Extending Interpretation of the Complex Plane

C: We could say that the zeros are not a solution to the problem, but the problem itself, expressed in a domain that’s more difficult for us to access; the exact nature of this domain then becomes the real focus of interest.

MW: Yes, the zeros are the problem, and thus the problem’s been displaced to somewhere we’re much less familiar with.  Counting, you know...Ancient Greeks and earlier people could count pebbles out on the ground, subdivide them into piles and contemplate different types of  numbers – ‘perfect numbers’, ‘triangular numbers’, prime numbers – and they were able to develop a certain amount of theory.  But that’s just one side of the coin. On the other side, there was no way they could have contemplated the Riemann zeros: (a) you need a theory of ‘functions of a complex variable’, and (b) in order to calculate more than the first handful of them you need a pretty powerful computer. It reminds me of the central image in the film 2001: It’s as if we’ve dug this monolithic thing up, it’s been there for aeons, as a structure it’s overwhelmingly impressive, and everyone concerned is flabbergasted, asking themselves how did that get there, you know: it comes from somewhere else, somewhere beyond, and it induces a sense of almost religious awe.  One suspects that if a mathematical structure underlying or ‘explaining’ the Riemann zeros were to emerge – that is, if in fifty or a hundred years someone comes up with something new which ‘explains’ the zeros in the way the zeros ‘explain’ the primes – then that new structure is just going to open up another even deeper mystery. Paul Erdös, who published more mathematics papers than anyone else ever, and who was primarily a number theorist, said that it’s going to be at least a million years before we understand the primes, and even then we won’t really understand them.

C: Is it a properly transcendental problem, relating to the limits of our thought: the more that we think, the further the problem moves away from us?

MW: Well, again, we don’t know that yet: it may be, but then who knows – maybe it’ll all neatly tie up somehow. But it feels to me that the problem has a quest-like quality.  The fact that the metaphorical image of the Holy Grail has been invoked a few times in the literature, as well as a lot of language poetically invoking the feminine and generally suggesting an ‘otherness’, suggests that I’m not the only one thinking like this.  I’ve had an interesting dialogue with some Jungians about this aspect of RH. The problem of the primes isn’t just different from other mathematical problems, it precedes them.  All other mathematical problems rely on the fact that there are positive integers.  Without the set of positive integers, those other mathematical problems couldn’t exist.  So the problem of the primes is the problem in a sense, it’s beyond the most basic, it’s there before all the others are there.  As soon as you’ve got counting, as soon as you’ve got any notion of repetition, then the problem of the primes is there waiting to be discovered. If we don’t understand the prime numbers, we don’t understand the positive integers.  And if we don’t understand the positive integers, then I don’t know if we understand anything at all, because all science is entirely built on measurement, and you can’t measure anything until you can count.  All our rational scientific thought relies on these very basic ideas of order and counting.   One of the most important quotations that I’ve reproduced on my website is this, from Gerald Tenenbaum (Institut Élie Cartan): As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the  subject of study of arithmetic is not the human mind itself.  From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such  formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant.

So, in probing the mystery of the prime numbers we’re effectively on a sort of journey to the centre of the mind, or of the collective human psyche, and ultimately to the point where that interfaces with the physical world which it finds itself inhabiting.  That quote perhaps best conveys some feeling as to why I’m so gripped by this stuff.

C: The story of the modern theory of primes begins with Gauss’s initial success in predicting approximately the distribution of primes.   How do we get from there to the pioneering interdisciplinary work that your web archive charts?

MW: Gauss – although he didn’t publish, he supposedly got there first – Gauss and Legendre noticed that there was at least a ‘statistical’ thinning out of the primes that you could quantify.  Riemann later uncovered the zeros of his zeta function – the Riemann zeros – and so was able to pin it down much more rigorously.  But there’s a fifty-year gap where...actually, I don’t know what mathematicians felt during that time. Practically, they were trying to refine the approximations; Chebyshev and others improved the approximation of how many primes you’ll find in any given chunk of the number line.  But whether there was an expectation that eventually someone would find a way to make this exact, or whether there was a general feeling that ‘this is the best we’ll ever do’, I don’t know, and I can’t recall seeing anything in the literature of that period where feelings about this matter were expressed.  Once Riemann’s work came along then no one was really interested in what people used to think. The history of mathematical ignorance isn’t as well documented as the history of mathematical discovery. There are some parallels with the situation we’re in now, where there’s a mystery about this proposed ‘Riemann dynamics’, this hypothetical dynamical system underlying the Riemann zeros.

C: The complex plane is the most important mathematical support of RH itself.  And here already a transformation takes place – de Sautoy talks about it as a sort of magic mirror we step through — which seems to unfold things we thought we knew, in a completely different space — as it turns out, very fruitfully for mathematics and the physical sciences alike. There’s obviously something very powerful about the complex plane itself which, at the very least, corresponds in some way to physical reality, and so the fact that it was also the complex plane which facilitated Riemann’s insight into the prime distribution is itself suggestive.

MW: The complex plane appears to have a life of its own.   Complex numbers are absolutely necessary to describe quantum-mechanical phenomena.  Electricians use the complex unit i just to work with AC electricity, so something as ‘nuts-and-bolts’ as the National Electricity Grid depends on the complex plane.  And yet it is this supremely mysterious thing.  I mean, all those fractals that started to circulate in the 1980’s – a lot of people don’t realise what they’re looking at, but those are things that naturally inhabit the complex plane.  Without the complex plane you wouldn’t be able to see such objects, that’s their natural domain.  And then the Riemann zeta

function, with all its strange properties; Riemann’s big step was to take a function which Euler had looked at and ask, what would that do if we extended it into the complex plane?  And what it was found to do then spawned the great mystery of the Riemann zeros. Another strange thing worth mentioning: One tends to think of temperature as existing on a linear scale, a one-dimensional scale.  But in statistical mechanics, by constructing a function of temperature, the ‘partition function’, and extending it out to the complex plane, you find that it has a set of ‘singularities’ , off the familiar real number line, in this other two-dimensional region that doesn’t seem to have anything to do with temperature or any other aspect of practical measurable physical reality. Yet these singularities correspond to phase transitions of the system.  Without the complex plane you’d never have known they were there.  The same thing happens with the zeta function, it’s got a set of singular points in the complex plane, the Riemann zeros off the real line.  From the behaviour of the zeta function on the real line, you would never have guessed they were there.  Various people have put forward models of two dimensional time – imaginary time certainly gets used, complex time.  Such models can be used in attempts to explain otherwise inexplicable phenomena, but none of this can be applied to our normal experience of reality, you can’t really do anything with it.  I would say that the complex plane is still deeply mysterious.  It’s ‘behind the scenes’ of reality as we experience it.

C: And historically, complex numbers had been discovered long before there was any sense of their ultimate utility.  Only later did it became evident that something which seemed to have been a mathematical fiction, was hugely important to work in these fields.

MW: Absolutely, the word imaginary, you know – you’ve got the ‘real’ numbers and the ‘imaginary’ numbers – it’s a very unfortunate name, but it’s simply because of the history of the thing.  For quite a while, no-one thought these things had any ‘reality’ to them, primarily because they didn’t correspond to anything experiential in the way ‘real’ numbers were seen to.

C: It’s difficult to ignore this experimental evidence that complex numbers relate to something in reality: we have to take account of these things which just impress themselves upon us.  The traits of the complex plane are obviously real, but they don’t correspond to any actual object, any actual thing we can get hold of.  They’re distributed through reality itself.

MW: Yes, the system of complex numbers is there, I don’t know ‘where’ it is, but it’s not just something we invented. And, interestingly, it’s most directly evident at the subatomic level.  As I said, the theory of AC electricity relies on it, but then ultimately that’s a quantum mechanical phenomenon, scaled up to the level where we

can, say, run a toaster on it.  Functions of a complex variable get used in statistical mechanics, aerodynamics, etc., but those are fairly indirect manifestations of something very deep, I feel.  The fact that the complex plane relates so closely to quantum mechanics means that in macroscopic reality, it permeates everything, as you say, and yet nobody had a clue it was there until relatively recently. Even after it had been mathematically brought into consciousness it was still seen as just a fiction.  As for the primes, you can’t understand the distribution of primes until you’ve grasped the Riemann zeros.  And the Riemann zeros live on the complex plane, inarguably.  The ‘nontrivial’ zeros, the ones RH concerns, inhabit a narrow vertical strip in the complex plane.  The RH simply says that they all — the entire infinite set of Riemann zeros — lie on the ‘critical line’ which runs up the middle of this ‘critical strip’. Now, to prove RH would be an exact mathematical task, so RH gets a lot of press – there’s the whole fame-and-fortune thing, literally a million-dollar prize, this idea of something like winning the ultimate intellectual gold medal, you know – but you’ve either done it or you haven’t, it’s very clear-cut.  But I’m more interested in the less clear-cut questions – what are the Riemann zeros, from where do they originate? To answer this we may need something else as new and unexpected as the complex plane was when it was first introduced, something we haven’t thought of yet, a new mathematical ‘environment’ in which these things will become perfectly clear.   But that may well lead to another body of questions which are even more baffling. But “from where do the zeros originate” – what does that mean?  They’re seemingly vibrations of something, but what?  What is that thing going to be – is it going to be a mathematical model of a dynamic system that people may or may not be able to physically manifest?  If it is possible to physically manifest it and someone does...what then are we confronted with? One gets a very strong feeling that until we understand the what the zeros ‘are’, we won’t be in a position to prove RH.  These two issues are tied together. But the former isn’t yet a precise question, whereas ‘is the RH true’ is.

PC As you say, none of our present discussion would be possible without the complex plane that provides a new environment which the Riemann zeta function can properly inhabit. This then entails the Riemann Hypothesis with its contention that the zeta zeros all lie on an imaginary critical line.

However from a holistic perspective, this requires that one can suitably interpret the meaning of the complex plane. And when this is done it can be readily appreciated that the complex notion of number (with real and imaginary aspects) is intimately associated with everyday life. So this massive failure of realisation is once again rooted in a limited interpretation of number (where the holistic aspect is reduced in analytic terms).

So with the complex plane we have both positive and negative axes in real and imaginary terms.

I have already gone some way to explaining the holistic notion of a negative number.

When numbers are understood in the conventional rational manner, they are thereby literally posited in experience giving them an independent identity. The unconscious (intuitive) direction arises through corresponding negation, which causes a dynamic switch in polarities of understanding e.g. from recognition of number as an external object to its corresponding internal perception. Depending on the degree of interaction, rather like the fusion of matter and anti-matter particles in physics, the negative direction then combines with the positive in the generation of a psycho-spiritual energy state i.e. intuition. And it is this intuitive appreciation that directly provides knowledge of the interdependent shared aspect of number.

In conventional quantitative terms the imaginary number i is defined as the square root of – 1. In corresponding holistic terms i relates to an attempt to express the 2-dimensional notion (where the fusion of opposite positive and negative polarities takes place) in a linear rational fashion. So this reduction from 2-dimensional to 1-dimensional understanding constitutes the inverse holistic equivalent to that of a square root.

Therefore we can express in a more succinct fashion the imaginary notion i, as the indirect analytic (conscious) representation of the holistic (unconscious) notion of 1.

Expressed in an equivalent manner, the imaginary notion represents the indirect rational expression of the qualitative aspect of number, or alternatively, the indirect representation in an independent quantitative manner of the qualitative notion of interdependence.

And when appreciated in this fashion, it can be seen that the imaginary notion necessarily pervades all experience of reality.

In scientific terms reality is reduced to what can be interpreted through the use of reason. So what is deemed real therefore directly correlates with such rational interpretation.

However at a deeper level we are aware that experience of reality also entails a holistic aspect relating to the unconscious quest for meaning.

For example, if one purchases a new car, clearly this can be given a rational conscious identity; however the car may equally serve a vaguer holistic desire as in some way enabling one to achieve a greater sense of fulfilment.

So therefore when we identify the local conscious aspect of this purchase in direct terms as “real”, then the unconscious holistic aspect is indirectly of an “imaginary” nature.

Because of the dominance of the (real) conscious aspect in our culture the corresponding imaginary (unconscious) aspect is frequently misinterpreted, where it is blindly projected on to conscious phenomena. And this is another area that is dealt with very well in the Jungian literature!

A purer form of imaginary understanding would relate to a key archetypal notion that is understood in a transparent spiritual fashion. In this case, though some localised symbol may indeed be involved in mediating the archetype, its holistic meaning is not rigidly identified with the symbol.

This qualitative notion of the imaginary intimately applies to all mathematical understanding (which necessarily entails the interaction of conscious and unconscious). However, as we have seen, the holistic (unconscious) aspect is completely ignored with respect to formal interpretation and reduced in a blind manner to its analytic counterpart.

When reading “The Emperors New Mind” by Roger Penrose some years ago, I was struck by the manner in which he repeatedly referred to the magic of complex numbers. By this he implied that complex numbers so often possess remarkable holistic properties. For example, when an analytic function is defined in one small region of the complex plane it can thereby be defined for the whole plane.

However though at one level such holistic properties are indeed recognised, in formal terms, mathematicians persist in attempting to understand complex numbers in a merely analytic (quantitative) manner. This then leads to a great lack in philosophical appreciation of what the imaginary truly entails. 

As you rightly say, the imaginary notion plays a key role in quantum mechanics and a little reflection should show clearly why this is the case.

At the macro level of reality objects appear to possess a rigid independent identity, which then becomes especially amenable to the quantitative rational approach.

However the sub-atomic level of reality is very different, where particle interactions are inherently dynamic, with the independence of individual particles increasingly difficult to identify.

At this level, individual particles have strictly no meaning divorced from a web of interdependence with other particles.

Because in effect the overall holistic aspect of particle interactions is now so prominent, their interpretation becomes amenable to complex notions involving the use of imaginary as well as real aspects.

However though the imaginary notion can now be employed to great effect, it remains confined to mere quantitative interpretation in physics.

Thus the entire field of quantum mechanics remains deeply-non intuitive with respect to present scientific understanding, as the constructs used are confined solely to analytic as opposed to holistic interpretation.

And this problem cannot be properly addressed without the holistic aspect of interpretation becoming fully incorporated in mathematical understanding.

You also refer to the manner in which fractal images can be generated from a simple function defined with respect to the complex plane.

The beauty of these images does not arise from consideration of their component parts considered in an independent manner. Rather, their remarkable quality arises from the wonderful holistic relationship involved (entailing the interdependence of these parts). And this is why the generating function is defined with respect to the complex plane (including the imaginary part of number).

You also mention how with partition functions, phase transitions of the system involved. can naturally occur as singularities on an imaginary axis.

A phase transition occurs when one state turns into another (e.g. water into ice).

So therefore the point of transition embraces both states. Whereas such recognition cannot be identified with real dualistic understanding (based on clear separation) it can however be directly associated with holistic intuitive understanding (where their mutual interdependence is directly appreciated).

And once again the imaginary notion entails the attempt to represent such holistic understanding indirectly in a rational manner.

   

So properly appreciated, all experience of reality is necessarily complex combining both real (analytic) and imaginary (holistic) aspects.

One important implication of this is with respect to our customary understanding of physical dimensions. In fact our experience already necessarily entails imaginary as well as real space and time. And these dimensions are intimately related to the holistic notion of mathematical dimensions (that indirectly are expressed through the quantitative roots of unity).

For example from a holistic 4-dimensional perspective, we live in a reality where both space and time have four directions (that are given by the axes of the complex plane).

So for example the interaction of external and internal polarities gives rise to two real dimensions of space and time that are relatively positive and negative (and negative and positive) with respect to each other.

Then the interaction of the analytic and holistic aspects of experience equally gives rise to these two directions being real and imaginary (and imaginary and real) with respect to each other.

At a deeper level, these dimensions relate to the manner in which both external and internal and whole and part aspects of phenomena, dynamically interact in experience (in both physical and psychological terms). 

Jung came very close to this interpretation in the way that he dealt with his four functions. So he identified two functions i.e. thinking and feeling as rational and the other two, sensation and intuition as irrational respectively (with each pairing constituting in dynamic terms complementary opposites). This thereby entailed that when one function – say – thinking is unduly dominant with respect to conscious experience that the opposite function of feeling remains largely unconscious, typically entering consciousness in an involuntary manner (generally in the form of blind projections).   

Following extensive reflection on his approach, I came to the realisation that what Jung termed rational and irrational corresponded better to the holistic notions of real and imaginary respectively. So thereby, the four functions (with minor modifications) constitute – relatively – real and imaginary polarities in positive and negative terms.

Jung based his theory of 8 Personality types on these four functions. This theory was then later extended to 16 types in the Myers Briggs Type Indicator.

Some time later directly using the holistic mathematical approach, I discovered 8 missing types thus bringing the total to 24. The basic rationale here was to define each personality as comprising four components (akin to the four functions) related to the four extremities of the complex plane, varying along a spectrum from the most dominant to the weakest inferior component.

So just as with four letters we can have 24 distinctive permutations, likewise all possible arrangements of the four key complex components would lead to 24 personality types.

It then occurred to me that each personality type could be validly viewed as representing a unique manner in which space and time dimensions are experienced.

This then led on to an unexpected complementary connection with the physical world of strings, which in the earlier bosonic version, was postulated to vibrate in 24 dimensions.

It is unrealistic to view each of the four dimensions in a separate manner at the level of strings. Rather in this context, each dimension can be better appreciated as representing a unique configuration of the four dimensions (that properly separate at the macro level of reality).

So from this perspective, just as we are enabled to define 24 personality types with respect to psychological behaviour, equally we are enabled to define 24 “impersonality types” with respect to physical reality. So again each (asymmetrical) vibration of the string therefore entails a unique dimension (of space and time)

Other versions of string reality have since emerged.

The original bosonic version was based on 26 dimensions (where two additional dimensions were added). This was then reduced to 10 (with 8 unique asymmetrical vibrations of the string and again two dimensions added). And a newer version associated with M-theory leads to an additional dimension so that we have 11 dimensions in all. However the same basic point holds, that all these models, which seem non-intuitive in terms of our customary understanding of space and time, can best be understood as representing varying entangled configurations of the original 4 dimensions.

So I am using this one example to demonstrate how the holistic mathematical approach can open up completely new and unexpected ways to look at reality, especially in forging complementary links as between the physical and psychological domains.  

As you rightly suggest, proper interpretation of the Riemann zeta function is awaiting a new more comprehensive framework.

Euler started with the zeta function that was defined in an analytic (quantitative) real manner.

Then Riemann extended the interpretation of this function over the complex plane entailing both real and imaginary aspects in an analytic (quantitative) manner.

So the obvious next step is now to extend this function over the complex plane by defining real and imaginary notions in terms of both analytic (quantitative) and holistic (qualitative) aspects of interpretation.

And in doing this, appreciation of the Riemann zeta function is utterly transformed.

I remember earlier on in my investigations being initially puzzled by the fact that for values of the function, where the dimensional number s ≤ 0, results appear deeply non-intuitive from the customary quantitative perspective.

For example for the first of the trivial zeros (where s = – 2), we apparently obtain through the Riemann zeta function,

1² + 2² + 3² + 4² + …    =  1 + 4 + 9 + 16 + …  = 0.

In conventional terms, one would give the value of this infinite series as ∞. So this leaves the considerable problem of how to properly explain why the alternative result arises?

And I found that all the technical mathematical explanations relating to analytic continuation, meromorphic functions and differing domains of definition failed to satisfactorily address the issue.  

Then to my delight I was able to directly use some of the holistic mathematical notions I had been developing to provide the requisite answer.

I have already dealt with the holistic nature of 2-dimensional understanding as the complementarity of positive and negative poles. In the contemplative literature, this is identified with the commencement of nondual understanding.

However frequently as the spiritual aspirant progresses, such understanding can become to a degree reduced to dual symbols with consequent possessive attachment arising.

So to fully erode such attachment, St. John of the Cross speaks of the need for “passive nights” where through sustained deep immersion in the unconscious regions of personality, such confusion can thereby be radically negated.

St. John interestingly states that the purpose of this cleansing is nada. Nada, which literally means nothing, can be validly interpreted as relating to a purely intuitive state of awareness (free of phenomenal attachment).

Interestingly when I was at Primary School in Ireland, the mathematical symbol for zero i.e. 0 was referred to as nought (or nothing).

So as I reflected on the first of the trivial zeros, I realised that 0 in this context of the Riemann zeta function refers directly to its holistic – rather than analytic – expression. And again in a holistic context, – 2 would imply the negation of any rigid phenomenal notion associated with 2-dimensional understanding, so that it fact it would now become purely formless and intuitive.

Thus the first of the trivial zeros results in a pure energy state from a psycho spiritual perspective. This implies a complementary energy state at the physical level, which would be represented well by the interaction of matter and anti-matter particles. 

Indeed the next of the trivial zeros at s = – 4, would apply in psychological terms the negation of any rigid element associated with imaginary as well as real understanding.

So this purer type of intuitive energy would entail for example a significant mastery with respect to involuntary projections (where the imaginary aspect is manifest).

Interestingly the physical equivalent to this would entail the energy generated from the clash of matter and anti-matter particles of a virtual kind that are extremely transient in nature.

Thus likewise in psychological terms, it becomes extremely difficult to sustain the intuitive psycho spiritual energy states in experience relating to the trivial zeros further out from zero on the negative real axis.

So again we have the two extremes of the absolute rational approach to number associated with the analytic aspect and the corresponding purely relative intuitive approach associated with the holistic aspect respectively.

As we know every value of the Riemann zeta function for s > 1 can be given an analytic (quantitative) interpretation that corresponds to our customary intuitions regarding number.

Then through Riemann’s reflection formula, all these values can be matched (using a symmetry line drawn .5 on the real axis) with corresponding values of the function for s ≤ 0. 

These latter values then appear nonsensical in terms of our customary intuitions and the reason for this is that they refer directly, in relative terms, to a holistic (qualitative) interpretation of number.

So properly interpreted, in a truly wonderful manner, the Riemann zeta function is now seen to show how the analytic and holistic aspects of number dynamically interact with each other throughout the entire complex plane.

Once again we map values for the function with the customary analytic interpretation on the positive real right hand axis with corresponding holistic values (through Riemann’s reflection formula) on the left.

Then for both analytic and holistic interpretations to coincide, the non-trivial zeros of the zeta function – by definition – must lie on the critical imaginary line through .5.

Therefore in a truly illuminating manner, these zeta zeros therefore serve as the condition for fully reconciling both the analytic (quantitative) and holistic (qualitative) interpretations of number. This implies for example (assuming the truth of the Riemann Hypothesis) that if all the zeros do indeed lie on the imaginary line, that the reduced quantitative approach which characterises conventional mathematical interpretation can operate in a consistent manner. Expressed another way, it provides the means for fully resolving the tension as between addition and multiplication.

Thus the key point regarding the Riemann zeta function is that for proper comprehension it must be interpreted in a dynamically interactive manner (entailing both the analytic and holistic aspects of number).

Indeed this point can be demonstrated in a startling fashion.

As we know, in analytic terms there is only one point where the Riemann zeta function cannot be defined and that is where s (representing the dimensional power to which the natural numbers in the function are raised) = 1.

Then in corresponding holistic terms, this implies that the only interpretation where the Riemann zeta function is undefined is 1-dimensional (which represents the accepted mathematical approach).

What this entails is that the number system is inherently dynamic by its very nature (entailing the interaction of analytic and holistic aspects). Therefore it cannot be properly understood in a merely absolute analytic manner (where the holistic aspect is not recognised).

So from a holistic point of view, conventional mathematical interpretation (with the dimension = 1) represents a unique limiting case, where the qualitative aspect is completed reduced to the quantitative, thereby avoiding all dynamic interaction with respect to number.

However associated with every other dimensional number (≠ 1) are alternative methods of dynamic mathematical interpretation (each of which has a partial relative validity).

So the potential scope of mathematics is immeasurably greater than we can imagine.

This new enlarged framework for the Riemann zeta function can also convincingly show why no proof (or disproof) is possible in the standard conventional manner.

As we have seen when we multiply numbers a shared qualitative resonance is thereby involved (which cannot be reduced in an analytic fashion). In psychological terms, this corresponds to a unique intuitive energy state; in complementary physical terms, it can then be also uniquely identified with a physical energy state.

Perhaps this can be even more easily understood in geometrical terms. For example, when we express 2.3 in geometrical terms we obtain 6 square units.

So therefore to make the assumption that 6 can now be treated as a number on the real line, we have to ignore the change in the qualitative nature of the units brought about through multiplication. So we thereby express the qualitative notion of two dimensions in a linear (1-dimensional) manner, Thus, when we place the composite natural numbers on the real number line, we are thereby reducing these numbers in a merely quantitative fashion.

Thus the bigger question now arises as to what justifies the treatment of multiplication in this reduced manner!

So we start with the assumption that all the natural numbers can be treated in independent quantitative terms as existing on the real line.

The Riemann hypothesis in fact is the expression of the unrecognised holistic aspect (which indirectly occurs in an imaginary linear fashion). And remember again that the imaginary serves the appropriate way for expressing the notion of the holistic interdependence of number indirectly in an analytic manner!

Therefore the requirement that all the natural numbers lie on the real line entails that all the Riemann zeros lie on a corresponding imaginary (shadow) line drawn through .5 on the real axis. The significance of .5 in a holistic context relates directly to the fact that proper comprehension of the zeros equally implies a dynamic balance in experience as between the external and internal aspects of number. Indeed it is through maintenance of this balance that the genuine notion of number interdependence (relating to its shared aspect) is realised.  

Then when this balance is achieved we can recognise how the qualitative aspect of number behaviour is indeed consistent with its corresponding quantitative aspect.

However conventional mathematical interpretation necessarily starts with the assumption that all the natural numbers lie on the real line.

Therefore it must already implicitly assume the truth of the Riemann Hypothesis.

Thus there is no way that one can satisfactorily prove a result that is already implicitly assumed in the starting axioms used to obtain that result.

One could also say that the Riemann Hypothesis must be true if the natural numbers lie on the real line. However because of the necessary interdependence of both real and imaginary lines involved, the truth of one implicitly involves the assumption that the other is true (and vice versa).

Therefore one cannot prove or disprove the Riemann Hypothesis in the conventional mathematical manner.

So properly understood, underlying the consistency of the whole mathematical edifice is a massive act of faith. For if the Riemann Hypothesis is not true then our belief in the consistency of the real number line is not strictly justified and with that the consistency of every mathematical result that has ever been obtained is in doubt.

One may suggest that that finding of a zeta zero off the critical imaginary line would indeed disprove the Riemann Hypothesis in a conventionally acceptable manner.

However the true position here is quite subtle.

Certainly it would disprove the Riemann Hypothesis. However it would not disprove it in the conventional manner.

Because the Riemann Hypothesis is already assumed in the conventional acceptance of the real number line, a zero found off the critical line, would imply that we can no longer accept the reduced notion that all natural numbers (as products of primes) lie on the real number line. This would then undermine our belief in the consistency of any conventional proof (or disproof) based on such an assumption.

From this perspective, the truth of the Riemann Hypothesis can be seen as a mathematical sine qua non with respect to all conventional proof. However it cannot itself be proved (or disproved) in the accepted conventional manner.

  

I believe due to a continual failure to solve the Riemann Hypothesis, the mathematical community will eventually waken up to the fact that there is something seriously wrong with the existing interpretation of number. Then eventually when the true dynamic interdependence of the number system gradually becomes better appreciated, it will become apparent as to why the Riemann Hypothesis cannot be solved in the conventional manner.

Though this may appear as a failure in some respects, with a revolutionary new understanding, the true scope of mathematics will be immeasurably increased, quickly opening up many exciting new domains that presently appear quite unimaginable.