Number as Energy

We return to the important of issue of number representing energy states!

From the psychological perspective the key to such appreciation is the recognition that the actual experience of number entails both a rational (analytic) and intuitive (holistic) component.

Now it should be relatively easy to accept that intuition does indeed represent a psycho spiritual energy state. Therefore, purely intuitive awareness equally represents pure spiritual energy (in this sense).

Put another way pure holistic appreciation corresponds to the appreciation of number as a pure energy state!

Properly understood in dynamic interactive terms, number entails the continual interaction of (rational) form and (intuitive) energy which thereby represents a perpetual transformation in the nature of number.

From this perspective, the conventional mathematical interpretation of numbers as representing abstract unchanging forms (in absolute) terms, greatly misrepresents their true dynamic nature. And once again this conventional appreciation is based on the reduced notion of number as merely quantitative, whereas more correctly, it entails the dynamic interaction of both quantitative (analytic) and qualitative (holistic) aspects.

And as – in dynamic terms – the physical and psychological aspects of reality are complementary, this likewise entails that number equally in holistic terms represents a psycho physical energy state!

Now once more, this exposes the limitations of the conventional scientific viewpoint which attempts to misleadingly abstract physical processes in nature from their complementary psychological aspect.

And when one reflects calmly on the issue, it should be readily apparent that it is impossible as human beings to attain a knowledge of the physical aspect of reality that is not necessarily intertwined with the corresponding psychological mode of  its interpretation.

What is of crucial significance here is that we begin to understand number in a more comprehensive dynamic manner, that it immediately becomes apparent that by its very nature it is inseparable from the notion of energy (in all physical and psychological contexts).

It then becomes perhaps easier to understand that the very source of created evolution (and indeed the ultimate realisation of such evolution) in this more comprehensive sense resides in the very nature of number.

However we know in physical terms that energy (i.e. electromagnetic energy) represents just one of the four recognised forces. This would then lead us on to the view that number equally can be given a coherent holistic meaning in terms of the other physical forces. And in other work – mainly in my Integral Science blogs – I have shown how complementary psycho spiritual equivalents exist for each of the four physical forces.

So this would entail that eventually we will recognise how all physical motion (and corresponding psychological motivation) ultimately have their ultimate source (and goal) in the comprehensive appreciation of number.

However for the moment we will confine ourselves to the holistic aspect of number as representing energy (in both physical and psychological terms).

Now it has been recently recognised that strong connections apparently exist as between the Zeta 1 non-trivial zeros and corresponding physical energy states of a quantum chaotic nature.

However my own appreciation of this energy connection developed over many years through growing appreciation of the nature of the – as yet unrecognised – Zeta 2 zeros.

So ultimately in a more comprehensive appreciation the Riemann Zeta Function will be interpreted in dynamic interactive terms with clear recognition of both its quantitative (analytic) and qualitative (holistic) aspects.

So just as with respect to the complex plane,  “real” and “imaginary” have a recognised conventional mathematical interpretation, equally they also possess – an as yet –  unrecognised interpretation as relating to both quantitative (analytic) and qualitative (holistic) aspects respectively.

So again from this enlarged dynamic perspective, the great limitation of Conventional Mathematics is that it remains confined to merely “real” (i.e. quantitative) interpretation.

The Zeta 2 zeros arise from recognition that a distinctive holistic appreciation relates to the interdependent relationship as between the (individual) ordinal members of any number group (except 1).

So 2 for example has 1^st and 2^nd members which indirectly can be represented in a quantitative circular manner through the 2 roots of 1 as + 1 and – 1 respectively.

Now the holistic interdependence of these two members is represented through their sum = 0.

This interdependence (here of two complementary poles) represents the holistic appreciation of 2.

And just as the interaction of matter and anti-matter particles leads to a fusion in physical energy, likewise the fusion of + 1 and – 1 leads to a fusion (from both physical and psychological perspectives) in pure energy.  And  this equally applies to all numbers (as dimensions) each of which is associated with a distinctive fusion of energy. 

This new dynamic understanding of number will have dramatic implications for our understanding of the true nature of - what we know as - space and time.

In conventional terms we speak of only 4 dimensions (3 of space and 1 of time). But this understanding is rooted in an adequate - merely quantitative - approach.

In truth space and time represent the dynamic interplay of a potentially unlimited set of possible number dimensions (each representing the distinctive arrangement of its ordinal members). This means in effect that space and time possess both qualitative and quantitative features in the ceaseless interplay of their inherent number configurations.

Indeed this is precisely what enables one to recognise phenomena as possessing both quantitative and qualitative features.

We are able to locate such phenomena in space and time precisely because these underlying dimensions possess the same features in endlessly varying dynamic configurations (representing ultimately the interaction of numbers with respect to both their analytic and holistic attributes).

Now it might initially help to understand the Zeta 2 zeros as relating to the transcendent aspect of number reality.

This pertains to both the quantitative and qualitative features of number, representing mathematical dimensions. which then in turn serve as the true underlying basis for the nature of space and time (in both physical and psychological terms).  

The better known Zeta 1 zeros then in complementary fashion can be identified with the immanent aspect of number reality.

This directly relates to both the quantitative and qualitative features of number (as the ground base) as it were which now serve as the underlying basis directly for both the quantitative and qualitative aspects of all phenomena (located within space and time).

Much is made of the relevance of these zeros to energy states associated with the subatomic structure of various particles.

However it is the constant interaction of such energy with matter as form that literally leads to the continual transformation of phenomena (which at a macro structure results in the inevitable interaction in experience of both quantitative and qualitative characteristics).

If we attempt to look at phenomena from a mere quantitative perspective (as in Conventional Science), this then imposes on them an artificial rigid structure.

However if we view them from a true dynamic interactive perspective (entailing both quantitative and qualitative aspects) all phenomena at a macro level are revealed as in a process of continual transformation consistent with their sub-atomic properties (which in turn is consistent with their original number nature).

To sum up when we look at number in the conventional Type 1 manner, it appears to have a single unambiguous identity (in mere quantitative terms).

So the number 2  for example has just one unambiguous i.e. (Type 1) quantitative identity.

However when we look at number in a more authentic dynamic manner, it now appears as having twin aspects (in both Type 1 and Type 2 terms).

So the number 2 has now a Type 1 cardinal identity as 2^1 (where the focus is on its ground aspect) and a complementary Type 2 ordinal identity as 1^2 (where the focus is on its dimensional aspect).

Now like a left turn on a road, both Type 1 and Type 2 aspects can be given a quantitative interpretation (when viewed in isolation from each other). However when viewed as interdependent they are now quantitative as to qualitative (and qualitative as to quantitative) in relation to each other!

The key significance of the Zeta 1 and Zeta 2 zeros is that they represent the qualitative holistic number equivalent to the analytic interpretation of both Type 1 and Type 2 aspects.

Clearly they cannot be properly appreciated in the absence of a suitable dynamic interactive framework.

Quite simply, in this context, the conventional mathematical approach is no longer fit for purpose.


Therefore, we urgently need to make the most radical move yet in our mathematical history by adopting an inherently dynamic approach to the study of all mathematical relationships! 

Though indirectly these zeros still necessarily possess a quantitative meaning, their direct significance is in the representation of the hidden qualitative (i.e. holistic) significance of number.

And once again there are two complementary aspects involved.

So the Zeta 1 zeros represent the holistic (qualitative) counterpart of the Type 1 aspect of the number system; the Zeta 2 zeros then represent the corresponding holistic (quantitative) counterpart of the Type 2 aspect of the number system.

Once again both of these interpretations are ultimately fully complementary with each other. 

Correctly understood the Zeta 1 zeros are dynamically implied by the corresponding Zeta 2 zeros; equally the Zeta 2 zeros are dynamically implied by the corresponding Zeta 1 zeros.

I will give one interesting indication of this finding in a future blog entry!

Stunning Accuracy!

I have mentioned several times before the formula for calculating the frequency of the non-trivial zeros up to any given height on the imaginary number line.

This can be expressed for convenience as:

t/2π{(log t/2π) – 1} where t represents the required height on the imaginary scale.

This in fact represents the “circular” version of the corresponding simple "linear" formula for calculating the average number of primes up to t on the real number scale,

i.e. t/log t.

However the latter formula – though progressively improving as a predictor of the number of primes in relative percentage terms as t increases – is hopelessly inaccurate as a predictor in absolute terms.

For example the actual number of primes up to t = 1,000,000 = 78,498.

However our simple formula predicts 72,382 (which undershoots the correct result by 6,116).

Though the formula will steadily improve as a predictor in percentage terms as t increases, the absolute deviation from the correct number of primes likewise also increases (at least for values of t that can be feasibly calculated)!

However by contrast the corresponding formula for predicting the number of non-trivial zeros is stunningly accurate, not only in relative percentage, but also in absolute terms.

Due to Andrew Odlyzko extensive tables of these non-trivial zeros have now been made available.

The accuracy  of earlier calculations with the formula impressed me greatly. However I did not realise until discovering these tables, how stunningly accurate is the formula apparently as a means of predicting the absolute number of these zeros to any required height.

The formula in fact seems to under-predict results on average by 1.

So the slightly modified version that I suggest is

t/2π{(log t/2π) – 1} + 1.

For example the number of zeros up to 100 is 29.

When we use this formula we obtain 29.1273… (which rounded to the nearest integer gives the exact answer).

Then the number of zeros up to 1000 is 649.

Again using the formula we obtain 648.7412 (which rounded gives the exact answer).

Using Odlyzko’s tables the 100,000^th zeros occurs at t = 74920.827… with 99,998 zeros up to 74920.

Using the formula with t = 74920, the number of zeros =  99998.2939… (which again rounded is the exact answer).

There are 10^12 + 1 zeros up to t = 267,653,395,649.

Using the formula for this value of t, the number of zeros = 1,000,000,000,000.616 (which once again rounded to the nearest integer gives the exact answer).

There are 10^21 + 5 zeros up to t = 144,176,897,509,546,973,539.

Using the formula the number of zeros = 1,000,000,000,000,000,000,004.02568… (which, when rounded is just 1 less than the correct answer). The slight underestimate here is due to the higher than average degree of clustering of zeros as between 8 and 9 with respect to the last digit of t!

Finally to illustrate there are 10^22 + 3 zeros up to t = 1,370,919,909,931,995,308,227.

Using the formula for this value of t, the calculated number of zeros = 10,000,000,000,000,000,000,002.3376 (which when rounded is again just 1 less than the correct answer. Once again the explanation is the higher than average degree of clustering of zeros in the region of the last digit of t.

The accuracy of these predictions of the frequency of the imaginary part of non-trivial zeros is simply stunning! Indeed we could conclude that on average i.e. where typical levels of clustering are in evidence in the region of the number t in question that – when rounded – the exact answer will emerge.

This contrast therefore sharply with the corresponding prediction of primes, with the absolute deviation (from the actual number of primes) tending to increase (as t increases) to very large numbers.

Indeed this throws fascinating light on the distinction as between the primes and corresponding non-trivial zeros.

The primes are characterised at an individual level by their uniquely independent features while equally being characterised at a general level by their universal common behaviour (with respect to the natural numbers).

However this implies an inevitable clash as between the accuracy of their prediction in relative and absolute terms.

The common universal pattern tends to dominate for large values of n leading to ever more accurate predictions of their frequency (in relative percentage terms).

However the individual independent features of the primes likewise tend to accumulate - at a slower rate - with larger n leading to growing deviations from their actual frequency (in absolute number terms).

I have frequently stated that the non-trivial zeros represent the shadow system to the primes.

So the very nature of these numbers is that they represent the reconciliation of both the analytic (independent) and holistic (interdependent) aspects of the primes with respect to the natural numbers.  Expressed another way, they represent the continual smoothing out of the deviations arising from the distinctive behaviour of the individual and universal features respectively of the primes.

Now this process - as it attempts to bridge both the discrete and continuous aspect of the number system - can never be perfect. Therefore we would always expect some small deviations to remain in locally confined areas of the number system.

So we cannot therefore guarantee absolutely precise predictions from this simple formula of the frequency of non-trivial zeros. However because their frequency increases as t increases, deviations (of the individual from the general behaviour of the primes) become confined to ever smaller local areas of the number system.

We therefore would not expect the absolute deviation (where it arises) from actual results to grow in any systematic manner, as by definition these zeros continually strive to reconcile the individual (analytic) with the universal (holistic) features of the primes.  

Put yet another way - as I have repeatedly highlighted in these blog entries - both the cardinal and ordinal features of the primes are uniquely determined according to two distinct aspects of the number system (Type 1 and Type 2 respectively).

The non-trivial zeros thereby represent the reconciliation of both  Type 1 and Type 2, which occurs in the combined dynamic complementary interaction of both aspects i.e. Type 3.

This highlights yet again the futility of the conventional mathematical approach in attempting to understand prime number behaviour in mere Type 1 terms!  

Mathematics at the Crossroads

In the last blog entry, I highlighted once again how in our experience of number, two complementary interpretations are necessarily intertwined, which are quantitative (cardinal) and qualitative (ordinal) with respect to each other.

So when properly interpreted, the inherent nature of number is dynamic and interactive. Thus, from the quantitative perspective numbers have a relatively distinct independent identity (whereby they can be clearly separated from other numbers). However from the equally important qualitative perspective, numbers have an overlapping interdependent identity, whereby they can be coherently placed in an ordered relationship with each other.

In this dynamic interactive sense number behaviour is necessarily conditioned by polar opposites i.e. quantitative and qualitative of a complementary nature.

Equally, number behaviour is dynamically conditioned by external and internal polarities that are - relatively - objective and subjective with respect to each other.

In other words the external individual "object" of  an  number has no meaning in the absence of a corresponding mental perception that is - relatively - of an internal nature. Likewise the general "object" class of number as external, has no meaning in the absence of the corresponding mental concept of number as - relatively - internal.  

So once again in a dynamic interactive sense we cannot divorce the external reality of number  from corresponding mental interpretation . Number behaviour therefore has no absolute basis. (This arises from the fallacy of attempting to view such behaviour as somehow independent of interpretation)!
Therefore all number behaviour is dynamically conditioned by two key sets of polar opposites.
Likewise - by extension - every mathematical notion is conditioned by these same two sets.

Thus we can no longer attempt to view mathematical relationships in absolute terms as solely quantitative i.e. analytic in nature. 
All mathematical relationships possess a qualitative i.e. holistic meaning that is of equal importance. Therefore comprehensive mathematical understanding requires the balanced interaction of both aspects. 

Likewise we can no longer view the external (objective) aspect of mathematical relationships as somehow separate from corresponding mental interpretation of a - relatively - internal (subjective) nature. For we now have not just one default absolute interpretation but a wide range of possible relative interpretations of mathematical reality. And in a very valid sense each of these differing interpretations now corresponds to a distinctive external reality! So when we change our mental interpretation, the objective mathematical world to which it relates likewise changes!



The very manner in which we look at the prime numbers subtly changes from the dynamic perspective.

The conventional mathematical perspective is strongly built on the notion of the prime numbers as absolute number quantities with inherent objective properties (as somehow independent of our interaction with them). However this is strictly meaningless as we cannot understand numbers without such mental interaction!

Likewise it is strictly meaningless to attempt to view the prime numbers as independent of the natural numbers.

So from the dynamic perspective we no longer attempt to understand the prime numbers (as independent) but rather as the relationship of the primes to the natural numbers.

And right away from the dynamic perspective, this is seen to have two complementary directions i.e. the relationship of the primes to the natural numbers and the corresponding inverse relationship of the relationship of the natural numbers with the primes.

So from the former perspective each natural number is seen in cardinal terms, through multiplication, as the unique expression of prime factors (as building blocks).
However from the complementary perspective, each prime number is seen in ordinal terms (indirectly) through addition, as the unique expression of a succession of natural numbers (as building blocks).

Therefore the notion of any prior causation in the origin of the primes and natural numbers loses its meaning from this dynamic standpoint. Both mutually give rise to each other in an ineffable manner (with the appearance of some prior causation arising from their phenomenal unfolding in space and time). 

Likewise the (external) physical aspect of the primes and natural numbers cannot be divorced from the (internal) psychological means of their interpretation. The deeper significance of this relates to the key fact that all physical and psychological processes in nature are encoded in number (as the original secret of their inherent nature).

 So from this perspective nothing indeed is more important than number! 

Once again I mentioned in the last blog that without explicitly incorporating a qualitative (holistic) aspect to Mathematics that we have no adequate means of establishing relationships between numbers.
Because - quite misleadingly - in Conventional Mathematics, numbers are viewed in absolute terms as independent entities, subsequent attempts to establish relationships as between such numbers (which  implies interdependence) must necessarily be of a reduced - and thereby distorted - nature.

This equally of course applies to the relationship as between the primes and the natural numbers. Here the attempt to view it from a merely quantitative perspective blinds us to its fundamental nature.

Though I have repeated this analogy on countless occasions before on these blogs, it remains of the utmost importance as appreciation of its significance highlights the key overriding limitation of the present mathematical approach.

When we fix the frame of reference e.g. with one travelling North on a road and encountering a crossroads, a left turn can be given an unambiguous meaning.

Then we fix the frame of reference in terms of the opposite pole of reference i.e. with one travelling South on the same road encountering the same crossroads, again a left turn can be given an unambiguous meaning. 

In each case here we are operating within independent frames of reference!

However though the designation of a left turn is unambiguous from each independent frame considered separately, in terms of both frames as interdependent,  it is rendered paradoxical. So clearly in this simultaneous context, a left turn implies its opposite pole i.e. right turn and a right turn likewise implies its opposite (i.e. a left turn).

It is exactly analogous in terms of the attempted study of prime numbers.

When we adopt an independent frame of reference (regarding the individual behaviour of primes with respect to the natural numbers) we can unambiguously identify numerical results of a quantitative nature.
Then when we adopt the opposite independent frame of reference (of the general behaviour of the primes with respect to the natural numbers) again we can unambiguously identify numerical results of a quantitative nature.

However when we attempt to relate both frames as interdependent (with respect to both their individual and general behaviour) numerical results of a merely quantitative nature are rendered paradoxical.
So just as a left turn in this context implies its opposite pole (in a right) and a right its opposite in a left turn, here with respect to the relationship between the primes and the natural numbers, quantitative interpretation implies qualitative and likewise qualitative interpretation implies quantitative respectively.

In other words it is strictly futile attempting to deal with the two-way interdependence of the primes and the natural numbers in a merely quantitative manner!

This two-way interdependence is indeed now recognised in Conventional Mathematics, with the non-trivial zeros understood as encoding the behaviour of the primes and the primes in turn encoding the behaviour of the non-trivial zeros. 
However what is not yet appreciated is that this interdependence necessarily implies qualitative (holistic) as well as quantitative (analytic) interpretation.

Put another way the relationship between the primes and natural numbers implies both quantitative (cardinal) and qualitative (ordinal) aspects.

However because these aspects must be properly considered in a dynamic relative manner, the cardinal has a qualitative while the ordinal also has a quantitative aspect.

So whereas the analytic (quantitative) appreciation is indeed of an unambiguous linear nature (when considered within either separate frame of reference), the holistic (qualitative) appreciation is of a paradoxical circular nature (when both frames are considered as interdependent).

We can appreciate this instinctively in the context of road directions at a crossroads. However we have yet to realise that the same appreciation intimately applies to the very nature of Mathematics.

Addition and Multiplication - Cardinal v Ordinal Interpretation of Number

I keep returning to this central point.

The mystery of the relationship of addition and multiplication, with respect to the number system, relates simply to the corresponding mysterious relationship as between its cardinal and ordinal aspects.

Cardinal relates to the quantitative aspect of numbers (considered as independent entities); ordinal - by contrast - relates to the corresponding qualitative aspect (where interdependent relationships as between numbers occur).


Properly understood therefore, the very nature of number is inherently dynamic representing the continual interaction of both its quantitative (cardinal) and qualitative (ordinal) aspects.

Remarkably however for several millennia, we have sought to interpret number - and by extension all mathematical activity - in a highly reduced manner (i.e. solely with respect to its quantitative aspect).

Though this has indeed enabled remarkable progress with respect to the specialisation of one valid aspect, overall it has left us with a severely limited - and thereby gravely distorted - appreciation of the true nature of number (and indeed of the true nature of Mathematics).

As this simple truth regarding the nature of our number system is of the most fundamental nature possible, it has the capacity - I firmly believe - to ultimately lead to the greatest revolution yet in Mathematics (and by extension all the sciences).


Though the inherently dynamic nature of number can be explained without reference to the Riemann Hypothesis, proper appreciation of the Hypothesis however does require such dynamic understanding.

Indeed the Riemann Hypothesis can be validly expressed as the central condition necessary for achieving the ultimate identity of both the cardinal and ordinal notions of number. Looked at in an equivalent manner it serves as the key requirement for the full reconciliation of quantitative and qualitative meaning (with respect to all created phenomena).


The significance of the primes and natural numbers in this context is that the key relationship of quantitative to qualitative is mediated through the two-way relationship as between both sets of numbers.

From the cardinal (quantitative) perspective i.e. Type 1, the prime numbers appear unambiguously as the building blocks of the natural number system. However from the equally valid - though totally neglected ordinal (qualitative) aspect i.e. Type 2 - the natural numbers equally appear in unambiguous terms as the building blocks of each prime number!

Then in the balanced and refined experiential interaction of both cardinal and ordinal aspects i.e. Type 3, both the prime numbers and natural numbers are understood in a transparent two-way complementary manner approaching simultaneity, as perfect mirrors of each other in their common identity (which is of an absolute ineffable nature).

Indeed one could validly maintain without hyperbole that the essential secret underlying all creation relates to this original identity of both the prime and natural numbers (and natural numbers and primes) which inevitably is separated through the process of phenomenal evolution.


One could equally maintain that the very purpose of such evolution is to finally understand (in an ultimately ineffable manner) its original greatest secret.

Again, when viewed from this perspective this true nature of number could hardly be more important as it is thereby inherently encoded in all created phenomena (both physical and psychological) as the original source of their very nature.

Thus in this sense everything in creation represents but a veiled expression of the dynamic interaction of number unfolding in relative space and time (whose ultimate identity however is ineffable).

So the critical fact that we have committed ourselves for so long to a reduced and distorted interpretation of number represents an issue of the very first magnitude. As many of the present great problems facing our world ultimately have their roots in this problem they cannot be solved without a radical new appreciation of its inherent dynamic nature.


I will briefly highlight here once more the essential nature of addition and multiplication.

To do this a number must be defined with respect to both base (or ground) and dimensional aspects.

From the Type 1 perspective the base aspect of number varies while its dimensional aspect remains fixed as 1. By contrast from the Type 2 perspective, the base aspect remains fixed as 1 while the dimensional aspect varies.

So 2 from the Type 1 aspect is represented as 2^1. By contrast from the Type 2 perspective 2 is represented as 1^2.

Thus the Type 1 and Type 2 aspects of number are direct inverses of each other with respect to their base and dimensional aspects respectively.

Properly appreciated the actual experience of number keeps switching as between its Type 1 and Type 2 aspects serving as the very means by which both cardinal (quantitative) and ordinal (qualitative) interpretation arises.

So every number e.g. 2, has a dynamic dual identity whereby meaning keeps switching as between its cardinal and ordinal identities respectively both of which correspond to distinct modes of interpretation. By extension every mathematical notion with an established quantitative (analytic) can be equally given an - as yet unrecognised - qualitative (holistic) interpretation. So properly understood, comprehensive mathematical interpretation represents the balanced interaction of both types of meaning.


Once again the Type 1 aspect of number relates to the quantitative interpretation of number in cardinal terms as comprising a whole unit.

Conventional Mathematics in formal terms operates within a reduced Type 1 interpretation of number that relates solely to its cardinal (quantitative) aspect.

From the Type 1 perspective 2 = 1 + 1 (where each of its units is viewed as homogeneous and thereby identical in quantitative terms). In other words each unit is thereby viewed as without qualitative distinction!

Thus when fully represented in Type 1 terms 2^1 = 1^1 + 1^1. So the cardinal notion of 2 is expressed in terms of the addition of its two constituent units (in quantitative terms).


However as repeatedly stated in my blogs, when we define number merely in this reduced quantitative manner, we lack the means therefore of enabling a qualitative distinction of the units to take place (in ordinal terms).

In other words defining 2 = 1 + 1 provides us with no means of distinguishing the 1st from the 2nd unit in ordinal terms (as this entails a qualitative ranking of units).


Therefore the ordinal nature of the individual units of 2 relate to a distinctive Type 2 interpretation - relating directly to multiplication - where 2 = 1 * 1.

Now to appreciate this properly, we must again define number fully with respect to both base and dimensional values.

So from this perspective, 1^2 = 1^1 * 1^1.


We could also of course write 1^2 = 1^(1 + 1).

So interestingly what represents (pure) multiplication with respect to the Type 1 system i.e. 1 * 1 represents (pure) addition with respect to the Type 2 i.e. 1 + 1.

This points to the very essence of addition and multiplication, with these operations inherently relating to the critical distinction as between the cardinal (quantitative) and ordinal (qualitative) interpretation of number.

It also implies that quantitative and qualitative have a merely relative arbitrary distinction. Thus when we fix the frame of reference with the base number as quantity, the dimensional value is then qualitative (in this relative context). However within its own frame of reference, the dimensional value can likewise be given a quantitative interpretation, whereby the base number is thereby - relatively - qualitative in nature!


To properly appreciate this ordinal nature of 2, we must indirectly represent it in a circular manner through obtaining the corresponding two roots of 1.

So in ordinal terms the two (qualitatively) distinct members of 2 relate to its 1st and 2nd members respectively.

Therefore, to indirectly represent these distinctive members in a quantitative circular manner, we must obtain the roots of 1^1 and 1^2 respectively.

So the square root of 1^2 = + 1 and the square root of 1^1 = - 1 respectively.

Therefore the 1st and 2nd members of 2 can be indirectly represented in a circular manner (as equidistant points on the circle of unit radius in the complex plane) as + 1 and - 1 respectively.


It must be appreciated that understanding here is inherently of a dynamic relative nature!

Thus the very means by which we are enabled to distinguish 1st and 2nd, implicitly entails a continual process of (conscious) positing and (unconscious) negating in experience.

In other words to consciously recognise the 2nd member as distinct from the 1st, we must unconsciously negate in experience exclusive identification with the 1st member.
By this means we literally are enabled to place the two numbers in relation to each other (which is what the qualitative aspect directly implies).

The important implication of this is that the ordinal appreciation of number is necessarily of a dynamic relative nature in experience (entailing unconscious recognition).

And as the cardinal is inextricably linked with its ordinal aspect, this means likewise that cardinal appreciation of number is necessarily also of a merely relative nature.

In other words, properly understood, natural numbers have both a cardinal identity (as relatively independent) and an ordinal identity (as relatively interdependent) respectively.

Thus the accepted absolute appreciation of numbers in rational objective terms as abstract independent entities represents a grave distortion of their true nature.

And it is this distorted interpretation that has dominated Western thought now for several millennia!


With this in mind let us go back again to the fundamental relationship as between the primes and natural numbers (and the natural numbers and the primes).


From a Type 1 perspective, the primes are seen as the building blocks of the (cardinal) natural numbers (in quantitative terms).

So from this perspective the number 6 for example is represented uniquely as the product of its two prime factors i.e. 6 = 2 * 3.

When we represent this more accurately with respect to both base and dimensional aspects,

6^1 = (2*3)^1.

However from a Type 2 perspective, the natural numbers are seen as the ordinal building blocks of the primes (in a qualitative manner).

So again from this perspective the prime number 3 for example is uniquely composed in natural number ordinal terms of a 1st, 2nd and 3rd member!

Thus from this alternative perspective, the number 6 is represented uniquely as

1^6 = 1^(2*3).

So to indirectly represent this dimensional notion of 6 we obtain the 6th root of 1!


Thus the uniqueness of this 6th root of 1 (indirectly representing the ordinal nature of 6) is that it lies as a distinctive point on the circle of unit radius (in the complex plane).

Therefore we have now two complementary notions of the prime numbers as building blocks.
From the first quantitative perspective each natural number in cardinal terms is uniquely built from the product of prime factors (representing base values).

From the 2nd qualitative perspective - indirectly represented in a quantitative manner - each natural number in ordinal terms is likewise uniquely built as the product of prime factors (representing dimensional values).


However the relationship of primes to natural numbers (and natural numbers to primes) is diametrically opposite in both cases.


Therefore when we combine these approaches (in Type 3 understanding) both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system simultaneously arise with the primes and natural numbers ultimately seen as entirely interdependent with each other (in an ineffable manner).


And this is the true mysterious nature of our number system where prior causation
(in terms of either primes or natural numbers) has no strict meaning.

Once again properly understood the Riemann Zeta Function (especially through its Functional Equation) represents the dynamic relationship as between the cardinal and ordinal aspects of number. The Riemann Hypothesis then represents the condition for the ultimate identity of both aspects.

As the very nature of this identity greatly transcends conventional mathematical interpretation there is no way of course of proving (or disproving) the Hypothesis (within its axioms).

The very fact that this is not yet clearly recognised indicates how limited in fact is our current appreciation of the number system!

Filling in the Picture (3)

We now will bring the various elements together to show how the relationship as between the primes and the natural numbers (and natural numbers and primes) is one of true interdependence (thereby revealing itself in dynamic terms through a precise form of two-way complementarity).


Once again from a Type 1 (linear) perspective, the primes are viewed as the basic building blocks for the natural number system in a merely quantitative manner.

Thus from this perspective, each natural number (in cardinal terms) represents a unique combination of prime factors.

However we have indicated many times the key problem with this approach whereby uniqueness in - solely - quantitative terms, strictly rules out any distinctions of a qualitative (ordinal) nature.

Again, a cardinal prime is defined by its collective whole nature (in quantitative terms).
So the prime number 3 is thereby defined uniquely in terms of unit parts that are completely homogeneous in nature (i.e. lacking qualitative distinction). So 3 = 1 + 1 + 1.


Therefore the cardinal approach to this key relationship - of the primes to the natural numbers - leaves us with no means of making ordinal distinctions of a qualitative nature. And without making such distinctions we cannot relate numbers and thereby achieve order with respect to the number system!

Thus in Type 2 terms, the natural numbers are viewed - in reverse fashion - as the building blocks of each prime number.
Therefore from this counter perspective, each prime number (in ordinal terms) represents a unique combination of natural number members.

So the prime number 3 is now viewed uniquely in terms of its natural number members i.e. 1st, 2nd and 3rd in ordinal terms.

Indirectly, each of these members can be expressed in quantitative terms through the corresponding 3 roots of 1.
Then their true qualitative significance (as interdependent) is revealed through combining (in dynamic terms) these - relatively - separate members = 0. So the significance of 0 in this context reveals the true qualitative meaning of the number 3 (which - literally - is nothing in quantitative terms).


From the Type 2 (circular) perspective, all prime numbers are defined (except 1) by a unique set of natural number members in ordinal terms (indirectly expressed in a circular quantitative manner by the roots of the prime number).

And the corresponding sum of these unique set of roots, representing the true holistic meaning of the prime number (as the dynamic interdependent notion of the number in question) = 0.


We now come full circle!

We started in Type 1 terms by defining the natural numbers as representing unique combinations of prime number factors (in quantitative terms).

Fro example the natural number 30 = 2 * 3 * 5 (from this perspective).

Now strictly we should express this Type 1 representation as 30 ^ 1 = (2 * 3 * 5) ^ 1


However we can equally express 30 as representing a unique combination of prime number factors (in a qualitative manner)

So in Type 2 terms, 2 * 3 * 5 = 30 is represented as 1^(2 * 3 * 5).

This means in effect that 30 is equally defined in a qualitative manner with respect to its 30 corresponding roots!

And as we know when we obtain the n roots of 1 (where n is any natural number ≠ 1) the sum of roots = 0.

Therefore when we interpret this in Type 2 (circular) terms, the prime numbers equally represent the qualitative building blocks of the number system (where each prime number is uniquely defined by its natural number members).


So once again, a matching qualitative (i.e. ordinal) interpretation exists both (internally) within each prime and and (externally) for the number system as a whole for the recognised quantitative interpretation.

Therefore let us now summarise the full picture.

1. In linear Type 1 (quantitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors.

2. Again in linear Type 1 (quantitative) terms, each prime number is defined (internally) in terms of a unique combination of homogeneous units i.e. 1 + 1 + 1 +... Unique in this quantitative sense simply means that each unit = 1.

3. In circular Type 2 (qualitative) terms, each prime number is defined (internally) in terms of a unique combination of ordinal number members i.e. 1st, 2nd, 3rd,... (indirectly represented in quantitative terms as the corresponding roots of 1). Since 1 is common to all roots, strictly uniqueness in this context implies all roots ≠ 1st. So even here, we have in the definition of qualitative uniqueness, complementarity with the quantitative (internal) definition!

4. Again in circular Type 2 (qualitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors (based on the corresponding ordinal number of roots).


So - when properly understood - the ultimate relationship of the primes to the natural numbers (and the natural numbers to the primes) is one of perfect complementarity.

In other words the primes and natural numbers (and natural numbers and primes) are fully interdependent with each other externally and internally (in both quantitative and qualitative terms) totally mirroring each other in an ultimate identity that is ineffable in nature. In other words in this ultimate mutual embrace, their objective (external) nature cannot be divorced from their corresponding (internal) interpretation! likewise quantitative cannot be divorced from corresponding qualitative identity!


Thus the mystery of the primes in relation to the natural numbers (and the natural numbers in relation to the primes) essentially relates to the manner in which both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system interact.

Put another way the very manner in which the quantitative (analytic) and qualitative (holistic) aspects of the number system are mediated is through this key relationship of the primes to the natural numbers (and natural numbers to the primes).

So in dynamic terms, the qualitative represents the - initially unrecognised - shadow of the corresponding quantitative aspect; the quantitative likewise represents the - initially unrecognised - shadow of the corresponding qualitative aspect.

The zeta non-trivial zeros (both Zeta 1 and Zeta 2) represent these shadow number systems, which mediate the ultimate perfect relationship of quantitative and qualitative aspects.

In the case of the Zeta 2 zeros, each natural number is given - through the unique interdependence of its prime individual members - a corresponding qualitative dynamic meaning. So the perfect relationship here as between quantitative and qualitative aspects is established internally ultimately with respect to each natural number.

In the case of the (recognised) Zeta 1, the zeros represent the direct shadow correspondent of the prime numbers which in their totality - which can only be approximated in finite terms - perfectly mediate the dynamic interaction of quantitative and qualitative aspects for the natural number system as a whole.

Clearly once again, it is futile trying to appreciate the ultimate nature of the number system in merely quantitative external terms as in Conventional Mathematics. In dynamic terms the number system has both external and internal interpretations (with both quantitative and qualitative aspects) as I have sought to demonstrate in this blog entry.

The significance of the Riemann Zeta Function cannot be properly appreciated in this limited conventional manner. Likewise the Riemann Hypothesis, which relates to this central relationship as between quantitative and qualitative aspects (externally and internally), cannot of course be proved within its limited axiomatic system (which gives no formal recognition whatsoever to these key polar distinctions).

Filling in the Picture (2)

In yesterday’s blog entry, I emphasised how all experience - including mathematical - is conditioned by two fundamental polarity sets that are (i) external and internal and (ii) quantitative and qualitative with respect to each other.


These polarities in fact are the basis for the alternative Type 2 aspect of the ordinal number system, where numbers are represented as equidistant points on the circle of unit radius (drawn in the complex plane).


So the first set of external and internal poles - initially with respect to conscious understanding - are represented on the (horizontal) real axis as + 1 and – 1 respectively.


Once again the cardinal (Type 1) aspect of the number system treats a number e.g. “2”, as a collective unit in quantitative terms. So if we attempt to sub-divide it as the sum of number parts, we must represent these in a homogenous manner (i.e. without qualitative distinction) as 1 + 1.


The ordinal (Type 2 aspect) then treats the number 2 in complementary fashion with respect to its distinctive individual components in a qualitative manner.

So from this qualitative perspective, 2 is composed of a 1st and 2nd member that are uniquely distinct in a relative manner. Now we can represent these 1st and 2nd members, indirectly in a quantitative manner, through obtaining the corresponding two roots of 1 i.e. + 1 and – 1 respectively.


However the true qualitative recognition of these two polarities as interdependent (which defines the qualitative aspect) comes from simultaneously combining both directions.

So this would be represented as + 1 – 1 (in indirect quantitative terms) = 0 (from a direct qualitative perspective).

Now the recognition of 0 in this context is directly of an intuitive rather than rational nature representing the Type 2 appreciation of the number “2”.

Thus in summary the Type 1 aspect is - directly - of a (linear) rational nature geared to the quantitative interpretation of “2”.

The Type 2 aspect by contrast is - indirectly - of a (circular) rational nature, i.e. paradoxical, culminating in direct intuitive recognition. This provides the corresponding qualitative interpretation of “2”.


Now with respect to our experiential understanding, both of these aspects necessarily interact in continual fashion.

However the qualitative aspect is then completely edited out in terms of accepted formal mathematical interpretation.


Therefore though the true understanding of number is thereby inherently dynamic in nature, Conventional Mathematics is built on a significantly reduced - and thereby greatly distorted - interpretation (i.e. that recognises merely the quantitative aspect).

Put another way the Type 2 aspect is inherently geared to appreciation of the manner in which the fundamental polarities (underlining all experience) interact.

Therefore associated with each number from this perspective is a corresponding set of individual ordinal members (as distinctive directions with respect to the two basic sets of polar co-ordinates).


So for example the number “4” is associated with four ordinal members represented by the four equidistant points on the unit circle. So once again along the real axis we again have + 1 and – 1 and now two additional points along the vertical axis i.e. + i and – i respectively.


Now + 1 and – 1 along the real axis are identified with understanding of a direct conscious nature. + 1 literally relates to the unitary direction of experience whereby phenomena are posited in conscious manner (which in our present scientific framework are thereby identified as “real”).

– 1 then relates to the (unconscious) negation of such phenomena which serves as the very means by which we are thereby enabled to switch polar direction (e.g. from external to internal) in experience.


When such dynamic switching takes place in a flexible manner, significant amounts of intuitive energy are generated (through the complementary interaction of both poles) which thereby enables understanding of a creative nature.


However when little dynamic switching occurs, understanding becomes ever more rigid in nature whereby existing assumptions are constantly re-affirmed.


This is why I would expect considerable resistance to the views that I am expressing.

Once again conventional mathematical interpretation is strongly 1-dimensional in formal terms. This means therefore that the conscious rational direction (+ 1) is solely recognised. Though implicitly, a degree of unconscious intuition informally takes place, it operates solely within the accepted paradigm.


However there will always be some - not necessarily professional mathematicians – operating at the margins, that perhaps suspect a fundamental problem may indeed exist with present Mathematics. And it this audience that I am mainly addressing!


Just as the notion of “real” can be given - according to the Type 2 aspect - a holistic (qualitative) mathematical meaning (i.e. as corresponding to linear rational interpretation) the notion of “imaginary” can be given a vitally important holistic interpretation.


As we know important national and religious symbols can convey a holistic significance whereby they embody an unconscious desire for meaning.

If for example we compare national flags there is little to distinguish one from another (from a mere rational perspective). However when we accept that a flag can embody deep notions of identity, we can then perhaps recognise that the significance is more of an unconscious than conscious origin.

If we generalise, then all local symbols of a conscious kind necessarily also embody projections of an unconscious universal nature.


Now in a precise mathematical manner, the very notion of “imaginary” relates to the indirect linear rational attempt to convey meaning that is properly of an (unconscious) holistic nature.

And as the Type 2 aspect of the number system is indeed properly of such a holistic nature, one could accurately express this in qualitative terms as the “imaginary” aspect of the number system.


In quantitative (Type 1) terms i is expressed as the square root of – 1.

It is similar in qualitative (Type 2) terms. – 1 here represents the negation of (conscious) understanding. A dynamic fusion thereby results through interaction with the existing positive direction leading to the generation of spiritual intuitive energy (that is inherently 2-dimensional in nature).

The resulting attempt to explain such holistic understanding indirectly (in a linear fashion) entails the notion of a square root (in a qualitative manner).


So again Type 2 represents the “imaginary” counterpart to the recognised Type 1 aspect of mathematical understanding.

Therefore we can perhaps now appreciate that just as we can define both real and imaginary aspect to numbers in quantitative terms, equally we can define real and imaginary aspects in a qualitative manner.


So a comprehensive paradigm for Mathematics is necessarily of a complex rational nature (with real and imaginary aspects).

The great limitation of Conventional Mathematics is that it is solely interpreted in a real rational manner!

So these two imaginary directions (+ i and – i) represent – in Jungian terms – the archetypal nature of number (as embodying a holistic qualitative element) now indirectly expressed in a rational manner. And once again this precisely defines the nature of the Type 2 aspect.

So we can only posit the qualitative aspect of number in an indirect conscious manner, as the true nature of holistic interdependence is unconscious in origin.

And as this qualitative nature continually alternates between the whole (in relation to the parts) and the parts (in relation to the whole) negating as well as positing with respect to number must continually take place.


Therefore the Type 2 nature of “4” relates to this more refined interaction as between both its real and imaginary co-ordinates (that are positive and negative respectively).


In principle any number “n” can be indirectly defined in Type 2 terms with respect to its n individual roots (the full combination of which represents its true interdependent appreciation)

And the sum of the n roots of 1 (except 1) = 0. So this circular interdependence of the all the ordinal members of n represents the Type 2 interpretation of number (in its pure qualitative appreciation).


In physical terms as nature becomes ever more dynamic at sub-atomic levels, an increasing number of directions (i.e. dimensions) is involved with respect to polar interactions.

Likewise in psycho spiritual terms as contemplation becomes ever more refined, appreciation with respect to a growing multiple of directions can be explicitly brought into conscious awareness.


In fact what we are talking about here - in Type 2 terms - is the direct appreciation of each number as representing a pure energy state.

And - as always - we have complementary directions in both physical and psychological terms.


Thus in Type 2 terms, every number - in principle - has a direct physical (or more correctly psychophysical) relevance as a pure energy state.

Likewise every number has a direct psycho spiritual relevance as a pure (intuitive) energy state.


Therefore in the dynamics of experience, intuition and reason implicitly interact enabling one to appreciate (to some degree) both cardinal and ordinal aspects with respect to number .

However because explicitly our subsequent formal interpretation is merely rational, we misleadingly identify the ordinal with the cardinal aspect.


Therefore we think that 1 (as cardinal) implies 1st (as ordinal), 2 (as cardinal) 2nd (as ordinal), 3 (as cardinal) 3rd (as ordinal) and so on! In fact the very process enabling us to make these connections entails the whole mystery of how the primes are related to the natural numbers (and the natural numbers to the primes) which entails two sets of zeta zeros (as complementary shadow systems).


So in this important respect our understanding of number still remains greatly confused.

Filling in the Picture (1)

As I have repeated often in these blogs, the true nature of number (as indeed all mathematical activity) is of an inherently dynamic interactive nature. Unfortunately Conventional Mathematics provides but a reduced and thereby distorted interpretation of number.


Firstly number inherently has both external (objective) and internal (subjective) aspects.

We cannot externally envisage a physical number “object” in the absence of the corresponding psychological mental perception of the number. So properly understood these two aspects necessarily continually interact in a relative manner with respect to experience.

Put another way, experience necessarily entails the interaction of two aspects of number that are physical and psychological with respect to each other.

Once again Conventional Mathematics gives but a reduced interpretation of this interaction.

Now a professional mathematician if sufficiently pressed might eventually concede that we cannot form knowledge of the number “object” in the absence of its corresponding mental perception. However the necessary interaction thereby involved is then completely ignored with interpretation taking place in a misleading absolute fashion. Thus the erroneous notion of numbers as abstract objective entities still dominates conventional thinking.

Because of its linear 1-dimensional nature Conventional Mathematics can only handle such dynamic interactions in a reduced manner whereby the subjective mental aspect is identified with the objective (which is predominant) or alternatively the objective aspect wirh its mental perception. In either case we then get an absolute rather than - more correctly - a truly relative interpretation of the nature of number.

Therefore to repeat once more the true dynamic nature of number necessarily entails twin interacting elements that are external (physical) and internal (psychological) with respect to each other.

This means in effect that once we identify for example – as recently with the (Type 1) non-trivial zeros – their physical similarity to certain quantum chaotic processes, this automatically entails that they must necessarily also have an equally important significance in complementary psychological terms.

However because Conventional Mathematics is completely lacking in dynamic interpretation it thereby places no emphasis on such complementary type relationships.
Therefore the extremely important psycho spiritual significance of the non-trivial zeros still remains completely unrecognised by the conventional mathematical community!

The other key distinction is with respect to the quantitative and qualitative aspects of number!

If we take the number “3” to illustrate we cannot experientially identify the cardinal nature of this number without implicitly recognising that it necessarily contains a 1st, a 2nd and 3rd member in ordinal terms. Thus the quantitative recognition of “3” implies corresponding ordinal recognition of its 1st, 2nd and 3rd members in a corresponding qualitative manner. And in reverse manner we cannot form knowledge of the ordinal members of a group without implicitly recognising its cardinal (quantitative) identity.


So all mathematical experience is fundamentally conditioned by the dynamic interaction as between opposite sets of polarities.

Chief among these are the external/internal that operate in a horizontal manner and the quantitative/qualitative that operates in a corresponding vertical manner.


In fact it may help to initially recognise the relationship as between them as like a compass with the four directions East and West along the horizontal axis and North and South along the vertical axis respectively.


However we can give a firmer mathematical rationale to these locations (in terms of the Type 2 aspect of the number system) by recognising these four equidistant points as corresponding to the four roots of 1. So the external and internal polarities are – relatively – complementary in a real manner (with directions that are + 1 and – 1 with respect to each other).

The qualitative therefore has likewise two directions that are + i and – i with respect to each other.

Quantitative and qualitative polarities are thereby real and imaginary with respect to each other.


What this means in effect is that the individual members of a number set have a unique qualitative meaning i.e. in their ordinal identity. However the overall set – which we initially identified in quantitative terms as cardinal likewise has an ordinal identity when related to other numbers.

So for example 2 and 3 are prime numbers (in a cardinal manner). However they equally enjoy a qualitative ordinal identity as the 1st and 2nd prime numbers respectively.


So the individual members of a cardinal prime number such as 3 enjoy a unique qualitative identity (in terms of its 1st, 2nd and 3rd members).

Thus the cardinal prime is ordinally defined in terms of its natural number members (in a corresponding qualitative manner).


However the same prime number 3 enjoys a unique collective qualitative identity as the 2nd prime in the natural number system. And all natural numbers represent a unique combination of these prime number factors.


Thus when properly understood, the all important relationship as between the primes and the natural numbers represents the fundamental manner by which their quantitative and qualitative aspects are related.

And as always – in dynamic interactive terms – there are two complementary ways in which this relationship can be understood:

1) whereby the natural number system as a whole is collectively defined through unique combinations of its prime number members.

2) whereby each prime number is uniquely defined through a collection of its natural number members.


Putting it bluntly therefore the conventional mathematical attempt to define the relationship as between the primes and the natural numbers misses the crucial point that this relationship entails a dynamic two-way complementarity as between its quantitative and qualitative aspects.


From one perspective, it is quite extraordinary how we have remained blind to this key relationship for so long!

We have tried to convince ourselves that Mathematics is solely concerned with the quantitative aspect of number. However strictly speaking we cannot even begin to identify the quantitative aspect without implicit recognition of its corresponding qualitative aspect.

Thus Mathematics is properly – in dynamic terms – as for example here with number, concerned with the relationship as between its quantitative and qualitative aspects.

Likewise – in relation to the other polarity set - we cannot form an objective knowledge of number (as external) without a corresponding mental interpretation (that is - relatively - internal).


Thus again in dynamic terms, Mathematics is properly about the relationship of objective type results to the corresponding interpretations (through which they are viewed). And from this perspective there is not just one absolute type interpretation that is valid but potentially an unlimited number (each enjoying a partial relative validity).

Once again the relationship as between the quantitative (analytic) and qualitative (holistic) aspects of number fundamentally points to the corresponding relationship as between the primes and natural numbers (and natural numbers and the primes).


And mediating this relationship are two important sets of zeta zeros (corresponding to the Type 1 and Type 2 number systems respectively).

As I have stated on a number of occasions, these zeros essentially can be viewed as the shadow of our one-sided quantitative view of number (i.e. where the qualitative aspect is directly confused with its quantitative expression).


So we can fruitfully view both sets of zeros as a means of giving two distinctive expressions (in a related complementary fashion) to the long unrecognised qualitative aspect of the number system.