INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber



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Peter CollinsPeter Collins is from Ireland. He retired recently from lecturing in Economics at the Dublin Institute of Technology. Over the past 50 years he has become increasingly convinced that a truly seismic shift in understanding with respect to Mathematics and its related sciences is now urgently required in our culture. In this context, these present articles convey a brief summary of some of his recent findings with respect to the utterly unexpected nature of the number system.

Part I | Part II | Part III

Dynamic Nature of
the Number System

Part II: Holistic Role of Zeta Zeros

Peter Collins

Introduction

When asked once what was the most important problem in Mathematics, the great German mathematician Hilbert replied!

"The problem of the zeros of the zeta function. Not only in mathematics. But absolutely most important" [1]

Hilbert was referring here to Riemann's Hypothesis (which directly relates to the zeta zeros).

However, apart from their crucial role with respect to the precise distribution of prime numbers, a considerable mystery exists in current Mathematics with respect to the very nature of these zeros.

So I hope to demonstrate that these indeed are crucially important - not just for Mathematics - but in fact for all created existence as we know it!

I hope by now, after the first of this trilogy of articles, that you can appreciate that cardinal and ordinal notions of number are quite distinctive, corresponding - relatively - to the quantitative and qualitative aspects of the number system respectively.

Whereas the cardinal notion initially lends itself to linear interpretation, the ordinal is associated with more circular (paradoxical) appreciation.

The deeper significance of this is that it takes a corresponding circular logic to properly unravel ordinal type notions.

And once again as Conventional Mathematics is defined within a linear logical framework, it is quite unsuited to this task.

Momentary refection on the matter will indicate that ordinal notions are in fact relative (depending on context).

Indeed the paradoxical nature of interpretation involved can be most easily demonstrated when we have a group of 1 member.

So what is first here is inseparable from the polar opposite notion of last. So by definition, in a one horse race, first is indistinguishable from the last past the post.

Then, when we increase the group of members involved in a number group, the ordinal meaning associated with earlier members continually changes.

Thus 2nd for example in the context of 2 is distinct from 2nd in the context of 3 which again is distinct from 2nd in the context of 4 and so on.

So, an unlimited number of qualitative meanings are associated with the notion of 2nd (and by extension all ordinal numbers) each of which possesses a relative validity (depending on context).

It is a short step to realising that if the ordinal notion of number is relative, likewise the cardinal notion must be relative, for in dynamic terms, the cardinal always possesses an implied ordinal meaning (and the ordinal an implied cardinal meaning).

Now we will look at the deeper basis of this relativity with respect to cardinal numbers later!

So once again, numbers have only the appearance of being absolute when we reduce qualitative (ordinal) notions in quantitative (cardinal) terms!

Two Different Perspectives on Primes

Now a key issue then arises as to the conversion of qualitative to quantitative (and quantitative to qualitative) format within the number system.

Once again the very ability to use numbers in a consistent fashion depends on such seamless conversion.

So firstly in this context, the question arises as how to express the ordinal notion of number (with its qualitative connotations) indirectly in a quantitative manner.

We have in fact already provided the answer to this question in the context of a group of 2 members!

So to express the qualitative notions of 1st and 2nd indirectly in a quantitative manner, we obtain the two roots of 1 i.e. (of 1^1 and 1^2 respectively)

So when x2 = 1^2, x = 1 (i.e. + 1)

And when x2 = 1^1, x = – 1

Then to express the qualitative notions of 1st, 2nd and 3rd (in the context of 3 members) we obtain in a similar manner the 3 roots of 1, i.e. of 1^1, 1^2 and 1^3 respectively.

So what we are doing here is converting numbers from the Type 2 to the Type 1 aspect of the number system.

And the remarkable fact is that implicitly this necessarily happens in everyday experience in the very recognition of ordinal type distinctions!

This leads to a very distinctive notion of the importance of prime numbers (which again is completely overlooked in conventional interpretation).

From the conventional (Type 1 perspective) the primes are viewed as the quantitative building blocks of the natural number system (except 1) in a cardinal manner.

Thus each natural number (other than 1) can be uniquely expressed in terms of the product of primes.

So for example 6 = 2 * 3, and this combination of primes is unique for the natural number 6.

Mathematicians tend to view the primes as “the atoms” of the number system (which is very misleading).

In fact, there is an equally valid way of looking at the relationship of the primes to the natural numbers (which represents the inverse complementary opposite of the Type 1 aspect).

Thus from the Type 2 aspect, each prime is uniquely expressed in terms of a combination of natural numbers in an ordinal manner.

So for example 3 is a prime and as we have seen from the Type 2 (qualitative dimensional) aspect, this is uniquely expressed by its 1st 2nd and 3rd members.

Strictly speaking, the 1st is not in fact unique, for as we have seen with the 1st dimension, qualitative is reduced to quantitative meaning. Thus when we get the various roots of 1, 1 will always be included as a root (which thereby can be viewed as a trivial root). All the other roots excluding 1 thereby can be expressed as the non-trivial roots!

So when we say that the 2nd, 3rd, 4th, … pth roots are unique for any prime p, this simply means that their values will never be repeated in the roots of 1 (for any other prime)!

We again have complementarity here, for 1 is likewise excluded from the Type 1 perspective.

So we have two directly opposite perspectives here!

From the Type 1 standpoint, each natural number (except 1) is uniquely viewed in terms of the cardinal composition of primes; however from the Type 2 standpoint, each prime is uniquely viewed (except the 1st) in terms of the ordinal composition of all other natural number members involved.

The great fallacy associated with the conventional perspective, is its sole consideration of the cardinal notion of number (where it enjoys an independent status).

However, whereas the primes are indeed the most independent of numbers from the cardinal, equally they are the most interdependent from the corresponding ordinal perspective.

When we properly appreciate this complementary two-way relationship between the primes and natural numbers (and natural numbers and the primes), with respect to both cardinal and ordinal aspects in a dynamic interactive manner, a radical change in perspective is involved.

It then becomes quite apparent that both the primes and natural numbers are inextricably linked with each other in a manner that is ultimately totally interdependent.

So the mystery of the primes is now more correctly viewed as the two-way interdependence of the primes and the natural numbers (and natural numbers and the primes).

Ultimately the primes and natural numbers are seen in refined experiential terms as perfect mirrors of each other in a manner that approaches an ineffable spiritual state of realisation.

So solving the mystery of the primes - as it might be crudely put - in a physical external manner, is inseparable from the corresponding task of attaining ultimate spiritual realisation (in an internal psychological fashion).

And this is just to highlight once more the inherent dynamic nature of the number system which is inseparable from development (in physical and psychological terms).

So the true importance of the two-way relationship as between the primes and natural numbers is that this serves as the very means by which both the quantitative and qualitative aspects of the number system are transmitted (in a complementary manner).

And as we shall see in the closing article in this trilogy, ultimately this serves likewise as the means by which quantitative and qualitative aspects with respect to all phenomenal evolution are likewise transmitted.

Zeta 2 Zeros

We are slowly zoning in on the zeta zeros.

In conventional mathematical terms, the Riemann Zeta Function is associated with the Type 1 aspect of the number system.

Now f(x) = 1 + x is an example of a simple function.

x here is the independent variable; f(x) is then the function (whose value depends on x).

Thus, when we give x a value, f(x) assumes a corresponding value.

For example in this case, if x = 1, f(x) = 2.

Now Zeta, which is a Greek letter, provides a distinctive way of designating the special Function used by Riemann in studying the number system. (To avoid web translation problems, we will refer to Riemann's Zeta Function as the Zeta 1 Function, denoting it as Z1(s). Again it is customary to use s (to represent the dimensional value with respect to the Function).

The zeros of a function are simply the values of the independent variable for which the corresponding value of the function = 0.

So for example, our simple function f(x) = 1 + x has one zero for x = – 1. In other words when we insert this value for x, f(x) = 0.

You probably will not be surprised at this stage to hear that associated with the Type 2 aspect is a corresponding Function - of equal importance to the Riemann Zeta Function - the relevance of which again is not recognised in conventional terms.

To maintain the desired complementarity with the Riemann Zeta Function, I refer to this as the Zeta 2 Function (with the Riemann Zeta Function referred to as Zeta 1).

So the Zeta 1 (Riemann) Function is directly associated with the Type 1 aspect of the number system; the (unrecognised) Zeta 2 Function is then directly associated with the Type 2 aspect. Again for simplicity, the Zeta 2 Function is represented as Z2(s) with s in this case representing a base quantitative value.

The considerable advantage of this (unrecognised) Zeta 2 Function is that it is much easier to demonstrate the basic role of its zeta zeros from this perspective. This Function is directly associated with obtaining the roots of 1

So to obtain the t roots of 1, we set 1 = s^t (strictly 1^1 = s^t, 1^2 = s^t, …1^t = s^t). Therefore 1 – s^t = 0

As we have seen the first root s = 1 (i.e. where 1 – s = 0) is trivial.

So we divide 1 – s^t by 1 – s to define the Zeta 2 Function i.e.

Z2(s) = 1 + s^1 + s^2 + s^3 …..s^(t – 1) = 0

What is clear here is that the natural numbers in ascending order (from 1 to t – 1) appear as dimensional powers.

The zeta zeros of this Function (Zeta 2) are thereby the values of the base number s that satisfy this equation.

The simplest case of the equation is for t = 2

The equation then is:

Z2(2) = 1 + s = 0; i.e. s = – 1

So this is the 1st of the non-trivial zeros, which provides the quantitative conversion (in a 1-dimensional manner) for the qualitative ordinal notion of 2nd (in the context of 2).

The other non-trivial zeros likewise correspond to such quantitative conversion of ordinal notions in all other possible group contexts.

For example in the important case where t = 4,

Z2(4) = 1 + s^1 + s^2 + s^3 = 0

s = – 1, + i and – i.

So we have here the 3 non-trivial zeros, representing in an indirect quantitative (Type 1) manner the notions of 2nd, 3rd and 4th (in the context of 4 members). [2]

I want to highlight here the vital significance of these Zeta 2 zeros. (The corresponding nature of the recognised Zeta 1 zeros - though much harder to intuitively appreciate - can then be understood in a complementary manner).

Remarkably, what we have found here, with the Zeta 2 zeros, is a magical form of number alchemy, whereby we can (indirectly) convert numbers, with respect to the Type 2 aspect of the number system, through its corresponding Type 1 aspect.

So again in the simplest case, 2-dimensional includes 1st and 2nd members (i.e. 1^1 and 1^2) in the Type 2 aspect of the number system.

1st (in the Type 2) is the same as 1 (in the Type 1) as by definition both are defined as 1^1. So this represents the trivial conversion.

However 2nd in the Type 2 (represented as 1^2) is now represented in a non-trivial manner by – 1 (with respect to the Type 1 aspect). [3]

So, through the simple equation for the Zeta 2 Function, we have the means to translate ordinal meaning (in any group context) defined by the Type 2 aspect of the number system, indirectly in a Type 1 quantitative manner.

Furthermore, we also are provided with the means to perfectly harmonise both quantitative (analytic) and qualitative (holistic) interpretations.

Therefore when we consider all the zeta solutions (including the trivial) for any particular group of numbers as interdependent, the corresponding sum in terms of our quantitative representation = 0.

For example in the context of 4 members, 1, – i, – 1 and i are the quantitative solutions (i.e. roots) relating to 1st, 2nd, 3rd and 4th. Now in relative isolation, a quantitative element is necessarily involved in all ordinal recognition. This is why we must always include 1. However when we see these [4] members as fully interdependent in a qualitative manner, as simultaneously related to each other, no separate independent element remains.

What this entails in terms of actual understanding, is the ready 4-way recognition (entailing multiple perspectives) that the fundamental polar frames of reference (dynamically conditioning phenomenal experience) switch both in real and imaginary terms, so that each of the four numbers can represent any of the ordinal positions (depending on context).

And then in a corresponding (indirect) quantitative manner, we replicate this through demonstrating that the sum of the 4 solutions = 0.

So the key importance of the zeta zeros is that they maintain a perfect correspondence as between both quantitative and qualitative interpretation with respect to the number system.

We realise thereby that both the Type 2 (geared to holistic interpretation of a qualitative nature) is fully compatible with the Type 1 aspect (geared to analytic interpretation of a quantitative nature).

And in the context of the Zeta 2 zeros, this coincidence with respect to both meanings is demonstrated through converting the Type 2 aspect (geared directly to ordinal interpretation of a qualitative nature) indirectly in a Type 1 quantitative manner!

Putting it more simply, what is completely missing in conventional mathematical interpretation is the crucial realisation that both cardinal and ordinal interpretations relate to two distinctive aspects of the number system respectively i.e. Type 1 and Type 2 respectively.

Furthermore the relationship between both aspects can only be properly appreciated in a dynamic interactive context (where they are viewed in complementary terms).

And the Zeta 2 zeros, from the perspective of the top-down conversion (i.e. from Type 2 to Type 1) provide the important solutions where both aspects directly coincide.

I can say all this with considerable confidence, as for about 40 years I have been deeply aware of the enormous importance of this latter system and have spent much of my intellectual life attempting to trace out its implications for Mathematics and the other Sciences (esp. Physics and Economics).

In fact, knowledge of this system provided the direct key for my realisation of what the Riemann Hypothesis is truly about, as it was a short step towards recognition that a strong complementary relationship exists (when properly understood in dynamic interactive terms) as between the Zeta 1 (Riemann) and these Zeta 2 (unrecognised) zeros.

In fact, neither set of zeros can be properly interpreted in the absence of the other!

Of course the quantitative importance of the roots of unity (to which the Zeta 2 zeros relate) is well recognised in conventional mathematical terms. However, the holistic significance of the symbols involved - which is vital for recognition of their importance in terms of the ordinal number system - is completely missed.

Implications for Psycho-spiritual Development

There is another - perhaps unexpected - relevance of these zeta zeros which may be of special interest to readers of this site.

As some may recall with respect to “Integral Studies” I have always been critical of the Wilberian treatment of the “higher” stages of transpersonal development, largely explained in terms of intuitive states (that borrows heavily from the Eastern mystical traditions).

My position all along has been that for proper balance is to be maintained in development, intuitive states of a spiritual nature, must be properly matched with corresponding (refined) cognitive and affective structures.

And with respect to Mathematics and the Sciences, this especially requires the appropriate mapping of the refined rational structures associated with each of the “higher” levels!

However the proper mapping of these (refined) rational structures of a circular paradoxical nature (reflecting the increasingly interactive nature of experience at “higher” levels) has not yet been undertaken. This reflects in turn an intellectual approach to these levels that is greatly lacking from a dynamic perspective.

Therefore little recognition yet exists that the various bands and levels on the Spectrum of Development apply not just to spirituality (as traditionally understood) but equally to Mathematics and all its related Sciences.

In other words Mathematics does not end with specialised linear understanding but - rightly understood - continues its development through all stages of the spectrum (with its very nature thereby radically changing in the process).

In fact, from a holistic perspective, I have maintained for several decades now, that when mathematical symbols are appropriately understood at advanced levels in a dynamic qualitative manner, they serve as the perfect scientific means for the mapping of the various refined rational structures that emerge.

So this holistic mathematical appreciation is in fact enshrined in the very nature of the Zeta 2 zeros (that I am outlining here).

In the very best sense, the Zeta 2 zeros (again when appropriately understood) provide the ready means of grounding all the refined states and structures of “higher” spiritual development (without reductionism) in everyday life.

And in a complementary manner, the Zeta 1 (Riemann) zeros provide the corresponding means of bringing all the primitive instincts and projections of “lower” physical nature to conscious light (without undue censorship) whereby they can be properly integrated with everyday experience. Again it must be clearly understood that just as both Zeta 1 and Zeta 2 zeros are fully interdependent, these related psychological tasks are likewise interdependent!

Again this is a major criticism I would have with the general Eastern approach to “higher” development in that it does not deal sufficiently with this important need for proper grounding of contemplative states in phenomenal reality (and the corresponding need for proper integration of instinctive behaviour with everyday life).

Indeed the zeta zeros (Zeta 1 and Zeta 2) ultimately provide the perfect answer towards the desired marriage of contemplation and activity where “higher” spiritual states and their corresponding refined phenomenal structures (both cognitive and affective) can be harmoniously integrated with “lower” physical instincts (and associated structures) through the medium of ordinary life.

And once again this demonstrates the important point that proper appreciation of the ultimate nature of the number system (in an external physical manner) is dynamically inseparable from the corresponding task of full psychological integration (in an internal psychological fashion).

Further Considerations

Now obviously in a short article like this I cannot go into many other important issues associated with the Zeta 2 Function.

For example I have initially defined it in finite terms. Now it can be extended in a (potentially) infinite manner.

I will just briefly deal therefore with important issue that has a direct bearing on the corresponding nature of the Zeta 1 zeros.

So for example let us go back to the simplest case where:

Z2(2) = 1 + s = 0.

Now remember the zeta zero corresponding to this equation is s = – 1

If we extend the Zeta 2 sequence in a (potentially) infinite manner, we get:

Z2(s) = 1 + s^1 + S^2 + s^3 + s^4 +……..

Now the question arises as the value of this (potentially) infinite series for s = – 1

So now Z2(– 1) = 1 – 1 + 1 – 1 + 1 – ……

Now there are two possible answers here!

If we take an even number of terms Z2(s) = 0

However if we take an odd number of terms Z2(s) = 1

In the circumstances it might seem reasonable to get an average (reflecting the perfect balance of both possibilities) so that Z2(s) = 1/2.

Thus the quantitative analytic interpretation is easy to appreciate.

However the corresponding qualitative holistic appreciation is also vitally necessary.

So the mid-point as between 1 and 0, here in analytic quantitative terms is replicated through the corresponding balance as between (conscious) analytic interpretation of a quantitative kind and (unconscious) holistic interpretation in a qualitative manner.

(Remember with our crossroads example, 1 in the positing of an isolated pole or reference, represents an unambiguous independent existence. However 0 represents the complete paradox of combining opposite poles in a circular interdependent manner).

The Riemann Hypothesis, as we shall see, implies that all the Zeta 1 zeros (which we will presently look at) lie on the straight line through 1/2.

Interpreted from a dynamic holistic perspective, this implies that the very recognition of such zeros requires approximating as closely as possible through experience, the perfect balancing of both conscious (analytic) and unconscious (intuitive) appreciation in their very recognition. 4

This proposition therefore clearly transcends conventional mathematical interpretation (which is formally based on mere analytic type appreciation)!

In fact the Riemann Hypothesis is pointing to the need for a new more complete Mathematics where both conscious (analytic) and unconscious (holistic) interpretation are equally recognised in a dynamic interactive manner.

This again is why it is futile believing that the Riemann Hypothesis can be proved (or disproved) in a conventional mathematical manner.

Indeed it would be a tragedy if somehow it could be proved - which would in fact point to an unrecognised problem in the use of its axioms - as this would condemn Mathematics for a much longer period of time to the poverty of its current reduced interpretations.

However I am confident that growing inability to prove the Hypothesis, will gradually prepare the way for a fundamental reconsideration of the very nature of the number system.

Riemann Zeros

Now we will move briefly on to the Riemann Zeta Function, which I refer to as the Zeta 1 Function.

This is expressed as

Z1(s) = 1^(– s) + 2^(– s) + 3^(– s) + 4^(– s) +….

For example when s = 2,

Z1(2) = 1^(– 2) + 2^(– 2) + 3^(– 2) + 4^(– 2) +….

= 1/(1^2) + 1/(2^2) + 1/(3^2) + 1/(4^2) + …

= 1 + 1/4 + 1/9 + 1/16 + …

This series converges to a finite value which Euler famously showed = (π^2)/6.

The key point that I wish you to observe is the Zeta 1 function represents the inverse complement of the Zeta 2.

Whereas in the Zeta 2, the natural numbers 1, 2, 3, appear as powers (i.e. dimensional values) with respect to the various terms, here they appear as the base values.

Likewise whereas the unknown value s, which is solved by setting Z1 (s) = 0, is a base value with respect to the Zeta 2, here with the Zeta 1, in inverse fashion, it is a dimensional value.

So, just as the Zeta 2 corresponds with the Type 2 aspect of the number system, the Zeta 1 corresponds with the Type 1 aspect.

When s is given a negative value, from a common sense analytic viewpoint the value of the series should diverge to infinity.

For example when s = – 1,

Z1 (– 1) = 1 + 2 + 3 + 4 +….. which clearly diverges in a conventional manner to infinity.

However using a technique known as analytic continuation, mathematicians are able to give a finite answer for all these negative values of s.

In fact, in the Riemann Zeta Function,

Z1(– 1) = – 1/12.

So there is clearly something strange going on!

Now the simple answer, which again cannot be properly recognised in conventional mathematical terms, is that all these non-intuitive answers (corresponding to negative values of s) relate directly to an ordinal (Type 2) rather than cardinal (Type 1) interpretation.

There also exists a famous formula which Riemann established for connecting Z1(s) with Z1(1 – s). So, for values of the Function where s > 1 on the RHS of the real axis, we can thereby calculate corresponding values of the Function where s < 0 on the LHS.

Therefore when correctly interpreted, what this formula shows is the manner of relating Type 1 (cardinal) to Type 2 (ordinal) interpretation.

What is especially interesting in this context is the value for s where both Type 1 and Type 2 values coincide. And this is clearly where s = 1/2.

So the Riemann Hypothesis (from a complementary perspective) is thereby pointing to the condition necessary for the mutual coincidence of both Type 1 and Type 2 aspects of the number system, i.e. quantitative (independent) and qualitative (relational) aspects.

There is only one value of s, for which the Riemann Zeta Function cannot be defined (in cardinal terms). This is where s = 1.

The simple holistic explanation of this could not be more damning for the status of Conventional Mathematics!

Remember by its very nature, Conventional Mathematics is precisely defined in a 1-dimensional manner!

This therefore means that - by definition - Conventional Mathematics is unsuited for appreciation of the true nature of the Riemann Zeta Function!

As we have seen, the Function is appropriately understood in terms of the dynamic interconnections as between cardinal (Type 1) and ordinal (Type 2) aspects of the number system.

Now every dimensional number (except 1) holistically interpreted, implies the dynamic interaction as between distinctive cardinal and ordinal aspects.

It is only when s = 1, that we get the extreme absolute limiting case, where qualitative (ordinal) meaning is directly reduced in a quantitative (cardinal) manner.

So clearly we cannot study the interaction of the Type 1 and Type 2 aspects of the number system, when we insist on recognition solely of its Type 1 aspect!

Thus the Riemann Zeta Function thereby remains (uniquely) undefined for s = 1, i.e. in terms of standard 1-dimensional mathematical interpretation (as it completely lacks a dynamic interactive framework).

Thus once again, our treasured number system - or rather our distorted interpretation of this system - on which everything else stands, is simply not fit for purpose.

However, it requires standing outside the present hermetically sealed mathematical chamber, by embracing a much larger developmental perspective, to be able to see this clearly.

Number Alchemy

The Zeta 1 zeros reflect the solutions for Z1(s) = 0.

The (non-trivial) zeros, which are especially relevant, come in complex number pairs and are of the form 1/2 + it and 1/2 – it respectively.

For example for the 1st pair s = 1/2 + 14.134725 and 1/2 – 14.134725 respectively.

Again the Riemann Hypothesis postulates that all such zeros (that are potentially infinite in range) have a real part = 1/2.

It is not my intention to get into the analytic detail of issues surrounding such zeros. My purpose here is more of an integral philosophical nature in providing a coherent explanation of the precise nature of these zeros.

A very quick way of gaining the basic idea of what is involved is through a complementary interpretation to the Zeta 2 zeros (which we have already encountered).

So we saw earlier that the (finite) Zeta 2 zeros represents a wonderful kind of number alchemy in providing the magical means of converting from the Type 2 (qualitative) to the Type 1 (quantitative) aspect of the number system. This thereby ensures a perfect correspondence as between quantitative (analytic) and qualitative (holistic) type aspects.

Now, more specifically this perfect correspondence relates directly to the natural number sequence of ordinal numbers with respect to any given prime.

Once again the prime numbers are unique in this respect in that all natural number roots of 1 (except 1st) can never be replicated with respect to other primes.

For example 5 is a prime number. Therefore the 5 roots of 1 (except 1) cannot be replicated with respect to any other set of prime roots of 1.

The (potentially) infinite set of Zeta 1 zeros represents an inverse kind of number alchemy, in providing the equally magical means of converting from the Type 1 (quantitative) to the Type 2 (qualitative) aspect of the number system. This again - from the complementary direction - thereby ensures a perfect correspondence as between quantitative (analytic) and qualitative (holistic) type interpretation.

Whereas with the Type 2, we were looking at the unique internal structure of each prime (in ordinal natural number terms), here we are looking at the corresponding unique external nature of the natural number system as a whole (with respect to the constituent prime number factors of each number).

So this would explain the finite and infinite relationship between both sets of zeros. Indeed in a deeper sense, we have already identified the quantitative notion with finite and the qualitative - relatively - with infinite appreciation respectively, where base and dimensional values are quantitative and qualitative with respect to each other.

And as the Zeta 1 zeros directly relate to dimensional number values, they thereby are identified with infinite (rather than finite) notions.

You might be initially tempted to ask why we should need such a conversion in the first place!

Surely the cardinal number system, which we all swear by, can stand on its own quantitative merits?

Well! This in fact is not the case and my recognition of the problem here had its roots in that childhood disquiet, I mentioned in the 1st article, regarding the conventional treatment of multiplication.

We might for example consider a natural number such as 6 as unambiguously representing the cardinal notion of number in quantitative terms!

However 6 is uniquely derived as the product of two primes (2 * 3) and as we have seen - though ignored in conventional terms - this represents a qualitative, as well as quantitative, transformation with respect to the result 6.

So just as in Zeta 2 terms, all primes are composed of unique natural number ordinal building blocks (except 1st), in a complementary inverse manner in Zeta 1 terms, all natural numbers (except 1) are composed of unique prime building blocks (in cardinal terms).

Thus with the Zeta 2, we try to reflect the qualitative meaning of ordinal numbers indirectly in a quantitative manner (while maintaining a perfect correspondence as between quantitative and qualitative meanings).

With the Zeta 1, in reverse, we try to reflect the quantitative meaning of cardinal numbers indirectly in a qualitative manner (with again a perfect correspondence as between quantitative and qualitative type meanings).

Indeed, though we take it for granted, it would not be even possible to order the cardinals on the number line in the absence of the Zeta 1 zeros!

So this mysterious order to the cardinals (which is of a relational qualitative nature) is provided through the Zeta 1 zeros, just as in reverse fashion, with the Zeta 2, a unique quantitative identity is required for every ordinal ranking, before successful interdependent relationships can arise.

Put more simply, in dynamic terms, both quantitative and qualitative aspects of the number system are inseparable. And we can approach this realisation from two directions (a) by moving from qualitative to quantitative (b) by moving from quantitative to qualitative.

In fact the Zeta 1 zeros can accurately be expressed as the holistic basis of the cardinal (Type 1) number system, which reveals its hidden qualitative aspect.

The Zeta 2 zeros can likewise be accurately expressed as the holistic basis of the ordinal (Type 2) number system, which reveals its hidden quantitative aspect.

So once again, the Zeta 1 zeros represent the magical means of converting from the Type 1 to the Type 2 aspect of the number system; the Zeta 2 - in reverse - represent the corresponding means of converting from the Type 2 to the Type 1 aspect.

So in the former case the direction is from quantitative to qualitative; in the latter it is from qualitative to quantitative And the essence of both conversions is that a perfect correspondence can be thereby maintained (though the zeros) as between both quantitative and qualitative interpretation.

Alternatively we can express this as the perfect correspondence between notions of independence and interdependence with respect to the number system. Perhaps, most simply, we can again express it as the perfect correspondence between cardinal and ordinal notions of number respectively.

Of course, ultimately both the Zeta 1 and Zeta 2 zeros are simultaneously determined in a manner approaching a completely ineffable state.

In fact, these zeros in their purest expression serve as the most refined bridge possible as between the phenomenal and ineffable realms, where they dovetail seamlessly with both the prime and natural numbers.

Number in Space and Time

No one, who has ever lived, has had knowledge of number independent of human experience! So, of course, for number to be meaningful, it must take place - as with all experience - in a dynamic context of space and time with respect to both its physical and psychological aspects (which are complementary).

So, number therefore necessarily unfolds in space and time.

Though I have not the space here to develop such ideas at length, corresponding to every notion of number in an analytic sense is a corresponding framework of space and time (with a holistic mathematical meaning).

For example linear notions of time and space correspond with the (positive) rational interpretation of number. However we can have negative (as well as positive), irrational (algebraic and transcendental) as well as rational and imaginary (as well as real) numbers.

Now, when appropriately interpreted in a dynamic interactive manner, associated with all such number types is a corresponding experience of time and space in a complementary holistic manner. [5]

For example the Zeta 1 zeros are complex numbers with real part 1/2 and an imaginary part representing transcendental numbers.

Therefore the true experience of such numbers takes place in a corresponding holistic mathematical framework of complex space and time, with a real fractional and imaginary transcendental aspect.

As much of my adult life has involved the holistic clarification of the nature of number (and its relationship to the understanding of space and time), I had already given a great deal of consideration to such issues before encountering the Riemann Hypothesis.

Indeed I had earlier identified the most refined form of contemplative type experience (compatible with the phenomenal realm) as of a transcendental imaginary nature. In other words, such rarefied experience unfolds in space and time that is understood in holistic mathematical terms as being of a transcendental imaginary nature.

Put more simply, this would approach as far as is possible in the phenomenal realm to the complete unification of both the (conscious) quantitative and (unconscious) qualitative aspects of experience.

So as the Zeta 1 zeros represent this finest bridge between the phenomenal and ineffable realms, it is not surprising that their ultimate understanding requires approximating this pure identity in number terms of both its quantitative (analytic) and qualitative (holistic) aspects. And this also implies the pure identity of both analytic and holistic notions of space and time.

This would in turn imply at the complementary physical extreme that the earliest bridge to phenomenal evolution (as the inherent encoding of the number system) would equally be identified with the Zeta 1 zeros (and their matching complex notions of space and time)!

So when we understand in a truly dynamic interactive manner, notions of number become inseparable from the corresponding nature of space and time (which they inhabit). In this sense, we can see how phenomenal reality is indeed ultimately encoded in number!

In fact the two sets of holistic zeros (Zeta 1 and Zeta 2) and two sets of analytic numbers (cardinal and ordinal) represent the two complementary extremes of the number system.

Thus, at the analytic level these numbers represent form in an absolutely fixed manner.

However at the holistic level, these numbers represent energy states both in physical and psychological terms (in an incredibly dynamic manner).

This relationship can be understood very well in terms of Jungian Psychology.

Therefore, the recognised conscious nature of number (in analytic terms) has its perfect shadow in the corresponding (unconscious) nature of number (in holistic terms).

From this perspective, an enormous problem hangs over Conventional Mathematics in an almost complete inability to recognise the nature of its own hidden shadow.

In psychological terms, it is most unhealthy to reduce signals from the unconscious (necessary for our integration) in a merely conscious manner.

In mathematical terms, it is equally unhealthy to reduce the holistic meaning of its symbols (necessary for integrated understanding of the number system) in a merely analytic manner. And the zeta zeros - when appropriately understood - represent the holistic counterparts to the conventional cardinal and ordinal aspects of the number system (in analytic terms).

We are now accustomed to understanding Einstein's famous equation regarding the equivalence of mass and energy.

However we have an even prior equivalence here with respect to number itself which can manifest itself as both form and energy. And it is the dynamic interaction of such form and energy that leads to continual transformation of the number system throughout evolution (with respect to both physical and psychological processes).

We will say a little more about this in the final article.

Notes

1. As quoted on in the book “Hilbert” by Constance Reid, p. 92.

2. I have used this example of 4 roots as it is comparatively easy to illustrate. However it is important to recognise that the prime numbers serve as building blocks in an alternative manner in the Type 2 system.

So in Type 2 terms (representing the dimensional notion of number) 4 is made up of prime building blocks i.e. 2 * 2.

What this means in effect is that the non-trivial roots for such composite numbers (serving as dimensions) will never be fully unique.

Thus when we take the 4 roots of 1 (i.e. of 1^1, 1^2, 1^3 and 1^4) we get + i – 1, – i and + 1. So the first 3 roots here are non-trivial. However, one of these i.e. – 1 is not unique as it already appears as the non-trivial root of the 2 roots of 1.

3. The distinction between the Type 1 and Type 2 aspects of the number system ultimately relates to the dynamic interactions as between whole and part (and part and whole) notions. Now once again this relationship is dealt with in a grossly reduced - merely quantitative - fashion in Conventional Mathematics (where the whole is viewed merely as the sum of its parts).

In this context, it is fascinating how the first (and most important) of the Zeta 2 zeros (i.e. – 1) is intimately connected with the means through which one switches from the whole notion of a number (as an integer) to its part notion (as reciprocal).

So the whole integer notion of 2 is more fully expressed as 2 ^ 1

So the number is expressed with respect to the default dimensional notion of + 1 (where qualitative is reduced to quantitative meaning).

However when we change this dimensional number to – 1, we are thereby enabled to switch from the whole to its corresponding part notion.

So 2 ^ (– 1) = 1/2.

So in the dynamics of experience, this very negation of the default quantitative interpretation of the quantitative notion of the number 2, thereby enables the switch of recognition to its corresponding part notion. Now though the result of 1/2 is again conventionally reduced in merely quantitative type terms, the very ability to make this switch depends implicitly on qualitative type recognition (arising in experience from the dynamic negation of what is conscious).

So together with the customary analytic recognition of number, we always have in the background a corresponding holistic type recognition of the same symbols which is vital in enabling number interactions (that are dynamic in nature) to take place. However though this holistic understanding necessarily implicit in conventional mathematical experience, it remains at an extremely underdeveloped level (whereby for the most part we remain completely unconscious of its very existence).

Worse still the very acceptance of conventional mathematical assumptions only serves to further weaken this holistic aspect.

This situation is equally the case with the number system as a whole, where the zeta zeros (both Zeta 1 and Zeta 2) represent the holistic correspondent of our analytic notions of the number system (indirectly converted through the opposite aspect of the system).

Thus the Zeta 1 zeros represent the holistic appreciation of the (Type 1) cardinal number system (translated through the Type 2 aspect).

The Zeta 2 zeros represent the corresponding holistic appreciation of the (Type 2) ordinal number system (translated through the Type 1 aspect).

What I believe was most unusual regarding my own development - and why I am now writing these articles - is that I could clearly see into the nature of mathematical reductionism at a very early age. This was experienced with such conviction that it subsequently drove me on to redefine the true dynamic nature of Mathematics (as I saw it) in a manner properly consistent with human experience.

Indeed it is fascinating how the reduction of ordinal (qualitative) to cardinal (quantitative) type notions is directly revealed in the very language we use to refer to fractions (as reciprocals of whole numbers).

For example in customary language we could refer to 1/3 in cardinal type terms as (one over three) or alternatively in an ordinal type manner as one third. Now one partial exception would be 1/2 which we could refer in cardinal terms as one over two or alternatively as a half (or one half).

Then 1/4 could be expressed as one over 4 or a quarter (possibly also one fourth though this would not be so common. It is perhaps the fact that these two exceptions represent the most widely used examples of fractions that special terminology has developed. This exclusiveness is replicated in terms of typewriter usage with 1/2 represented as ½ and 1/4 represented as Ό.

However for all other numbers there would be a direct matching as between both cardinal and ordinal type terminology. So these language conventions only go to show how deeply ingrained this reduction of qualitative to quantitative type meaning has become with respect to the number system.

4. Strictly speaking, the value of 1/2 on which all the non-trivial Riemann (Zeta 1) zeros represents a probable value. As we have seen the Riemann Hypothesis properly understood transcends the nature of conventional mathematical proof. Thus, acceptance of the Riemann Hypothesis ultimately represents an act of faith that subsequent mathematical activity is indeed meaningful. But in existential terms, faith, no matter how well-founded, entails uncertainty.

5. See “Multidimensional Nature of Space and Time (1-20)” from my Integral Science blog.

References:

Riemann Hypothesis - This is my aforementioned blogspot.




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