjective) status i.e. as a mental perception that also attains a relative independence. So the actual experience of number necessarily entails both of these poles (as relatively independent) in rational terms and also a recognition of their combined interdependence (in a directly intuitive manner).


So the key point to recognise is that both a quantitative recognition of independent identity and also the qualitative recognition of interdependent identity are necessarily involved in all number experience.


In this manner we are enabled to understand number with respect to both its (quantitative) cardinal and (qualitative) ordinal nature.


However with conventional mathematical understanding, gross reductionism operates at every turn.

Because such mathematics is formally based on mere rational understanding, only the independent quantitative aspect is recognised. This, misleadingly leads to the notion of a number as having an absolute objective identity (independent of subjective interpretation). Then when focus is placed on the mental theoretical side of recognition, again this is given an absolute identity (independent of objective circumstances).

When the independent aspect is treated as absolute this leaves no proper role for the interdependent aspect of recognition (arising from the dynamic interaction of opposite poles) which is inherently of a qualitative - rather than quantitative - nature.


So quite simply no distinct recognition of the ordinal (qualitative) nature of number can exist in Conventional Mathematics. It is thereby grossly reduced in a merely quantitative manner (i.e. as the rankings of cardinal numbers).


Even as a child of 10, I could vaguely see something fundamentally wrong with the manner in which the number system is conventionally understood.

In a sense, my subsequent intellectual journey has involved an on-going attempt to rectify this problem (which I see as central to the very nature of Mathematics, Science and indeed of life generally).


Frankly I am amazed that such little apparent questioning with respect to such a fundamental issue takes place for certainly from my perspective I see clearly the basic structure of Mathematics as deeply flawed and in urgent need of the most radical revision.

In fact the nature of the number system cannot be properly addressed within Conventional Mathematics. For the fundamental paradigm that defines such interpretation, misrepresents its nature in the most fundamental manner possible!


So to sum up! Mathematical experience is of a dynamic interactive nature entailing distinctive quantitative and qualitative aspects.

The true nature therefore of the number system is likewise of a dynamic interactive nature with both quantitative and qualitative aspects.

The Riemann Zeta Function - when correctly decoded - provides a wonderful map of the relationship as between the quantitative and qualitative aspects of the number system.

However, crucially, the one dimensional value for which the Riemann Function is undefined is where s = 1.

As in qualitative terms, Conventional Mathematics is defined by its 1-dimensional rational paradigm (based on isolated poles of recognition) the Riemann Function cannot be coherently interpreted from this standpoint.

Once again when appropriately understood, the Riemann Zeta Function establishes the dynamic complementary relationship as between both quantitative and qualitative aspects of the number system (with the Riemann Hypothesis the central condition for attaining the mutual identity of both aspects).

So clearly the Function (and of course the Riemann Hypothesis) cannot be properly appreciated in conventional mathematical terms, as it defines number in a merely quantitative manner with no formal recognition whatsoever of its equally important qualitative aspect!

Interpreting Riemann Function Denominator Values (for Negative Odd Integers of s)

As we have seen the basis of the Zeta 2 Function (in obtaining prime roots of unity with complementary qualitative interpretations) is to provide the unique individual structure of each of the natural number ordinal members of this prime set (in what I refer to as unique circles of interdependence).

Now the interdependence of such prime roots is exemplified by the fact the sum of the roots of 1 = 0. In other words the circular interdependent nature of these roots (when combined) rules out cardinal features, which would imply an independent quantitative aspect!


However a reduced linear measurement of such interdependence can be given basically by ignoring the distinction as between positive and negative values (and also real and imaginary values). And as we have seen this very approach leads to both an alternative prime number theorem and Riemann Hypothesis. So the mean average for both (absolute) cos and sin values (as the measurement of roots) tends to 2/π = i/log i). And the the ratio of deviations of cos to sin average values from 2/π approaches .5!


As we have explained earlier when we express Zeta 2 in terms of Zeta 1 values we have to allow a gap of 2. Therefore s = - 1 for s with respect to Zeta 1 corresponds with s = 3 (with respect to Zeta 2).

Therefore with respect to explaining the denominator (in absolute terms) of ζ(- 1) we refer to the 3 roots of 1 (with respect to Zeta 2).

Now the sum in absolute terms here of these three roots approximates 12/π.

Then converting this to a linear expression of number as a measurement of cardinal independence we decircularise the numerical expression by multiplying by π to obtain 12.

However because this really relates to an initial expression with respect to the ordinal interdependent nature of a group of numbers (as roots) its true nature remains hidden within the context of the standard Zeta 1 Function.


So the best way of looking at the number 12 as the denominator of ζ(- 1) (in absolute terms) is - not as a single number measurement - but rather as a way of numerically measuring interdependence (among several relatively independent numbers).

Not surprisingly, denominators of the Zeta 1 Function (with respect to negative odd integers) tend to be very rich in combinatorial terms.

Now going back to the Zeta 2 Function, the sum of the 3 roots of 1 (representing the natural numbers in ordinal terms i.e. the 1st + 2nd + 3rd roots) = 0.

Then when we add these three natural numbers in cardinal terms they are factors of 12. In other words 1 + 2 + 3 = 6 is a factor of 12.

Now this is a feature that tends (with qualifications) to characterise denominators of the Zeta 1 Function (where the prime numbers are factors).

For example the denominator (absolute) of ζ(- 3) is 120 which is divisible by 5.

And 120 is likewise divisible by 1 + 2 + 3 + 4 + 5 = 15.

The denominator (absolute) of ζ(- 5) is 252 which is divisible by 7. and 252 is divisible by the sum of the first seven natural numbers (= 28).

Then the denominator of ζ(- 9) is 132 which is divisible by 11. And 132 is divisible by the sum of the first 11 natural numbers (= 66).

Finally the denominator of ζ(- 11) is 32760 which is divisible by 13. And this number is divisible by the sum of the first 13 natural numbers (= 91).

This clear pattern which applies only where prime numbers are concerned, breaks down after this (but still holds when suitable small modifications are made).


For example the denominator of ζ(- 15) is 8160 which is divisible by 17. However 8160 is not directly divisible by the sum of the first 17 natural numbers (= 153). However when 8160 is multiplied by one of its prime factors (3) it is indeed divisible!


The sum of the first five roots of 1 (in reduced linear terms) approximates 20/π. Once again we decircularise this expression by multiplying by π to get 20 (and 20 is a factor of 120).

Now 120 is especially rich in combinatorial significance (as the product of the first 5 natural numbers (which - as we have seen is also divisible by the sum of the first 5 natural numbers). Also 120 = (2^3) * 3 * 5. So as a product it includes all the prime factors from 2 to 5 (inclusive).


However there is another quite remarkable feature to this number that directly highlights the cardinal/ordinal interdependence of this number with respect to the primes.

113 (as cardinal number is the 30th prime (as ordinal number); 127 (as cardinal number is the 31st prime (as ordinal).

120 therefore lies exactly half way as between the 30th and 31st prime number.

So so with respect to the 30th it is + 1/2 and with respect to the 31st - 1/2.

Now 30 in turn (now as a cardinal number) lies exactly half way as between the 10th and 11th primes (as ordinal) i.e. 29 (as cardinal) is the 10th prime (as ordinal) and 31 (as cardinal) is the 11th prime (as ordinal).

Now converting once again 10 (now as a cardinal number) lies between - though not exactly half way - between the 4th and 5th prime numbers (as ordinal).

So taking 4 now as cardinal it lies half way between 3 and 5 (i.e. the 2nd and 3rd primes).

So 2 now as cardinal is the 1st prime.


So with respect to the prime numbers involved here there is a there is a perfect cardinal ordinal relationship that winds itself back to the 1st prime.

So 127 (as cardinal) is the 31st prime (as ordinal);

31 (as cardinal) is the 11th prime (as ordinal).

11 (as cardinal) is the 5th prime (as ordinal).

5 (as cardinal) is the 3rd prime (as ordinal)

3 (as cardinal) is 2nd prime (as ordinal)

And finally,

2 (as cardinal) is the 1st prime (as ordinal).

Meanwhile these primes are associated - in the manner demonstrated - with the numbers 120, 30, 10, 4, 2 and 1 (which are all factors of 120).

Now this pattern cannot be shown as clearly with other denominators (which seems related to the fact that not all prime numbers are included in the denominator).

For example the denominator for ζ(- 5) i.e. 252 = (2^2)*(3^2)*7. So 5 is missing as a prime factor here!


So once again the denominators of the Zeta 1 (Riemann) Function for negative odd integers of s do not correspond with normal linear interpretation (where distinct cardinal values of a merely quantitative nature apply).

Rather these numerical values pertain in varying ways to the collective relationship of both ordinal and cardinal type features.

The trivial zeros for negative even integers of s do directly apply to a purely ordinal interpretation of interdependence among numbers of a qualitative nature. However denominator values of the Zeta Function for negative odd integers of s are more complicated in nature as they combine once again the relationship of both ordinal and cardinal type features of number.

Prime Numbers and Zeta 2 Function

Yesterday we explored one way in which the Zeta 2 Function can be used to show an extremely interesting connection with the prime numbers.


So again by using the series,

y = 1 + s + s^2 + s^3 +....+ s^(n - 1) then setting s = 1, by a process of continual differentiation of y with respect to s, we can devise a simple rule to determine whether a number is prime (or not prime).


Ultimately (by differentiating down to the linear form of the original expression for y), we derive a numerical result = (n - 1)! + (n - 2)!

Again when n is prime this entails that this result will be divisible (for n > 3) by all prime factors from 2 to n inclusive (and only these primes). It will also always be divisible (again for n > 3) by the sum of all the natural numbers from 1 to n inclusive!


Now this result is especially interesting in that it provides an inversion of the usual means of testing for primeness!

Conventionally to test if a number is prime, we look for factors for that number and if no such (reduced) factors can be found, we conclude that the number is indeed prime. The problem is that with very large numbers a huge combination of potential factors exist which makes testing for primality very time consuming!


However in this case we head in the opposite direction whereby we test for primes through establishing whether the number in question is itself a factor of some larger number (that is determined by the precise procedures in question).

So in this case using the Zeta 2 expression, by establish that n is a factor of each of the derivative expressions of y (where s = 1), we conclude therefore that it is prime!


Now we have already met another example of this procedure with respect to the denominators of the Zeta 1 expression (where negative odd integral values of s are concerned).


So here we start with the value s (representing the dimensional power of the Zeta 1 Function) that is even.

Then through the Functional Equation a direct link is established with the corresponding dimensional value for the Zeta Function i.e. 1 - s.

And we then concluded that if the denominator of ζ(1 - s) is then divisible by s + 1 that s + 1 is thereby a prime number!


So for example when s = 10,

1 - s = - 9.

And the denominator (absolute value) of ζ(- 9) = 132.

Therefore if 132 is divisible by s + 1 (= 11), then s + 1 is prime.


And of course we can easily verify that that it is indeed true in this case!


However this would suggest that there are in fact deep close connections as between the Zeta 2 Function and the corresponding Zeta 1 Function (for negative values of s < 0).

Thus we cannot give a meaning in the standard cardinal sense to numerical results of the Riemann Zeta Function for values of s < 0 (indeed for values < 1).


However once we recognise that the corresponding ordinal notion of number is qualitatively distinct (relating directly to the Zeta 2 Function), then indirectly we can then interpret numerical values for the Riemann Function (s < 1) in a meaningful fashion.


To appreciate this properly we need to move to a dynamic relative notion of number. The cardinal aspect can then be identified with the standard quantitative interpretation (where number is now understood as independent in a relative manner).

The ordinal aspect relates to the corresponding relational aspect of number as interdependent (which constitutes the unrecognised qualitative aspect of interpretation).


In this context the trivial zeros of the Zeta Function have a pure ordinal meaning representing perfect interdependence (so that any distinctive cardinal aspect no longer remains).

So when 0 is used to represent the value of the Riemann Zeta Function for all negative even values of s, it relates to a pure ordinal meaning that is of a directly qualitative nature.

Now the values of the Riemann Zeta Function for all the negative odd values of s are more problematic in that distinct rational values arise. However here the denominators - rather than simply cardinal values - represent number relationships that have both ordinal and cardinal aspects.

In other words, pure cardinal meaning (as in conventional linear mathematical interpretation) and pure ordinal meaning (as in pure circular mathematical interpretation) represent two extremes.


And once again we have seen that the pure circular aspect arises for all even numbered roots (and corresponding dimensions) as perfect complementarity can be maintained as between positive and negative roots).

So when we look at the 4 roots of unity, 1 has a complement in - 1; likewise i has a complement in - i. Thus a pure circular interdependence exists with respect to these 4 roots. In corresponding qualitative terms, pure circular interdependence applies to 4 as a dimensional means of mathematical interpretation! (This for example helps to explain, with respect to Jungian Psychology, why mandalas based on the geometrical structure of the four roots of unit can operate as powerful symbols of integration)!


Now once again the positive dimensional number implies the (indirect) rational understanding of interdependence (as the complementarity of opposite poles).

The negative dimensional number implies the direct intuitive recognition of such interdependence (which is nothing in phenomenal terms). So this is the true qualitative explanation of why all the trivial zeros are associated with negative even dimensional numbers for the Zeta Function!


However we do not have perfect complementarity with odd integer roots! Likewise we do not have perfect complementarity with the odd integer dimensions. Thus the intuitive attempt to grasp the nature of holistic interdependence with respect to the interpretation of the Riemann Zeta Function with respect to the negative odd integers (as dimensional values of s) necessarily implies also a rational aspect!


So in a future entry, I will look in detail at the significance of the numerical values for the negative odd integers of s (showing precisely how they are related to the Zeta 2 Function and how they combine both ordinal and cardinal elements with respect to interpretation.

Interesting Prime Result

We have already defined Zeta 2, the finite equation (complementary with Zeta 1) as

s^1 + s^2 + s^3 + s^4 +.......s^n = 0

Then dividing by the trivial solution i.e. s = 0 we obtain


1 + s^1 + s^2 + ....... s^(n - 1) = 0.


For any given value of n, we obtain the (n - 1) non-trivial roots of 1.

For example in the simple case where n = 2, we thereby obtain the equation

1 + s = 0. So s = - 1 represents the non-trivial 2nd root of 1. As + 1 will always be a root, regardless of the value of n, we refer to it as the trivial root!

Now, if n is a prime number, the non-trivial roots of 1 will be unique in nature (and the basis for all other natural number roots).


So in this complementary sense, we can understand the prime numbers as comprising unique circles of interdependence (with respect to their ordinal natural number roots).

Therefore for example with respect to the 5 roots of 1, we have a unique circle of interdependent roots (as points on the unit circle in the complex plane) with the 4 non-trivial roots arising as the solution to the equation,

1 + s^1 + s^2 + s^3 + s^4 = 0.

Now the series involved i.e.

1 + s^1 + s^2 + s^3 + .... + s^(n - 1) can be shown to have a fascinating relationship to the primes when we take the value of s = 1.


For example, once more when n = 5, we derive the series

y = 1 + s^1 + s^2 + s^3 + s^4

Now setting s = 1, we obtain y = 1 + 1 + 1 + 1 + 1 = 5 (which is the same prime number).


Now if we the differentiate this series (with respect to s) we obtain

dy/ds = 1 + 2s + 3(s^2) + 4(s^3);

Now setting s = 1, we obtain 1 + 2 + 3 + 4 = 10 (which is divisible by 5)

Then by deriving the second derivative we obtain the new expression

2 + 6s + 12(s^2)

So setting s = 1 we obtain 2 + 6 + 12 = 20 (again divisible by 5)

Differentiating once more we obtain

6 + 24s and setting s = 1 the sum = 30 (again divisible by 5).

Basically if n is a prime number when we set s = 1 in our series (= y) and repeatedly differentiate the expression with respect to s till we obtain a linear expression in s, the resulting sum will always be divisible by n.

If n is not a prime number then this conclusion will not apply.


For example when n = 6,

y = 1 + s^1 + s^2 + s^3 + s^4 + s^5

setting s = 1 we obtain y = 1 + 1 + 1 + 1 + 1 + 1 = 6.

Then differentiating (with respect to s) we obtain

dy/ds = 1 + 2s + 3(s^2) + 4(s^3) + 5(s^4).


And setting s = 1 we obtain 1 + 2 + 3 + 4 + 5 = 15

However 15 is not divisible by 6.

If 2 is a factor of n, when we obtain the 1st differential of our original expression for y (with respect to s) and set s = 1, the resulting sum will not be divisible by n.

Then if 3 is a factor of n and we obtain the second differential of the original expression for y (with respect to s) and set s = 1, the resulting sum will not be divisible by n.

For example, 3 is also a prime factor of 6 and the second differential of our expression (where n = 6) is


2 + 6s + 12(s^2) + 20(s^3).

So setting s = 1, we obtain 2 + 6 + 12 + 20 = 40 (which is not divisible by 6).

Thus in general terms, if k is a factor of n then we obtain the (k - 1)th differential of the original expression (with respect to y, and set s = 1, the resulting sum will not be divisible by k.

So therefore it is only when n is prime that the sums with respect to all differentials of the original expression for y (until the linear form is obtained) will all be divisible by n.


There are some other highly interesting aspects to the ultimate sum for the expression (when it is reduced to linear format).


For example when n = 5 the ultimate linear form of the expression = 6 + 24s.

So setting s = 1 the resulting sum = 6 + 24 = 3! + 4!

This is a pattern that universally holds, so the ultimate sum

= (n - 2)! + (n - 1)!

Alternatively it can be expressed as n * (n - 2)!


Also when we look at the factors of the sum (where again n = 5) we can see that it is 30 = 2 * 3 * 5 (which is the product of all the prime factors up to and including 5).


Again this universally holds. So if n is prime, the ultimate sum will be an expression involving the product of all the prime factors from 2 to n inclusive (and only these primes) with many of these factors perhaps recurring!


Also the ultimate sum also will always be divisible by the sum of all the natural numbers (up to and including n).

Where n = 5, for example the sum of the natural numbers = 1 + 2 + 3 + 4 + 5 = 15.

And of course 30 (the ultimate result) is divisible by 15!

Emergence of Zeta 1 and Zeta 2 Functions

We started with the two basic approaches to the natural number system representing the Type 1 and Type 2 aspects of mathematical understanding respectively.

Once again Type 1 is defined in terms of the natural numbers defined as quantities (expressed with respect to the invariant default number dimension of 1).

So here we have

1^1, 2^1, 3^1, 4^1,........

The Type 2 is then defined in complementary terms with respect to the natural numbers - relatively - expressing qualitative dimensions (expressed with respect to the invariant default base number quantity of 1).

So here by contrast we have

1^1, 1^2, 1^3, 1^4,......

When considered in relative isolation from each other the Type 1 can be directly associated with appreciation of the (specific) quantitative aspect of number relationships, with Type 2 then associated - by contrast - with the (holistic) qualitative aspect.


However in Type 3 understanding - where both quantitative and qualitative aspects progressively interact in understanding - the quantitative is seen to have a qualitative aspect and the qualitative a quantitative aspect respectively.


Bearing comparison with Quantum Mechanics, Type 1 provides the particle aspect of number understanding and Type 2 the wave aspect (in isolation).


However with Type 3 understanding the particle likewise has a wave aspect (and the wave aspect a particle aspect) respectively.

Now the findings of Riemann with respect to his Zeta Function that a harmonic wave aspect underlines the accepted particle approach to number is really a manifestation of what properly belongs to Type 3 understanding.

However because of the lack of any explicit Type 2 interpretation, Conventional Mathematics lacks the means to properly explain the true nature of this wave system.


Now the importance of the (accepted) Zeta Function is that it can be seen as a natural extension of the Type 1 system where the various natural number terms are expressed with respect to the dimensional number s (which can be any complex number) and then summed to infinity (in accordance with conventional reduced notions of the infinite)and set = 0.

So we now have 1^s + 2^s + 3^s + 4^s + ........ = 0

Now s is conventionally expressed (when the function is written in this form) with respect to a negative value.

i.e. 1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) + ..... = 0


I refer to this as the Zeta 1 Function (as it reflects simply the Type 1 approach).

Now the truly remarkable fact about this distribution is that the only value for which it remains undefined is where s = 1.

And as the the conventional Type 1 approach is defined in terms of the default dimension of 1, this actually implies that the Riemann Zeta Function cannot truly be understood from the conventional mathematical perspective.

And the simple reason for this is that the Riemann Zeta Function (when properly decoded) establishes the dynamic relationship as between the quantitative and qualitative aspects of the number system. So this clearly cannot be achieved within a merely quantitative interpretation of number!


However in line with the Type 2 approach is a corresponding Zeta 2 Function (that is complementary with Zeta 1).

So we start with the expression for obtaining the roots of 1,

i.e. 1 - s^n = 0

However 1 - s is a factor so that 1 - s^n = (1 - s){1 + s^2 + s^3 + ... + s^(n - 1)}.

Therefore,

(1 - s){1 + s^2 + s^3 + ... + s^(n - 1)} = 0

So dividing by 1 - s, we obtain

1 + s^2 + s^3 + ... + s^(n - 1) = 0.

Now once again this implies that 1 - s = 0 is redundant as a solution.

What this means in effect is that when we take 1 as the root of unity, it is not unique.


So for example, 1 represents one of the 3 roots of 1; however it equally represents one of the 5 roots of 1. Therefore though 3 and 5 are prime numbers the common root = 1 is not unique.

So again remarkably from the Zeta 2 perspective the one value for which the Function remains undefined is for s = 1.

So with respect to Zeta 1, The Function is undefined when s = 1 (with 1 here representing a qualitative dimensional value).

Then with respect to Zeta 2, the Function remains undefined when s = 1 (with 1 here - in complementary relative terms - representing a base quantitative value)!


So just as we cannot consider quantitative meaning (in isolation from qualitative (as with Zeta 1) we likewise cannot consider qualitative meaning in isolation from quantitative (as with Zeta 2).



So in the appropriate dynamic interpretation of the Zeta Function, both Zeta 1 and Zeta 2 aspects interact. So what is hidden from the perspective - say - of Zeta 1, can be explained with respect to Zeta 2 (and vice versa).

Indeed we could accurately say in Jungian terms, that when one aspect is made conscious and thereby known the other aspect remains hidden (and unconscious). Then when the latter aspect is made known the first aspect remains hidden.


Now just a couple of more points with respect to Zeta 2!

Strictly, to obtain direct comparability (in complementary terms) with Zeta 1 we must multiply by s. So s = 0 is also a solution (which really implies the quantitative answer in an approach that is intrinsically of a qualitative nature).


Secondly the Zeta 2 is necessarily of a finite - rather - than infinite nature (which again is complementary with Zeta 1).

Therefore with this modification

s^1 + s^2 + s^3 + ... + s^(n - 1) + s^n = 0

And when n is a prime number, the solutions to this equation (as roots) will produce unique values which then likewise have unique qualitative interpretations as the structure of their corresponding dimensional values.

To put this more simply, the solutions to the Zeta 2 function, establish the qualitative uniqueness of the natural numbers (excepting 1) among the primes.

So Zeta 1 relates to the quantitative uniqueness of the primes (among the natural numbers). Zeta 2 - in complementary fashion - relates to the qualitative uniqueness of the natural numbers (among the primes).

Thus properly understood - in dynamic interactive terms - the primes and natural numbers (and natural numbers and primes) are ultimately totally interdependent in an ineffable manner!

And this is the great central mystery that underlies the number system, the proper comprehension of which will change for ever the very nature of mathematical enquiry!

Through the Looking Glass of Number

Let us sum up for a moment the key issues with relation to the number system!

The basic reason why the relationship as between the primes and the natural numbers remains so unclear is because the very nature of the number system is fundamentally misrepresented in conventional mathematical interpretation.


In keeping with the linear rational approach adopted, numbers are treated in independent terms as absolute quantities that are fixed. This accords with the cardinal nature of number. However properly speaking numbers possess also a relational capacity of interdependence with other numbers which strictly is qualitative in nature. This in turn accords with the ordinal nature of number.


In conventional Mathematical terms the qualitative ordinal aspect is misleadingly reduced in quantitative terms leading to much confusion that simply cannot be properly addressed from this perspective.

So the basic starting point for a coherent interpretation is the recognition that the number system is properly of a dynamic interactive nature, with complementary aspects that are quantitative and qualitative with respect to each other.


Thus from one perspective, numbers possess a relatively independent status in cardinal terms as number quantities; from the equally important opposite perspective numbers equally possess a relatively interdependent status in ordinal qualitative terms (through their relationship to other numbers).

Whereas the independent aspect can be represented directly in linear terms, the interdependent aspect is revealed indirectly in a circular manner.

However it is vital to understand that this latter circular aspect corresponds to a distinctive circular logical approach to understanding!

I have already demonstrated that the ordinal notion of a number is highly ambiguous and dependent on an overall holistic context. Thus the meaning of 2 in ordinal terms (i.e. 2nd) will change depending on the size of the number grouping to which it relates!. So the 2nd of two members is relatively distinct from the 2nd of – say – 20 members!

And likewise this is true for all ordinal numbers (with an unlimited amount of possible interpretations).


Now one might still wonder in what sense the meaning of cardinal numbers is relative!

In what sense for example is 2 merely relative?

Well, to appreciate this, we must remember the dynamic relationship of whole and part (and part and whole) in experience. Thus any particular number such as 2 only has meaning in the context of the number concept which is potentially unlimited in scope.

Thus in this dynamic interactive context, the number “2” and indeed any specific number has no precise meaning as it is necessarily defined in terms of a concept (that is potentially unlimited).

In other words the very nature of number cannot de defined in an absolute finite manner and so must always remain to a degree indeterminate!


So strictly speaking all cardinal number quantities are merely independent in a relative quantitative sense; equally all ordinal numbers are merely interdependent in a relative qualitative sense.


Thus our actual experience of number is inherently dynamic entailing the continual interaction of both cardinal (quantitative) and ordinal (qualitative) aspects.

And crucially both of these aspects correspond to distinctive forms of mathematical understanding.


So the former (Type 1) aspect – once again – corresponds with rational interpretation of a linear kind.

The latter (Type 2) aspect corresponds directly with intuitive appreciation (that indirectly can be expressed in a rational circular manner). And with respect to each number as a dimension (other than 1) a distinctive configuration with respect to circular understanding can be defined (which always necessarily includes linear type understanding in a refined manner).


When the number system is appropriately understood in this dynamic interactive manner (with both quantitative and qualitative aspects that are relative) the true nature – though not the mystery – of the prime numbers resolves itself revealing a looking glass reality of mirror number reflections.

So again from the – relatively – quantitative (Type 1) perspective, the primes are seen as the essential building blocks of the cardinal number system with all natural numbers in cardinal terms representing a unique combination of prime components.

However again now from the – relatively - qualitative (Type 2) perspective, each prime is defined in ordinal terms through a unique combination of natural number components.


Then in Type 3 understanding, where both aspects are harmonised in complementary fashion, the prime numbers and natural numbers now increasingly are seen as perfect mirrors of each other (in an ultimate identity that is identical). And the Riemann Hypothesis simply points to this ultimate identity (of the quantitative and qualitative aspects of the number system).


However there is no way that this two-way mirror relationship as between the primes and natural numbers (and natural numbers and the primes) can be understood in – mere – conventional (Type 1) terms.


Thus the true significance of the Riemann Hypothesis is that its proper appreciation will eventually entail a profound revolution in the very manner the number system - and indeed all Mathematics - is understood.

Nature of Number System (6)

We will briefly return to a key issue that I raised earlier, i.e. that the ordinal notion of a number has a purely relative meaning (depending on context).

In a grouping of two objects, the 2nd member might seem unambiguous in meaning, in the sense that no other interpretation of 2nd can be given in this context. And indeed this is precisely how the dimensional number 2 is defined in the circular number system.

So 1^2 in this context (in qualitative terms) corresponds with the quantitative notion of 1^(1/2) = - 1.

And if we are to preserve the true qualitative meaning of the ordinal nature of a number (associated with interdependent relationships with other members) then we must thereby use the qualitative meaning of this circular result (i.e. - 1) in interpretation of the 2nd member (of a grouping of two members).


So what does this mean?

Well! Obviously if we have two objects in a group (in cardinal terms) we must first recognise these objects as relatively independent in a quantitative linear manner(using 1-dimensional interpretation).

However the ordinal recognition of the 2nd member (in relation to the first) requires establishing an interdependence as between the two objects (in qualitative terms).


So quite literally this entails the temporary negation in understanding of the 2nd object as independent (i.e. - 1).

In other words we start with two independent units that are posited in rational conscious terms i.e. + 1 and + 1. At this stage the very notion of 2 has no strict meaning in the recognition of each separate individual unit.

However once we can then negate the separate existence of one of these units (thereby establishing interdependence with the other unit), the notions of 1st and 2nd are thereby involved. So 2 in this ordinal relationship context of interdependence implies the negation of 1 (as independent).


Now all of this might seem initially strange to anyone approaching number in conventional mathematical terms!


However this is precisely the point for as remarkable as it might seem, no coherent interpretation of the ordinal nature of number is possible within this conventional framework. Because of its merely linear basis of interpretation, qualitative notions of interdependence are necessarily reduced in a quantitative manner. And when one investigates the implications of this approach, deep confusion arises!


So whereas we can give an unambiguous interpretation to the ordinal meaning of 2 (within the context of 2 objects) this inevitably changes when we switch to 3 or more objects.


In other words the meaning of 2nd in the context of 3 objects is necessarily distinct from the meaning of 2 within the context of 2 objects!

And of course we can can extend this indefinitely so that 2 (in an ordinal sense) can be given an unlimited number of possible interpretations. We could equally express this point by saying that the 2 has an unlimited set of possible qualitative interpretations (depending on the group context in which the notion of 2nd arises)!

Now to solve this problem of unlimited ordinal interpretations (with respect to number) we need to recognise a corresponding unlimited set of interpretations associated with each number (as representing a dimension). This then enables us, through the reciprocal nature of its root, to obtain a numerical value in the circular number system.


So to solve the problem of the ordinal nature of 2 (in the context of a group of 3 objects) we obtain 1^(2/3) to obtain the 2nd root of 3 (i.e. - .5 - .866i).

And we can see that this involves both real and imaginary aspects, which likewise entails a unique configuration with respect to independence and interdependence.

So the ordinal and cardinal nature of number is inevitably mixed in a dynamic interactive manner which explains the complex nature of results that generally arise.


To sum up this section therefore, the circular system of numbers (corresponding in quantitative terms to the various roots of 1), provides in direct complementary fashion the appropriate means for the ordinal interpretation of number.


So underlying the natural number system 1, 2, 3, 4, 5,.... (when interpreted in an ordinal manner) is a unique harmonic system of complex numbers. And these numbers are derived with respect to corresponding roots of 1 (derived from these numbers in quantitative terms). However then associated with the quantitative expression of each of these numbers is a corresponding qualitative meaning, that allows for their appropriate ordinal interpretation.


The importance of the prime numbers in this context is that they provide (among the natural number set of its ordinal members) unique circles of interdependent relationships!

So once again 5 for example is a prime number. Therefore with respect to the 5 roots of 1, a unique set of complex numbers (on the circle of unit radius) can be generated which define the appropriate qualitative interrelationship of these 5 members!


Therefore we can perhaps see, that once we allow for a distinctive ordinal interpretation of number (in dynamic relative terms) that we necessarily must recognise a dual structure to the number system (representing - in quantum mechanical terms - both its particle and wave aspects respectively).


However since the ordinal nature of number is necessarily related to its cardinal nature, once we accept the dynamic relative nature of the number system (from an ordinal perspective) we must do likewise from a cardinal perspective.

This therefore entails that the cardinal number system must likewise possess two aspects i.e. a particle like system of the recognised counting numbers and an accompanying wave system (representing the interdependence of primes among the natural numbers).


So the non-trivial zeros of Riemann Zeta function are the direct counterpart (from the cardinal perspective) of the various prime numbered roots of 1 (on the ordinal side).

The clear implication of all this is that the Riemann Zeta Function in fact establishes the precise relationship as between both the cardinal and ordinal aspects of number. Or to put it in the terms that I customarily use, it establishes the precise relationship as between the quantitative and qualitative aspects of number! And the Riemann Hypothesis is the essential condition required to establish the identity of these two aspects!


And once again I must state bluntly, that as the conventional mathematical approach allows for no distinctive ordinal treatment of number, it cannot thereby properly convey either the true nature of the Riemann Zeta Function or its associated Riemann Hypothesis!

Nature of Number System (5)

It is perhaps appropriate now to briefly explain the nature of this alternative circular approach to the number system where each number representing a dimension (power or exponent) is given a distinctive qualitative interpretation that is inherently of a dynamic interactive nature.

The basic starting point is that all relationships (and corresponding experience with respect to such relationships) are conditioned by sets of fundamental polar opposites.

The two key sets (with respect to such opposites) relate to the fact that very object (viewed as external) is related to a corresponding subject (that - relatively - is of an internal nature). Likewise all wholes (in collective terms) are necessarily related to corresponding parts (that - are relatively - specific in nature).


Now the relationship between these two sets can best be visualised - like a compass with four co-ordinates - through the unit circle (drawn in the complex plane with horizontal and vertical axes) with the horizontal axis representing the real relationship as between external and internal polarities and the vertical axis the imaginary relationship as between whole and part.

Given the predominance of reductionist linear type logic in scientific understanding, it might initially seem difficult to appreciate why the relationship as between whole and part is of an imaginary nature.

However customary logic leads to the notion of objects as whole-parts (where every object as whole is part of a larger whole) which represents reduced interpretation of their nature.


However more refined appreciation of this relationship (as I have expressed before on these blogs) requires recognition of the role of the unconscious in enabling the dynamic switch as between recognition of whole and part (or alternatively part and whole) in any context to take place. And the qualitative notion of the imaginary is the indirect means of representing in rational terms the nature of the holistic unconscious!

So, as those familiar with Jungian Psychology might appreciate, when this (imaginary) contribution of the unconscious is especially refined in understanding, "objects" can increasingly serve in qualitative terms as infinite archetypal symbols. Here the parts can increasingly be reflected through the whole (in collective terms) or alternatively the whole be reflected through each part (in a unique manner) without undue reductionism.

So the true dynamic relationship as between whole and part is represented through the use of imaginary polarities (that are positive and negative with respect to each other).


However just as a compass can be used to give an ever more detailed notion of co-ordinate directions, in like manner the unit circle (with real and imaginary coordinates) can likewise be used to give ever more detailed directions (representing the relationship as between the fundamental polarities).


So in short each dimension (as number) gives rise to a special direction with respect to the relationship as between polar co-ordinates.


Therefore with higher dimensional numbers (as qualitatively understood) one becomes better enabled to appreciate, in a refined interactive manner, the inherently dynamic relationship as between the fundamental polar coordinates.


And the actual structure of such dimensions is provided through corresponding appreciation of their roots (in quantitative terms).


So for example to find out the nature of 5 as dimensional number (in qualitative terms) we obtain the 5th root of 1 (giving a special configuration with respect to real and imaginary co-ordinates).

In psychological terms this would represent a distinct configuration with of both rational and intuitive type meaning (with respect to interpretation)!


However as we know conventionally with respect to 5 we have 5 roots (and not just one).

So we now express these 5 roots of 1 in a more refined manner as the 5 roots of 1^1, 1^2, 1^3, 1^4 and 1^5 respectively.


Thus with respect to these 5 roots, a unique system of circular interdependence exists.

This means in corresponding qualitative terms that when 1, 2, 3, 4 and 5 are used (in this context of 5) to represent dimensional understanding that a corresponding unique system of circular interdependence exists.

This could be expressed alternatively by saying that where the number 5 (as a cardinal number is concerned) a unique system of circular interdependence exists in ordinal terms as between its 5 members (i.e. 1st, 2nd, 3rd, 4th and 5th).

And right here we are in a position to perhaps appreciate the alternative qualitative significance of prime numbers!


Thus, corresponding to each prime number (as a dimension) is a unique set of ordinal members bound together in a circular interdependent manner that cannot be replicated through any other number.


So in quantitative terms the prime numbers are seen as the basic building blocks of the natural number system! However now in direct complementary fashion from a qualitative perspective, each prime number is seen as composed of a unique set of ordinal members in natural number terms!


Therefore associated with the prime number 5 (in this dimensional qualitative sense) is a unique set of numbers corresponding to its 1st, 2nd 3rd, 4th and 5th roots).


So we started out by seeing prime numbers as the most independent of numbers (with no factors) in quantitative terms.

However now in a complementary manner, the prime numbers are revealed as the most interdependent of numbers, composed of a natural number set of ordinal members (that comprise a unique circle of interdependence with respect to this number).

Nature of Number System (4)

As we have seen linear approach to number (Type 1) is based on the natural number system (where each base quantity is raised to the unchanging power i.e. dimensional quality of 1).


i.e. 1^1, 2^1, 3^1, 4^1,…


The corresponding circular approach to number (Type 2) is initially based on a complementary natural number system where – in reverse – the unchanging base quantity of 1 is raised to the natural numbers as varying powers representing qualitative dimensions.

i.e. 1^1, 1^2, 1^3, 1^4,…


So right away through defining the number system with two distinctive aspects in complementary terms (quantitative and qualitative), we pave the way for its understanding in a truly dynamic interactive manner (with a merely relative validity).

Equally this amounts to recognition of the complementary nature of both cardinal and ordinal type understanding of number!

Now the key to the circular nature of the second aspect of the number system comes through associating each dimensional number (D) in qualitative terms, with its corresponding root (1/D) in a quantitative manner.


So for example to represent the qualitative nature of 2 as dimension (1^2), we find the second root of 1 i.e. 1^(1/2) in quantitative terms which is – 1. This can then be geometrically represented as a point on the circle of unit radius (in the complex plane).

Furthermore because of the unique association of each root with a corresponding dimensional number, we get rid of the confusion in Conventional Mathematics whereby a root can be given a multiple number of solutions.

So – 1 as the 2nd root of 1 = 1^(1/2). However + 1, which misleadingly is given as the alternative square root of 1, is now properly explained as the 2nd root of 1^2 (which qualitatively is of a distinct nature).

(Put another way, multiple roots in Conventional Mathematics arise due to the lack of the explicit qualitative notion of a dimension)!

However as well as the Type 1 (quantitative) interpretation, – 1 is now likewise given a complementary qualitative interpretation as the negation of (linear) rational interpretation (that is literally posited in experience in a conscious manner).

And it is through such dynamic negation that holistic intuitive meaning (of an unconscious nature) unfolds.


So in admitting a 2nd as well as 1st dimension in qualitative terms, we likewise need to explicitly admit the interaction of holistic intuitive type understanding (of a qualitative nature) with standard rational interpretation that is quantitative.

Put another way the 1st dimension (of linear rational interpretation) is based on absolute recognition of meaning (based on independent poles as reference frames).
The 2nd dimension (of holistic intuitive appreciation) is based on relative meaning (based on the dynamic interdependence of opposite reference frames).


Thus the very negation of rational meaning (in a manner analogous to the behaviour of anti-matter in physics) creates a fusion through intuitive type energy serving as the basis for understanding of holistic interdependence (with respect to relationships).

So incorporating the very notion of interdependence with respect to interpretation in a meaningful way in Mathematics, requires acceptance of both quantitative and qualitative type appreciation of its symbols.

At a minimum this requires incorporation of the 2nd with the 1st dimension with respect to understanding (where a dimension is now rightly seen in holistic qualitative terms as representing an overall manner of interpreting symbols).


So once we move to the use of more than one dimension (with respect to interpretation), mathematical reality is now necessarily viewed in dynamic interactive terms, with both quantitative and qualitative aspects.


And just as the geometrical representation of the roots of 1, entail both linear and circular aspects, likewise with respect to understanding, the qualitative interpretation of the number dimensions entails both linear and circular aspects of understanding.


Indeed linear understanding can be correctly viewed as the very important special case where qualitative is reduced to quantitative type interpretation. In other words here the holistic circular aspect – in formal terms – disappears, in a merely linear rational type interpretation of mathematical reality.


Now in direct terms, intuition is of an infinite holistic nature that is empty (in phenomenal terms), However indirectly it can be expressed in a circular rational manner that is paradoxical (in terms of the normal use of logic).

So for example + and – in normal linear logic are understood as independent opposites that are separate from each other.

However from a circular holistic perspective + and – are understood as interdependent opposites that are complementary with each other (and ultimately identical). So once again this latter understanding of interdependence, when conveyed indirectly in rational terms, is paradoxical (in terms of the linear use of reason).


Thus to convey directly the intuitive meaning of interdependence, we must negate the indirect rational constructs used to convey its nature.

Thus whereas + 2 as qualitative dimension represents the rational interpretation of the interdependence of two opposite poles in experience, – 2 conveys the direct intuitive appreciation of such interdependence (which is nothing in phenomenal terms).

As we have already seen this plays a big role in appropriate interpretation of the nature of the first of the trivial zeros (for s = – 2) in the Riemann Zeta Function, = 0. Correctly interpreted the zero here (as with all trivial zeros) refers to the qualitative rather than quantitative interpretation of 0 (as a numerical symbol).

Nature of Number System (3)

To understand number in an appropriate dynamic interactive manner, we need to move to a circular rather than strictly linear appreciation. Now to be precise, such circular likewise entails linear appreciation in a refined manner.

In other words in such understanding all numbers arise from the interactions of various poles of understanding (which comprise the use of numbers in qualitative terms as dimensions).

Now as we have seen the essence of linear understanding is that it is - by definition - based on uni-polar reference frames. So as interpretation - in any context - is based on just one pole, it is - literally - therefore 1-dimensional in nature.

This thereby excludes any genuine notion of interaction and therefore any genuine notion of interdependence (which requires at a minimum two poles).

So for example in conventional mathematical terms the objective status of a number is unaffected by our subjective interaction with that number. In other words through linear interpretation, the objective pole is isolated in an absolute independent manner.


So the inherently qualitative notion of interdependence (arising from the dynamic interaction as between distinct poles) is thereby inevitably reduced in a misleading quantitative manner.

Therefore, in failing to recognise a distinctive qualitative aspect to understanding of its symbols, Conventional Mathematics cannot properly deal with the notion of interdependence (which once again is directly of a qualitative nature).

Now the starting point for the recognition of this refined circular/linear approach (representing the dynamic interactive nature of number) is – what I have referred to as – the Type 2 number system.


Now once again the Type 1 system with respect to the natural numbers is defined as
1^1, 2^1, 3^1, 4^1,….

So in this system each natural number (as base quantity) is defined with respect to an invariant dimensional number i.e. 1 (which is - relatively - of a qualitative nature).

From another perspective we can express this by saying that the base quantities represent cardinal numbers whereas the dimensional number - by contrast - is of an ordinal qualitative nature.

So 1, in this ordinal context, relates to the 1st dimension.

Now in the Type 1 system the pure nature of addition (as positing) can be isolated.
Thus when we add numbers in the Type 1 system as quantities, the dimensional number remains unchanged (in qualitative terms).

Thus for example 2^1 + 3^1 = 5^1.


In the Type 2 system we have a complementary number system where each natural number (as power) now represents an ordinal dimensional while the base quantity remains fixed as 1.

Therefore in this system, the natural numbers are represented as,

1^1, 1^2, 1^3, 1^4,…..

Now in the Type 2 system the pure nature of multiplication can be isolated
So when we multiply numbers, the base number remains unchanged (in quantitative terms).

So 1^2 * 1^3 = 1^5.

This demonstrates the very important fact that whereas addition (with respect to the Type 1 system) is of a quantitative nature, multiplication (with respect to the Type 2 system) is - relatively - qualitative in nature.


Thus the key problem of reconciling the nature of addition and multiplication with respect to the primes points to the fact that – properly understood – they relate to the quantitative and qualitative aspects of mathematical understanding respectively.

So once again Conventional Mathematics must necessarily deal with the operation of multiplication in a reduced quantitative manner.

Strictly speaking therefore when we multiply two numbers (with base number other than 1) both a quantitative and qualitative transformation is involved.

So, 3^1 * 4^1 = 12 in quantitative terms.

However the dimensional nature of units strictly has now changed from 1 to 2 (which we can easily visualise in dimensional terms as a rectangle with sides 3 and 4 units respectively).


However in conventional terms this dimensional change is ignored.

So from this linear perspective

3^1 * 4^1 = 12^1.

Thus - quite literally - in the linear approach when carrying out quantitative calculations, the dimensional nature of numbers is ultimately reduced to 1.


However the key to unlocking the truly circular nature of the Type 2 system requires an important transformation whereby each power or exponent of 1 (as representing a dimensional number that is qualitative) is directly related in complementary terms with its quantitative root!

So for example to establish the circular nature of 1^2 we obtain the second root of
1, i.e. 1^(1/2).

So more generally the number D (as qualitative dimension) is inversely related with 1/D (as quantitative root).

Therefore when the root (1/D) is interpreted in a quantitative numerical manner, the corresponding dimension (D) is interpreted directly in a qualitative manner!
This implies that associated with each number (representing a dimension) is a distinctive qualitative means of interpretation.


The implications of this are enormous!

Once again Conventional (Type 1) Mathematics corresponds to the qualitative interpretation of 1 as dimension (whereby in effect qualitative notions are reduced to quantitative)!

However corresponding to every other number as a dimension is a distinctive means of overall qualitative interpretation (that entails a unique configuration of both quantitative and qualitative type appreciation).

Now the only dimensional number for which the Riemann Zeta Function is undefined is where D = 1.

This therefore implies - that when appropriately understood - this Function establishes at all other defined points complementary relationships as between quantitative and qualitative type meaning. Thus the mystery inherent in the primes relates directly to the key relationship as between quantitative and qualitative type meaning inherent in number symbols.

So once again, this demonstrates why it is ultimately futile trying to approach this mystery merely in conventional mathematical terms (which is solely of a 1-dimensional nature)!

Nature of Number System (2)

It is customary to view primes as the basic building blocks of the number system; however this merely reflects the quantitative bias that is inherent in the conventional mathematical approach to number.

So from this approach numbers are viewed essentially as quantities (without qualitative distinction applying to individual members). And the natural numbers are then defined in terms of the product of unique combinations of prime numbers.


However the very notion of a prime number strictly has no meaning in the absence of its individual members.

So for example when I identify 3 as a prime number this implies a grouping with a 1st 2nd and 3rd member.

This ranking of members in an ordinal fashion relates directly to the qualitative rather than the quantitative notion of number.
Also it quickly becomes apparent that no absolute distinction can apply to such ordinal rankings which have merely a relative meaning depending on the number grouping in question.

So the 2nd of 3 objects in a group implies a different meaning than the 2nd of 5. Thus the very notion of a specific number (in this ordinal sense) keeps varying depending on context.

And if the meaning of what is ordinal (in terms of qualitative distinction) is merely relative, then this implies that the cardinal notion (in terms of quantitative distinction) itself is likewise relative.

Thus the very precondition for relating the cardinal and ordinal features of number i.e. the quantitative and qualitative is that both aspects are understood in relative terms.


Now the standard quantitative notion of a prime number is necessarily based on a reduced interpretation of its qualitative nature.

For as soon as we even admit to the very notion of individual members of the prime group, we must necessarily use the natural numbers in an ordinal sense.
So once again to illustrate this point 5 as a collective prime number quantity implies 1st, 2nd, 3rd, 4th and 5th individual members (i.e. the natural numbers from 1 – 5 in an ordinal sense).

So from the quantitative perspective we start out attempting to explain how the natural numbers are derived from the prime numbers (in quantitative terms).

However when we allow for the individual qualitative distinction that is necessarily implied through the ordinal rankings of the members of the prime number group, then we realise that the natural numbers are already uniquely contained within this prime number grouping.


Therefore when we properly allow for both the cardinal and ordinal nature of number (i.e. quantitative and qualitative aspects) we are presented with two complementary perspectives with respect to the relationship as between the primes and the natural numbers.

From the standard quantitative perspective the natural numbers are collectively understood in cardinal terms through unique combinations of prime number components.

However from the (unrecognised) qualitative perspective the prime numbers are collectively understood in ordinal terms through unique combinations of natural number components.


In other words when quantitative and qualitative are properly recognised in a balanced interactive manner, the prime numbers and natural numbers can be seen in complementary terms as perfect mirrors of each other. And the pure experience of this identity is ultimately of an ineffable nature (where no distinction as between quantitative and qualitative remains).


The Riemann Hypothesis is simply a statement pointing to the nature of this identity!

When seen in this light it is not only futile trying to prove the Riemann Hypothesis in a merely quantitative manner; it is futile even trying to understand its true nature in this manner.

The price to be paid in properly incorporating the qualitative aspect is that our conception of the very nature of Mathematics is in need of radical revision. However such a revision will then open up marvellous new vistas of understanding that are presently unimaginable.