The Holistic Nature of the Number System (11)

In this entry, I will once again attempt to communicate in an intuitively accessible manner the true startling nature of our number system.

Firstly, when properly understood i.e. in a dynamic interactive manner, number is directly related to human experience with two complementary aspects that are physical and psychological with respect to each other.

In other words the long accepted notion of number as absolutely existing in some abstract mathematical space is but a fiction arising from a reduced interpretation of its true nature.

More correctly, number is inherent in all processes (physical and psychological) as their most fundamental nature. So from this perspective, we cannot abstract number from either physical reality (or its corresponding psychological interpretation) as - by their very nature - these are dynamically encoded in number.

All the surprise that is being now expressed regarding intimate connections as between the Riemann zeros and chaotic quantum states thereby reflects the reduced existing interpretation of number. If number was properly understood i.e. in a dynamic interactive manner, no such surprise would occur!

 

As well as twin physical and psychological aspects (reflecting the necessary human interaction as between opposite polarities that are internal and external with respect to each other) number equally expresses the necessary dynamic interaction as between quantitative and qualitative aspects (reflecting in turn the relationship between wholes and parts).

There is huge confusion in present Mathematics with respect to this issue. Because of its merely quantitative bias in formal terms, the qualitative aspect is thereby reduced and distorted (from every perspective) in quantitative terms.

The cardinal and ordinal notions of number relate to the quantitative and qualitative aspects of the number system respectively.

This means in effect - because of reduced quantitative nature - that no coherent interpretation of the ordinal aspect of number can be given within Conventional Mathematics. (And because of their necessary interdependence, no coherent interpretation can likewise be given of  the cardinal aspect)! 

This Mathematics is solely geared to the analytic type appreciation of relationships (as independent). This corresponds in psychological terms with a merely (conscious) rational interpretation.

However when appropriately understood, the ordinal aspect of number relates to the holistic type appreciation of number relationships (as interdependent).

This likewise entails that such appreciation relates directly to  understanding of an (unconscious) intuitive rather than (conscious) rational nature.

The clear implication therefore is truly revolutionary in that the understanding of our number system and indeed all mathematical relationships (in a coherent fashion) requires a radical new paradigm with analytic and holistic type appreciation both recognised - though (conscious) reason and (unconscious) intuition respectively - as equal partners.

In a dynamic context, sole emphasis on the prime numbers (as the building blocks of the number system) is seen to be in error.

Rather what is now emphasised is the dynamic two-way relationship as between the primes and the natural numbers (and the natural numbers and the primes) through which both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system are mediated.

When looked at in a dualistic manner (through separate reference frames) the direction of causation as between both seems directly in conflict. Thus from the cardinal perspective, the primes appear as the building blocks of the (composite) natural numbers; however equally from the (unrecognised) ordinal perspective, the natural numbers appear as the building blocks of each prime!

Thus from the linear (quantitative) perspective, the primes appear as the most independent of all numbers; however equally from the circular (qualitative) perspective, the primes now appear as the most interdependent!

 

So when one properly appreciates this paradox with respect to the primes, the true fundamental requirement is then to establish the ultimate consistency of both quantitative and qualitative aspects.

And when properly understood, this is what the Riemann Hypothesis is truly about!

The zeta zeros relate directly to this mysterious identity in the number system with respect to both cardinal (quantitative) and ordinal (qualitative) aspects.

Again - when appropriately understood in a dynamic interactive manner - two complementary sets of these zeros can be seen to exist (which I refer to as Zeta 1 and Zeta 2).

The Zeta 2 zeros (which are the simpler to intuitively grasp) relate within each prime number (representing a group of members) to how ordinal identity - which is qualitative in nature -  can be coherently expressed (indirectly) in a quantitative manner. Therefore, though ordinal identity relates directly to the interdependence nature of a number group, as this is of a merely relative nature, indirectly it likewise has an independent identity!

The Zeta 1 zeros then relate to the corresponding issue of how ordinal identity with respect to the number system as a whole (arising from the combination of prime number factors) can likewise be coherently expressed through an unlimited set of numbers. In reverse manner, though cardinal identity directly relates to the independent nature of each individual number, again because this is of a merely relative nature, indirectly it likewise has an interdependent ordinal identity with respect to the number system as a whole (which is what the Zeta 1 zeros represent). 

 

What is crucial to understand is that both of these sets of zeros relate directly to the holistic - rather than analytic - interpretation of number (i.e. to the interdependence of both quantitative and qualitative aspects).

This likewise implies that their true appreciation can only come from the specialised development of the intuitive unconscious in higher dimensional contemplative experience of reality. However indirectly such intuitive appreciated can be given circular rational expression in a paradoxical manner.

And this cannot be provided through Conventional Mathematics, which by its very nature is defined in a linear (1-dimensional) conscious format!

Rather, from a comprehensive mathematical perspective, 3 - rather than at present 1 - relatively distinct areas are required:

1) The Type 1 aspect geared to traditional analytic appreciation in a quantitate manner. However crucially all relationships would now be interpreted in a relative - rather than absolute - manner.

This aspect of Mathematics is rational in a linear (i.e. 1-dimensional) fashion based on isolated polar reference frames.


2)  The Type 2 aspect geared to the (formally) unrecognised holistic appreciation in a qualitative manner. Typically, I have referred to in the past as Holistic Mathematics.

This aspect of Mathematics crucially entails viewing the polar reference frames (which condition all mathematical experience) as interdependent in a dynamic interactive fashion. In direct terms it is of an intuitive nature which indirectly can be expressed in a rational manner (which appears paradoxical in terms of linear reason).

So this aspect of Mathematics is rational in a circular manner (in all dimensions ≠ 1). However its simplest expression in 2-dimensional terms - based on the direct complementarity of opposite poles such as internal/ external and whole/part - in an important sense serves as a prototype for all other higher dimensional interpretations.

Since my late teens, I have been in the process of developing this completely neglected qualitative aspect of Mathematics (which rightly should be seen as an equal partner with its quantitative counterpart).

In particular I have spent many years using it to show that human development - with respect to the entire spectrum of its possible structures - can be scientifically best appreciated in a holistic mathematical manner.


3) The Type 3 aspect - which I formerly referred to as Radial Mathematics - is geared to the two-way comprehensive integration of both Type 1 and Type 2 aspects.
Of course, this Type 3 aspect cannot have meaning in the complete absence of formal recognitiont of the Type 2 (holistic) aspect of Mathematics.

In a very preliminary manner, I have been demonstrating in recent years on this blog how this Type 3 aspect is crucially necessary for coherent interpretation of the Riemann Hypothesis (and its many fundamental consequences).

Therefore, when correctly interpreted in a more comprehensive manner, the most important implication of the Riemann Hypothesis is the truly inadequate state of existing Mathematics.
Indeed continued failure to prove the Riemann Hypothesis may well serve to eventually act as the catalyst for a much needed new vision of Mathematics.
  

So underlying our common sense conscious interpretation of number (in analytical terms) is a deep unrecognised unconscious appreciation (of a truly holistic nature).

This holistic unconscious appreciation represents the great unrecognised shadow of Mathematics.

As it stands, our present formal appreciation of Mathematics is totally lacking a holistic integral perspective (which requires explicit recognition of the role of the unconscious). In other words our understanding of Mathematics at present is hugely unbalanced.

Though it may seem highly rigorous and specialised to many, present Mathematics is in fact based on greatly reduced - and thereby greatly confused - assumptions. This means in turn that accepted notions of science (which are intimately based on Mathematics) thereby suffer from the same reductionism and confusion.

If we cannot provide a coherent account of the number system (on which everything else is based) well then we cannot offer in truth a coherent account of anything!

 

Though I would expect that what I have said here will simply be ignored by practicing mathematicians, I write with a great conviction as one who already began to recognise the highly reduced nature of mathematical assumptions at an early age.

Now, after 50 years of on-going reflection, I am utterly confident of the position outlined here and that what I say here will eventually be accepted in the years to come.

Make no mistake about it! When it is clearly realised what is at stake here i.e. the true nature of Mathematics, will set the stage for by far the greatest revolution yet in our intellectual history. All previous developments simply pale by comparison.     

The Holistic Nature of the Number System (10)

In this blog entry, we will look directly at the holistic nature of the famed Riemann (Zeta 1) non-trivial zeros.

Now once again in terms of the Type 1 aspect of the number system, each (composite) natural number is viewed quantitatively in cardinal terms as comprising a unique combination of prime  factors.

So for example 6 (in cardinal terms) can be expressed uniquely as the product of  the primes, 2 and 3.

Therefore 6 = 2 * 3.

Then expressed more fully (with respect to the default dimensional value of 1),

6¹ = 2¹ * 3¹

However as we have seen before, whenever two numbers are multiplied together that a qualitative (dimensional) transformation is likewise involved.

So for example, if we were to represent the product of 2 and 3 in geometrical terms we would get a rectangle with units measured in square (2-dimensional) rather than linear (1-dimensional) units.

Now we can represent this latter qualitative aspect through the Type 2 aspect of the number system.

Therefore 1⁶ = 1^{(2 * 3)}

So just as the adding of two indices implies multiplication, the multiplication of two indices implies exponentiation.

                                             ₃

So therefore 1^{(2 * 3)}  = (1²)

Now as we have seen we indirectly express this in a circular quantitative manner by obtaining the corresponding roots.

So the sixth root of 1 i.e. 1^1/6 = 1^{(1/2) * (1/3)}

Therefore, just as the cardinal notion of 6 is derived from a combination of prime factors, the ordinal notion of 6 (i.e. 6^th) is derived from a similar combination of prime roots.

Thus the 6^th root of 1 (that indirectly expresses in a quantitative manner the ordinal notion of 6) is itself derived from a combination of the 2^nd and 3^rd roots of 1.

However, calculating this 6^th root involves a process of exponentiation, where we initially obtain the 2^nd root (1^1/2) and then raise the resulting value to 1/3.

So once again when we consider the multiplication of prime factors, two distinctive processes are at work.

1)      From the quantitative perspective, the (composite) natural number in cardinal terms that results, represents a unique combination of prime factors. This reflects the Type 1 aspect of the number system.

 So for example, again from the quantitative perspective, the (composite) natural number 6 in cardinal terms represents the product of the two primes (i.e. 2 and 3) 

2)      From the qualitative perspective, the (composite) natural number in ordinal terms that results, indirectly expressed in a circular quantitative manner, likewise represent a unique combination of these prime roots (of 1).

So now from the corresponding qualitative perspective the (composite) natural number 6 in ordinal terms (6^th) represents the product (of indices) of the two prime roots.   

So when we look at this process of prime number multiplication, two complementary aspects are always necessarily involved in a dynamic interactive manner, which are quantitative and qualitative with respect to each other.

From one quantitative perspective, the unique identity of each (composite) natural number in cardinal terms is achieved; from the complementary qualitative perspective, the unique identity of each (composite) natural number in ordinal terms is achieved.

So again for example, we cannot divorce the unique composition of 6 in cardinal from the corresponding unique composition of 6^th in ordinal terms.

And both of these involve distinct mathematical processes that cannot be reduced in terms of one another.

So from this more comprehensive dynamic perspective, the conventional mathematical attempt to explain the nature of the number system in a mere quantitative manner is ultimately seen as quite futile!

Thus when one accepts the complementary dynamic interaction of two distinct processes (with respect to both the cardinal and ordinal aspects respectively), then the key issue relates to the ultimate reconciliation of both aspects within the number system.

In other words this entails (i) the manner in which cardinal and ordinal aspects can ultimately be unified as mutually identical with each other and (ii) the expression of this mutual identity in an appropriate numerical fashion?

We already saw how this identity is achieved with respect to the Zeta 2 (non-trivial) zeros.

Once again the task here is to reconcile the overall (qualitative) interdependence of  the ordinal  natural number grouping comprising each prime number, with the separate quantitative independence of each individual member.

So in the simplest case, the two roots of 1 indirectly express in a quantitative manner the ordinal nature of 1^st and 2^nd (in the context of 2 members).

Each of these ordinal members enjoys a (relatively) separate identity in quantitative terms.

However the overall qualitative interdependence of the group is expressed through their collective sum = 0.

So we can perhaps see here how this unification of cardinal and ordinal aspects is achieved with respect to each prime number (and how this relationship is numerically expressed).

Now the Zeta 1 zeros achieve a somewhat similar task with respect to the number system as a whole.

When one understands the relationships as between ordinal and cardinal appropriately, a dynamic interdependence of both is necessarily involved.

In other words the very reason why each (composite) natural number can be expressed as the product of prime number factors in a cardinal (Type 1 ) manner is dependent on the corresponding fact, that each (composite) natural number can equally be (indirectly) expressed  as the product of prime number roots in ordinal (Type 2) terms.

And of course this also works in reverse with the Type 2 relationship intimately dependent on its Type 1 counterpart.

So the zeta zeros represent the simultaneous interdependence of both sets of relationships, where both quantitative (cardinal) and qualitative (ordinal) aspects for the composite natural numbers are directly reconciled with each other.

Thus the very nature of the zeta zeros is of a holistic nature.

Remember the nature of analytic interpretation (especially as used in Conventional Mathematics) is to attempt to absolutely separate, in a static fixed manner, quantitative and qualitative aspects (and then directly reduce the qualitative aspect in quantitative terms)!

However the nature of holistic is quite the opposite where the quantitative and qualitative aspects are always seen in a necessary dynamic relationship with each other and where one thereby seeks the harmonious identity of both in a relative approximate manner.

So once again the very nature of the zeta zeros is holistic and therefore not strictly amenable to conventional mathematical interpretation.

This time in the case of the Zeta 1, we have the separate identity (indirectly) of - relatively - independent zeros, combined with the overall relative interdependence of the zeros for the number system as a whole. Thus expressed through these zeros, we have the reconciliation for the composite numbers as a whole, of  both its cardinal (quantitative) and ordinal (qualitative) features.

Then also as we have earlier seen in the case of the Zeta 2, we have again the separate identity of - relatively - independent zeros (indirectly representing the ordinal nature of the natural number members of each prime group), combined with the relative interdependence of these ordinal members (within each prime number).

And once again, in dynamic interactive terms the solutions for the two sets of zeros is obtained in a simultaneous manner (with both intimately dependent on each other).


Thus the very nature of the holistic approach is to establish such complementarity (and ultimate identity) in a dynamic interactive manner as between relationships occurring in polar opposite reference frames.

And for their appropriate understanding, the zeta zeros - par excellence - require the extreme specialisation of this approach i.e. where one can simultaneously "see" relationships in terms of both complementary perspectives.

When one properly grasps this point, it is thereby again futile attempting to grasp the nature of the zeros in the standard fixed analytic manner (based on mere quantitative interpretation)

The Holistic Nature of the Number System (9)

I will use an interesting example here to demonstrate the true nature of the holistic approach.

Once again, by its very nature it requires going beyond the 1-dimensional standpoint of standard Mathematics where all symbols are given a merely quantitative (analytic) interpretation in absolute terms.

In the holistic approach, while symbols do preserve their independent analytic identity, they now possess a relative - rather than absolute - quantitative meaning. Most importantly they additionally acquire a new - hitherto unrecognised - qualitative identity which properly relates to interdependence with respect to relationships.

So from the holistic perspective, all symbols possess a - relative - independent identity in quantitative terms, combined with a relative interdependent identity from a qualitative perspective. And the very essence of the holistic approach is the integration of both of these perspectives!

Thus it is utterly pointless trying to judge the value of the holistic mathematical aspect from within the limited confines of what is formally accepted as valid Mathematics.

Rather I suggest that its great potential value can be better appraised through the manner, for example, that it can give a much more coherent explanation of the true nature of our number system!

I have long been fascinated by the nature of cyclic primes finding their properties quite magical!

Cyclic primes relate to the (repeating) decimal structure of the reciprocals of certain prime numbers.

Now we operate of course in a base 10 number system and not all primes are cyclic in this base. For example, 5 clearly is not cyclic. So the reciprocal of 5 is 1/5 and in a base 10 system this is .2!

However in the base 8 system for example, 5 is indeed fully cyclic with the repeating decimal sequence of 1463. Fully cyclic in this context implies that the number of digits in the repeating decimal sequence is 1 less than the number in question!

So indeed given an appropriate number base, all primes - and only primes - can be given a full cyclic sequence.

The first and best know cyclic prime in the base 10 system is 7.

So the reciprocal of 7 results in the repeating decimal sequence of the six digits 142857.

Now one of the remarkable properties of this number is that when we multiply by 2, 3, 4, 5 and 6 the same digits keep repeating with the circular order remaining unchanged.

142857 * 2 = 285714

142857 * 3 = 428571

142857 * 4 = 571428

142857 * 5 = 714285

142857 * 6 = 857142

The circular nature of such numbers becomes even more remarkable when we consider the larger cyclic primes.

For example 97 is a cyclic prime. So when we multiply its 96 repeating digits by 2, 3…..96 the circular order of these 96 digits will remain fully preserved!

So there is a wonderful magical order of a circular nature involved here which we can all admire!

And this represents just one fascinating aspect of the ordered structure of such primes!

However, within Conventional Mathematics though we may indeed marvel at the unique number nature of cyclic properties, because of the lack of an accepted holistic dimension, the deeper significance of all this for the prime numbers is completely missed.

Now once again in conventional linear terms, the prime numbers (which have no constituents factors other than the numbers themselves and 1)) are treated as the most independent of all numbers, indeed as the essential building blocks of the cardinal number system!

 So 7 is a prime number!

Written more fully (in a Type 1 manner) this is 7¹.or to spell it out even more clearly 7 ^{+ 1}   

Now the reciprocal of 7 can be written  as 7 ^{– 1} = 1/7.

As we have seen, holistic interpretation starts with 2-dimensional - rather than 1-dimensional interpretation - where its structure is indirectly expressed (in a quantitative manner) by the corresponding two roots of 1, i.e. + 1 and – 1.

However - and this is the very essence of holistic interpretation - corresponding to these two numbers + 1 and – 1, is a distinctive qualitative dimensional interpretation.

So as we have seen the holistic meaning of + is - literally - to posit in experience, which refers directly to conscious type recognition. However the holistic meaning of – is to (dynamically) negate (what has been posited in conscious terms). And this is the very means in experience through which unconscious recognition takes place!

So therefore when we look at prime numbers in 2-dimensional holistic terms,  we recognise that by their very nature they contain two extremes.

Thus from the conscious perspective (with respect to its quantitative nature) they do indeed appear as the most independent of all numbers serving as the basic building blocks of the cardinal number system (in a linear manner).

However when we look at prime numbers from the unconscious perspective (with respect to their qualitative nature) they now appear as the most interdependent of all numbers serving as the unique basis for the ordinal number system (in a circular manner).

 

So the true mystery of prime numbers relates to this marked independence with respect to their cardinal identity, coupled with an equally marked interdependence with respect to their ordinal identity.

Thus the very nature of prime numbers is inherently dynamic entailing the interaction of these two extreme tendencies.

Therefore once again we cannot hope to understand the nature of prime numbers - and indeed our number system - in mere quantitative terms (based on analytic type interpretation).

Rather the true nature of prime numbers entails twin aspects that are relatively analytic (quantitatively independent) and holistic (qualitatively interdependent) with respect to each other.

 

When seen properly in this dynamic context, the corresponding nature of the famed zeta zeros is easy to appreciate as a set of numbers that establishes the mutual identity of both aspects (quantitative and qualitative). The Zeta 2 - which are not even conventionally recognised - uniquely establishes this within each prime number; the Zeta 1 then establishes this identity for the number system as a whole.

When I read the conventional literature on the Riemann Hypothesis, there are constant references as to how mathematicians are so surprised with its connections to the physical world of quantum theory.

Well I am not at all surprised as from my dynamic appreciation of the nature of prime numbers, I always assumed that this was necessarily the case!

Likewise we have the same surprise expressed in relation to the fact that the zeta zeros strongly resemble some spectrum relating to the vibration of physical frequencies.

Well again I am not surprised as I have long considered this to be necessarily the case.

Indeed I would go considerably further in recognising a complementary (as yet unrecognised) psycho-spiritual spectrum for all these zeros (which in the future will play an extremely important scientific role with respect to the most advanced meditation practices).

 

Indeed properly understood the physical nature of the zeros cannot be properly understood in the absence of their corresponding psychological appreciation (as both aspects are fully complementary).

 

And as to what is ultimately vibrating, the answer is simply the number system itself.

You see, the existing paradigm - based on 1-dimensional interpretation - of the number system existing in some abstract unchanging space is ultimately completely flawed.

When we look on the number system is a limited reduced manner - as we have been doing for millennia - it certainly appears to have an absolute unchanging nature.

However this is not as it truly is: in other words properly understood, in a dynamic interactive manner, number serves as the deepest inherent encoding of all physical and psychological processes in creation (in quantitative and qualitative terms).

Thus as soon as creation vibrates in its first instants, the number system itself vibrates with respect to the relationship between its quantitative and qualitative aspects (which necessarily begin to separate to a degree with the birth of phenomena in space and time)!   

I have been aware for some 50 years now of the need for a massive paradigm shift with respect to Mathematics. And when I say massive, I mean massive as the whole discipline now needs to be rebuilt on completely new foundations (where holistic is fully recognised and then integrated with analytic interpretation and where the role of the unconscious in mathematical understanding is likewise fully recognised and then integrated with the conscious aspect).

Though I realised this long before becoming acquainted with the Riemann Hypothesis, growing acquaintance with this problem has only served to strongly confirm my initial diagnosis.                    

The Holistic Nature of the Number System (8)

Readers of this blog may have noticed that I have been running a companion blog for some time “Spectrum of Mathematics”.

Indeed as the true implication of the Riemann Hypothesis are so fundamental and far reaching for Mathematics, in effect the present has been used in many respects to overlap with material covered in that latter blog.

Though in some senses a bit vague, the word “spectrum” is now widely used in physics to refer - in any context - to a variety of related phenomena,

Perhaps its best known use is in connection with the “spectrum of light” whereby in passing through a prism, the various wavelengths of natural light can become manifest as different colours. And of course the colours of the rainbow produce a beautiful natural example of this phenomenon!

Then when it was realised that natural light represented just one small band with respect to electromagnetic energy, this spectrum notion was widened to include all its possible bands from very high frequency gamma radiation at one end to extremely long frequency radio waves at the other.

 

The notion of light in physical terms invites comparison with the corresponding notion of light in a psycho-spiritual context so that human development itself represents a spectrum of all possibilities with respect to the capacity for enlightenment.

Such enlightenment is directly related to the holistic aspect of intuition.

As we have seen mathematical activity itself, in dynamic terms, necessarily entails an interaction between reason and intuition. Indeed it will be readily admitted by mathematicians, that where especially creative work is involved, that intuition necessarily plays an important role.

However in formal terms, mathematical results are then presented in a merely reduced rational manner (with the important holistic aspect of intuition edited completely out of the story).

What is not readily recognised however is in a somewhat complementary manner to the electromagnetic spectrum, that the full range of intuitive possibilities allowed by the psycho-spiritual spectrum is much greater than generally imagined.

Indeed just as natural light forms just one small band of the electromagnetic energy spectrum, likewise - what we might call - natural intuition (that supports the linear use of reason in Conventional Mathematics) comprises just one small band of the full range of intuitive possibilities available.    

Detailed explorations of higher intuitive states have been undertaken in all the major spiritual contemplative traditions.

However what has been almost completely missing has been the attempt to study the relevance for Mathematics of such psycho-spiritual development.

 

So from a very early age, I could somehow see this area as my special goal in life, i.e. to marry the holistic possibilities opened up through spiritual contemplation, to the mapping of a new mathematical understanding (consistent with such development).

I then realised that inherent in the very notion of number, is a marvellous way of precisely detailing all possible stages of development in a holistic mathematical manner.

Some 20 years ago I wrote an online book  “The Number Paradigms” that basically outlines my earlier findings on this matter.

What struck me forcibly at the time is this!

From a quantitative perspective, they are now indeed many recognised number types that play a huge role in Conventional Mathematics.

So I outlined firstly what I termed the original numbers 1 and 0. Next on the list we have the primes, then the natural numbers and the integers (positive and negative).

These are followed by the important rational numbers.

Then we move on to irrational number (algebraic and transcendental), the imaginary and complex nos. and the transfinite.

 

Now this is the key point!

In conventional mathematical terms all these are understood within what may be correctly referred to  - qualitatively speaking - as the rational paradigm.

So, while we recognise a wide range of number types in quantitative analytic terms, we recognise just one  i.e. rational in a qualitative holistic manner. And as I have repeatedly stated in my blog entries the very nature of this rational paradigm is to reduce all qualitative meaning in merely quantitative terms!

Thus we have sectioned off just one very small restricted band of possible development (relating to specialised rational understanding) and defined Mathematics exclusively in terms of this band.

It is quite akin to sectioning off the small band corresponding to natural light and refusing to recognise the validity of all other energy wavelengths on the electromagnetic spectrum!

 

Indeed the parallels are very striking! It would be extremely difficult for one identifying exclusively with the present mathematical paradigm to even appreciate what is being said here.

When one literally “sees” exclusively in accordance with the natural intuition that informs conventional mathematical understanding, any appeal to different types of mathematical reality (based on holistic intuition relating to other bands on the spectrum) literally has no resonance.

Though my thinking on these matters have become considerably more refined in the last 20 years or so, I still would see the basic position taken in “The Number Paradigms” as fully sound.

Putting it simply, when we adopt a true holistic (i.e. qualitative) mathematical perspective, then all of human development  - and in complementary terms physical evolution - can be fruitfully seen as representing a number spectrum.

So giving a brief outline we have firstly the lower stages of development.

The basic point here is that these are marked by confusion as between conscious and unconscious processes.

So the first task is to successfully differentiate conscious from unconscious understanding. The earlier number types mentioned, original, prime, natural and integers in qualitative holistic terms are then associated with these stages.  

The middle stages represent the specialisation of differentiated conscious understanding (free of holistic unconscious confusion). Conventional Mathematics - par excellence - represents an extreme in terms of this understanding which of course in qualitative terms is associated with the rational numbers.

The higher stages then represent the process of now developing in a mature way the unconscious holistic aspect of understanding, which is qualitative terms is associated with irrational (algebraic and transcendental) and imaginary numbers.

One of the interesting findings I made at the time was that the most refined form of holistic understanding (where phenomena are now of a highly transparent and fleeting nature) are associated with transcendental structures of an imaginary nature.

As these directly relate to the very nature of the Zeta 1 (non-trivial) zeros, let me briefly elaborate.

Imaginary in a qualitative holistic sense, refers to the indirect expression (in a refined conscious) manner of what pertains directly to unconscious meaning.

So when we are understanding in a refined imaginary manner we can quickly recognise all phenomenal projections (though invested in consciously recognised objects) as directly expressive of holistic (unconscious) meaning.

Transcendental refers to a highly refined form of  understanding where objects are neither interpreted exclusively in a linear (conscious) nor circular (unconscious) manner, but in terms of the common relationship - which is ultimately ineffable - as between both.

So the ability to immediately interpret projections (as imaginary) in a transcendental manner, requires the most refined spiritual development possible (consistent with experience in the phenomenal realm).

Now the significance of this is that the zeta zeros necessarily unfold in the very first instants of phenomenal creation.

When number is appropriately understood in a dynamic interactive manner, it is no longer considered in abstract (merely rational) terms, but rather as the most intrinsic encoding of all phenomena (both in physical and psychological terms).

Without this encoding, enabling both quantitative and qualitative aspects of reality to coherently interact, no phenomena could ever become manifest.

Thus to properly appreciate what is - literally most primitive or prime - with respect to earliest creation, requires ultimately the most developed spiritual experience possible.

In the truest sense the zeta zeros come closest in the phenomenal realm to serving as the first bridge from the nothingness preceding physical evolution.

Therefore their mature understanding likewise requires reaching as close as possible to that final spiritual bridge to the ineffable.

The zeta zeros in fact represent the holistic (unconscious) counterpart to the analytic (conscious) aspect of the primes.

We look on the primes as perhaps the most rigid and immutable in absolute terms.

In fact, as is the nature of all experience, they cannot be totally absolute, though yet coming closest in the phenomenal realm to reaching this standard.

However as the zeta zeros represent the holistic (unconscious) shadow of the primes, they approximate closest at the opposite extreme to the most purely relative - and inherently dynamic - expression possible of the number system.

So the number system is the original and most important spectrum (with respect to both physical and psychological aspects). And of course it is an inherently dynamic spectrum that necessarily vibrates through the medium of physical and psychological interactions)..

The idea that this system is some fixed abstract entity frozen in space and time is just the product of a false and limited type of understanding.

In fact the number system represents all the possibilities for the evolution of phenomenal creation (as its inherent encoding in quantitative and qualitative terms).

The Holistic Nature of the Number System (7)

I will demonstrate briefly here the complementarity as between the linear and circular approaches to number.

The quantitative aspect in the linear approach is expressed through addition.

So for example 2¹ = 1¹ + 1¹

 

However the qualitative aspect - indirectly expressed in a quantitative manner - in the circular approach is expressed through addition.

So  0 = 1^st + 2^nd i.e. indirectly expressed in quantitative terms as 0 = + 1  – 1.

Then the qualitative aspect in the linear approach is expressed through multiplication.

So for example 1³ = 1¹ * 1¹ * 1¹

However when this is expressed in a circular manner - in an indirect quantitative fashion through the 3 roots of 1 - we arrive back at the quantitative aspect of number,

i.e.  1 * (– .5 + .866i) * (1 (– .5 – .866i) = 1

Thus for any prime number p (except 2) the product of its p roots = 1, while the sum of its p roots = 0.

Now the analytic interpretation of this is of course well known. However there is a much deeper corresponding holistic significance in demonstrating the complementary nature of addition and multiplication (when reflected through linear and circular frameworks respectively).

So whereas Conventional Mathematics can indeed analyse circular relationships in quantitative terms, it necessarily does so analytically from within a rational linear framework.

However to properly interpret these relationships from a corresponding holistic perspective, we must do so within a rational circular framework (where paradoxical complementary relationships indirectly reflect holistic understanding that is directly of an intuitive nature).

Once again this holistic aspect is entirely missing from Conventional Mathematics (as formally understood).

So when we appreciate mathematical relationships in an appropriate holistic manner, we are then enabled to seamlessly convert from quantitative to qualitative aspects respectively through continually switching in turn as between linear and circular modes of interpretation.   

 

Perhaps the most important holistic mathematical appreciation of all relates to the qualitative significance of the imaginary notion.

Imaginary numbers are indeed used mow extensively in quantitative analytic terms within Conventional Mathematics.

However the corresponding qualitative holistic interpretation of imaginary numbers - which is equally important - is entirely missing from conventional mathematical interpretation.

I have found my acquaintance with Jungian notions extremely helpful in arriving at the holistic mathematical meaning of the imaginary.

Again in Jungian terms, when a conscious function such as thinking is unduly dominant, its unconscious (shadow) complement is projected in a somewhat blind manner on to conscious phenomena.

Now in scientific terms, “reality” is defined in a merely conscious rational manner which is properly geared to analytic type interpretation.

However we are all perhaps aware that our actual experience is conditioned to a degree by unconscious type projections (which relate to a distinctive holistic meaning).

So when we experience objects there is indeed a specific local aspect (according with conscious recognition).

Likewise however there is a universal holistic aspect (according with unconscious meaning).

For example one might speak of a “dream house”. Now the house indeed can be consciously verified as an object; however there is also a holistic aspect here in the experience that serves a deeper (unconscious) holistic desire for meaning.

Now the “imaginary” in qualitative terms, simply refers to objects as indirectly representing this holistic aspect of meaning.

In this sense all reality is complex with objects having a - relatively separate - local identity (as real) and a whole relational identity - as imaginary.

 

As we have seen, this is equally true of all mathematical objects (such as numbers).

Indeed the ordinal nature of number properly relates to its qualitative holistic identity.

In direct terms this relates directly to an unconscious - rather than conscious - recognition that is then indirectly expressed rationally in a circular (paradoxical) fashion. It then can be converted, as it were, into linear type interpretation through giving it an “imaginary” rather than “real” identity.

As we have seen, in quantitative terms,  the imaginary number i is given as the square root of  – 1.

Now, in the context of the Zeta 2 Function, – 1 is all important, serving in qualitative terms as the expression of unconscious meaning. In other words, in experiential terms, the very way in which the unconscious is activated in experience, is through the dynamic negation of conscious type meaning.

So once again, in Jungian terms, if the emphasis on conscious experience is unduly dominant (as it most certainly is in Conventional Mathematics), then the unrecognised unconscious aspect will be blindly projected onto consciously understood phenomena (and then directly confused with them).

And of course this is  precisely what has happened in Mathematics. Though ordinal appreciation is qualitative - inherently relating to the unconscious - this is blindly projected on to the conscious recognition of mathematical objects (as quantitative) and directly confused with them.

So we are still labouring under the huge fallacy that the number system can be understood as quantitative in a merely reduced manner, though in truth we can give no meaningful order to this system without qualitative recognition.

And without such qualitative order in an overall context for relating mathematical objects, the quantitative aspect itself can have no strict meaning!

So again the definition in quantitative terms of i is the square root of – 1; 

In holistic terms, – 1 defines the 2^nd dimension. So to indirectly express this in terms of the 1st we again obtain the square root.

What this means in effect is that from a holistic perspective, the imaginary notion relates to the indirect expression (in a linear quantitative manner) of meaning that is properly of a qualitative nature.

If we then apply this to the Zeta 1 zeros, they are all postulated to lie on an imaginary line (through .5).

What this holistically implies, is that all these zeros directly represent a qualitative - rather than quantitative -meaning.   

 

In other words the very nature of the zeta zeros is that they represent an indirect quantitative way of expressing the qualitative (ordinal) nature of the number system as a whole.  

Now the Zeta 2 zeros express this holistic nature more directly (in a circular manner) thus demonstrating the unique qualitative nature of each prime number group (representing a individual natural number members in an ordinal relationship with each other).

The Zeta 1 zeros express this holistic nature more indirectly (in an imaginary linear manner) and demonstrate the unique qualitative nature of the (composite) natural numbers (representing a unique grouping of prime numbers in ordinal relationship with each other).

So put most simply, the zeta zeros represent, from two distinct perspectives, the (hidden) qualitative nature of our number system.

Though this representation is necessarily given an indirect quantitative expression, as its true nature is holistic, it cannot be successfully interpreted in a conventional mathematical manner (that is exclusively concerned with quantitative meaning).

The Holistic Nature of the Number System (6)

I want to illustrate here in a more detailed manner the true distinction as between cardinal and ordinal interpretation of number.

Once again, if we take the number 3 to illustrate in cardinal terms this is treated unambiguously in quantitative terms as representing a collective whole integer. Thus if we wish to break it into constituent units i.e. 1 + 1 + 1 this must be rendered in homogeneous terms (without qualitative distinction).

However if we look on 3 in ordinal terms, it takes on a very distinctive qualitative type meaning based on its relationship with other numbers in a group.

The simplest case would involve a group of 3 members. So the ordinal notion of 3^rd therefore implicitly entails setting up a relationship with the two other members in the group, which thereby can be designated in this context as 1^st and 2^nd.

To do this we must initially fix the position of the 1^st member. Now the fascinating thing about 1^st is that ordinal and cardinal meanings necessarily coincide! In other words with the 1^st member we have by definition no outside context yet (with other members). So in this sense the ordinal notion of 1^st must coincide with the cardinal notion of 1.

Once again this is precisely why the 1-dimensional paradigm employed in Conventional Mathematics ordinal and cardinal meanings coincide! So in effect ordinal notions are effectively reduced in cardinal terms! 

So the fixing of position of the 1^st member in a group implies the cardinal notion of 1.

Now this can be done in any of 3 different ways (as there is no distinction as between the three cardinal units of 3).

This means that in true circular terms we can have three distinctive arrangements of 1^st, 2^nd and 3^rd.

In other words this ordinal approach demonstrates the true interdependence of the group (in a merely relative manner).

Therefore, each of the 3 members of the group can be 1^st, 2^nd and 3^rd (depending on context).

So we have moved quickly here from one extreme to another.

In the cardinal definition, 3 has an unambiguous (linear) quantitative meaning that is independent of other numbers.

In the ordinal definition 3 (as 3^rd) has a merely (circular) qualitative meaning that is intimately dependent on its relative relationship with other numbers (as interdependent).

So depending on context, any of the 3 individual members of the number group can be designated as the 3^rd!

 

In 1-dimensional interpretation, as the ordinal relationship is confined to switching as between one member of a group, effectively it becomes indistinguishable from its cardinal identity.

Thus in 1-dimensional interpretation the cardinal numbers are literally represented as points (drawn at an equal distance from each other) with their corresponding ordinal identities assumed to follow from their cardinal positions.

This equally coincides with the fact that when the dimension of a number is 1, the corresponding (one) root of the number is thereby identical. When the dimension (i.e. exponent) > 1, then the structure of the corresponding roots becomes more complex. So for example when we have 3 roots, these serve, in an indirect quantitative manner, to provide the ordinal relationships between the 3 members of a prime group! 
 

Therefore, once a number is defined with respect to any other dimensional  number other than 1, we then have to draw a clear distinction as between cardinal and ordinal type interpretations with respect to number.

Once again this is of supreme importance with respect to interpretation of the Riemann Zeta Function which of course entails the natural numbers defined with respect to varying dimensional powers (i.e. s).

Now the one dimensional value for which the Riemann Zeta function is undefined is where s = 1.

From a holistic perspective, this is precisely because no distinction can be drawn here as between cardinal (quantitative) and ordinal (qualitative) type interpretations.

So this immediately suggest that - when appropriately defined - the Riemann Zeta Function, via the Functional Equation, relates to the intimate connections as between quantitative and qualitative type meaning. The significance of .5 in the context of the Riemann Hypothesis thereby relates to the condition necessary for the mutual identity of both cardinal and ordinal interpretations.

Therefore, the importance of .5  is that it represents the dimensional value relating to the Zeta 1 Function (with all the non-trivial zeta zeros presumed to lie on the imaginary line drawn through this point).

 

However we can demonstrate an equal remarkable significance to .5 in the context of the Zeta 2 Function (where it now represents a corresponding quantitative value).

As mentioned in the two previous blog entries we can initially define the Zeta 2 Function initially as,

ζ₂(s) = 1 + s¹  + s²  + s³  +….. + s^{t – 1} (with t prime)

And for the zeta zero solutions we set,

ζ₂(s) = 1 + s¹  + s²  + s³  +….. + s^{t – 1}  = 0

The question then arises as to what happens is we attempt to extend the Zeta 2 series in the(conventional) infinite manner!

1 + s¹  + s²  + s³  +…..

As we have seen in the simplest finite case case (the non-trivial second root of 1)

1 + s¹  = 0 with s = – 1.

Thus if we insert this value in the infinite series we get

1 – 1 + 1 – 1 + 1 –……

Now clearly there are only two options here for the value of the infinite series!

If we have an even number of terms then the sum = 0!

If we have an odd number of terms the sum = 1.

As the chance of an even or odd number of terms is similar, then we can say that the probable value of the series = .5.

 

There is a simple identity formula we can use to obtain this value

1/(1 – s) = 1 + s¹  + s²  + s³  +…..

So when we insert the value of s = – 1 on the LHS, the value on the RHS = ½ (i.e. .5)

What is remarkable here, is that the result of the formula for the first of these zeta zeros, strictly represents - not an actual - but rather a probable value!

 

What is even more remarkable is that the probable value of the infinite series for all Zeta 2 zeros is likewise .5.

One way of expressing this is that the sum of all non-trivial roots of 1 = – 1 and when we insert this value in the formula we get .5.

However I will demonstrate it more fully for the two non-trivial roots of the 3 roots of 1 (correct to 3 decimal places),

i.e. the solutions for 1 + s¹  + s²   = 0  i.e. s = –  .5 +.866i and  –  .5 +.866i.

Now there three possibilities with respect to the number of terms in the series here here

3n, 3n + 1 and 3n + 2. (where n = 1, 2, 3, 4,....)

If we have 3n terms the value of series in both cases = 0 (for each root value used). .

However if we have 3n + 1 terms there are two possibilities (for the two root values) with the answer in each case = 1

If we have 3n + 2 terms there are again two possibilities.

For the 1^st root value, the sum of series = 1 –  .5 +.866i  = .5 +.866i

For the 2^nd root value the sum of series = 1  –  .5 –.866i   = .5 - .866i

Thus the sum of these two possibilities = 1

Therefore for 6 different options (i.e. 2 root values applies to 3 different terms of series) the overall expected value = 2 + 1 = 3.

Therefore the average expected value = .5.

So remarkably for all Zeta 2 zeros (and indeed all non-trivial root values for the natural numbers), the expected value (or probable value) of the infinite Zeta 2 series = .5.

And this is what determines the real part of s for non-trivial zeros (in the complementary Zeta 1 series) where,

ζ₁(s) = 0.
 

So what constrains the non-trivial zeros of the Zeta 1 Function to lie on the imaginary line through .5 is the fact that the (probable) value of the Zeta 2 infinite Function for all its non-trivial zeros = .5.
And of course there is a circular relationship involved here, as the very reason why this value of .5 holds with respect to the Zeta 2 Function is because of its application in turn to the Zeta 1 Function.

So correctly understood, in dynamic interactive terms the value of .5 is simultaneously co-determined with respect to both Functions!

 

However what is important to point out is that this value of .5 cannot be interpreted in the standard absolute analytic manner.

In truth it represents a merely probable value. This in turn reflects an inherently dynamic interpretation of the series where both quantitative and qualitative aspects necessarily interact.

It is very reminiscent of the nature of interpretation within Quantum Mechanics.

Indeed at a deeper level, the very behaviour manifested by particles in quantum physical terms is rooted in the (true) inherent dynamic nature of the number system.