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Peter 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.

THE PROBLEM WITH MATHEMATICAL PROOF

Lack of an Integral Dimension

Peter Collins

In a previous article I indicated how lack of the qualitative dimension in Conventional Mathematics has prevented recognition of the true nature of the Riemann Hypothesis. This latest contribution shows how the same lack of the qualitative dimension has led to a reduced notion of mathematical proof.

Introduction

There is a central problem with mathematical proof which ultimately has far reaching consequences for the nature of both mathematics and science.

This problem relates to a fundamental confusion of the finite and infinite. Properly speaking these are qualitatively distinct notions relating to unique aspects of understanding. However because Conventional Mathematics is formally based on a merely (linear) rational mode of interpretation, it thereby reduces - in any relevant context - infinite to finite meaning. [1]

To show more clearly what is involved here I will illustrate with respect to the well-known Pythagorean Theorem, which states that in any right angled triangle, the square on the hypotenuse is equal to the sum of squares on the other two sides.

However when we probe into the assumptions behind this proof (and by extension any conventional mathematical proof) we find that a crucial uncertainty principle is at work.

Now the general proof of the Pythagorean Theorem has application to “all” right angled triangles.

However the use of “all” can be seen to be highly problematic.

In its true holistic sense the infinite is empty of phenomenal form with a merely potential meaning (applying in this case to all unspecified right-angled triangles).

However when we then attempt to identify this potential meaning of the infinite with the actual finite existence of such triangles, an inevitable uncertainty principle arises.

In finite terms the collective meaning of “all” is ambiguous as the very process of determining specific cases always requires that other cases remain undetermined.

Thus, no matter how many finite examples we identify to illustrate our “proof” an inexhaustible number of unidentifiable examples must necessarily remain.

Conventional Mathematics attempts to avoid this problem through a basic reductionism whereby the correct potential meaning of the infinite is then collapsed so as to be directly identified with the finite in actual terms.

This then leads to the misleading notion that somehow as a finite number becomes larger it approaches the infinite in size. Strictly speaking however this is meaningless representing a direct confusion of a quantitative with an inherently qualitative concept. And such reduced thinking lies at the heart of mathematical proof!

I do not question for a moment the great value of Conventional Mathematics. However my point here is that it represents but a limited special case of a much more comprehensive mathematical appreciation that properly integrates both quantitative and qualitative type concepts.

All mathematical understanding is rooted in experience (which is of a dynamic interactive nature). And with such experience the twin processes of differentiation and integration are necessarily involved.

Differentiation is associated with the conscious aspect whereby finite analytical distinctions can be made with respect to phenomena of form.

Integration by contrast is associated directly with the unconscious aspect whereby holistic connections (which are strictly infinite and empty of form) can likewise be made. Thus through the interaction of differentiated and integral aspects in experience, a continual transformation can thereby take place.

Such a transformation likewise applies to all mathematical processes.

However because Conventional Mathematics in formal terms seeks to ignore the holistic integral aspect altogether (through reducing qualitative to quantitative meaning) it thereby misleadingly offers a somewhat static interpretation of truth in absolute terms. And this is especially evident in its treatment of mathematical proof!

To move to a more satisfactory explanation of what is involved in mathematical understanding we need to explicitly recognise that quantitative and qualitative appreciation relate to two distinct modes of understanding both of which must be equally incorporated in interpretation.

Again the quantitative aspect is directly associated with the differentiated (analytic) aspect of understanding; the qualitative aspect by contrast is associated with the integral (holistic) aspect.

Whereas the quantitative is appropriated through conscious reason, the qualitative in direct terms is provided by unconscious intuition. However such intuitive appreciation can then be indirectly expressed through circular type reason (which is paradoxical in terms of standard logic).

Both the quantitative and qualitative aspects of mathematical understanding can be given a specialised expression.

Conventional Mathematics in its present form represents an extremely specialised development of the quantitative aspect; however Holistic Mathematics is equally designed to provide the specialised treatment of its qualitative counterpart.

Then in its most comprehensive interactive expression, the mature appreciation of both specialised quantitative and qualitative aspects can be combined in - what I refer to as - Radial Mathematics.

So properly understood, we should have three great branches of mathematical activity (Conventional, Holistic and Radial). However because of the present unbalanced emphasis on merely quantitative meaning, little or no recognition exists within the mathematical community of the latter two branches.

Some Uses of Holistic Mathematics

Before proceeding further I will illustrate briefly here some of the important uses that I personally have discovered through the exploration of Holistic Mathematics.

For example with respect to the (entire) spectrum of development, Holistic Mathematics has a remarkable potential relevance with respect to the precise scientific encoding of all stages. Indeed the underlying structure of each stage is mathematical (in this qualitative sense).

Likewise I have already referred to both linear and circular reason both of which must be employed in a more comprehensive mathematical interpretation. The (straight) line and the circle are represented with only minor modifications by 1 and 0 respectively. So in holistic mathematical terms, comprehensive mathematical - and indeed all scientific -understanding requires a qualitative binary system as a means of encoding transformation processes. This complements the quantitative binary system that digitally encodes information. Now it would be readily accepted that the use of a quantitative unary system (based on 1) would be highly inefficient as a means of encoding information. In like manner the use of a qualitative digital unary system (based on linear understanding) is highly inefficient in terms of encoding transformation processes. And Mathematics and Science - as we know it - is based on such a unary system!

In the true qualitative sense, a number (expressing a dimension) has an important holistic interpretation as a general means for interpreting mathematical symbols.

So linear reason which informs conventional mathematical interpretation, in qualitative terms is based on 1 as dimension (in what is literally 1-dimensional understanding).

However the important implication here is that every number has a corresponding qualitative significance as a coherent general means of overall interpretation.

So this leads directly to the remarkable finding that potentially we have an unlimited number of possible logical interpretations of mathematical symbols (all of which enjoy a certain partial validity).

So Conventional Mathematics, in being so firmly rooted in linear type logic as the only valid type, thereby excludes all other possible interpretations!

This 1-dimensional interpretation represents but an important special case where in formal terms, qualitative is totally reduced to quantitative meaning.

However for every other number dimension, interpretation entails a certain unique dynamic configuration of both quantitative and qualitative meaning.

Reformulating Mathematic Proof

Once we accept that mathematical symbols equally possess both quantitative and qualitative aspects (which are distinct) then the inevitable corollary is that both of these aspects must be provided in a comprehensive proof.

There are great similarities here with the nature of quantum mechanics. Physicists now recognise that at the sub-atomic level, phenomenal interactions entail both wave and particle aspects. Indeed even at the macro level of everyday experience, phenomena possess wave effects. However these are so small that in practical terms they can be ignored with objects considered as having an independent (particle like) existence. [2]

Likewise in Mathematics objects such as numbers have both particle (quantitative) and wave (qualitative) aspects. However at the conventional level of mathematical understanding, the qualitative aspect is simply ignored with numbers understood with respect to their independent quantitative characteristics.

Due to quantum mechanics, it is now understood that macro appreciation of phenomena represents but a convenient approximation that breaks down at the sub-atomic level. Likewise conventional mathematical appreciation represents but a convenient approximation that breaks down at a more dynamic interactive level of understanding.

And just as the Uncertainty Principle applies with respect to quantum mechanical behaviour, equally a corresponding Uncertainty Principle applies with respect to the true nature of mathematical proof.

So properly understood, every mathematical proof combines both quantitative and qualitative aspects (that are relatively distinct).

When we concentrate merely on the quantitative aspect in an attempt to approximate absolute type verification (as with Conventional Mathematics), this blots out recognition of the equally important qualitative aspect.

In reverse fashion when we concentrate merely on the qualitative aspect again with a view to absolute type appreciation (Holistic Mathematics) this blots out recognition of the quantitative aspect. [3]

So any mathematical proof in practice is of a merely relative approximate nature (entailing an inevitable compromise as between quantitative and qualitative type appreciation). [4]

Now this might come as a great shock to those trained to believe in the absolute nature of mathematical proof.

However momentary reflection should show that in practice this is not the case.

In an experiential sense, mathematical proof represents but a special form of social consensus. Even in the case of a well established “proof” a degree of uncertainty always applies with respect to the communication of such a truth with no two people understanding in exactly the same manner. [5]

Also, as demonstrated in the famous case of Andrew Wiles' proof of Fermat's Last Theorem, it is possible for a false consensus to develop. So when Wiles' proof was first announced in 1993, it was generally accepted as true. It was only later that a crucial flaw in reasoning was pointed out which Wiles himself initially had difficulty in accepting. Happily, with the help of a student he was subsequently able to rectify this problem in 1995. So as time goes by with no further errors arising we can accept with an ever greater degree of probability that Fermat's Last Theorem has indeed been proved (from a conventional mathematical perspective). However this truth is still strictly of an approximate nature (in similar fashion to the belief in independently existing objects).

However the criticism I am making is of a much more fundamental nature in that all accepted proof is necessarily based on a key reductionism (whereby qualitative is reduced to quantitative meaning).

So, just as conventional understanding of physical objects is no longer adequate at the more dynamic interactive level sub-atomic level of reality, likewise conventional understanding of mathematical proof is no longer adequate at the more interactive intuitively inspired appreciation of higher stages of reality (where quantitative and qualitative aspects are combined).

The Pythagorean Issue

Modern Mathematics is now amazingly specialised geared exclusively to quantitative type interpretation of its symbols.

Though this has indeed enabled enormous progress with respect to one valid aspect of enquiry, it has unfortunately blotted out recognition of an alternative qualitative aspect.

However it was not always this way. Indeed an important school associated with the Pythagoreans existed some 2,500 years ago where it was explicitly recognised that mathematical symbols possessed both quantitative and qualitative meaning.

We have already mentioned the famous theorem associated with the Pythagoreans. However it was this same theorem that was to lead to a crucial problem that they could not resolve.

Now the Pythagoreans adopted a worldview whereby they believed that a direct correspondence existed as between quantitative and qualitative aspects of meaning. In their scientific investigations they were already making good use of the rational method (which we still employ so fruitfully today). This qualitative method then seemed to correspond with the nature of mathematical quantities, which they believed were rational i.e. that could be unambiguously expressed as fractions.

However the simplest possible case of the Pythagorean triangle was to shatter this belief for when both the opposite and adjacent lines = 1, the hypotenuse = the square root of 2 (which is an irrational quantity).

At one level it was remarkable that the Pythagoreans were able to prove in quantitative terms that the square root of 2 was indeed irrational. However the problem was that they could not produce a corresponding qualitative “proof” of why it is irrational.

They could sense that some qualitative transformation took place through the squaring of a number but unfortunately could not successfully explain why this would lead to the subsequent generation of an irrational number.

In fact we can illustrate the truly linear nature of present mathematical activity in this context.

Because mathematics was still firmly rooted in experience at the time of the Greeks, they tended to associate numbers with physical lengths.

They would have appreciated easily therefore that a qualitative transformation takes place through squaring a number.

For example if we start geometrically with a line length of 1 unit, then when we square the number, we would represent this geometrically with a square (with each side 1 unit).

Now clearly the square is of a qualitatively different nature from the line. The square belongs to a 2-dimensional reality, whereas the line is 1-dimensional.

However subsequently in Mathematics this important distinction has become entirely lost through attempting to treat numbers as abstract objects (unrelated to experience).

So in conventional terms when we square 1, the result remains unchanged as 1.

Now strictly speaking though the quantitative aspect remains unchanged, a qualitative transformation in the nature of units necessarily takes place.

However this important qualitative change is totally ignored in conventional mathematical interpretation with the qualitative aspect thereby reduced to the quantitative.

So in a very precise manner we can see how Conventional Mathematics is based on linear rational understanding whereby all numerical values are ultimately expressed with respect to a reduced 1-dimensional quantitative interpretation.

When I first started to seriously engage with Holistic Mathematics, I could see that it was important to resolve The Pythagorean Dilemma - as I refer to it - by providing the qualitative aspect of this proof.

So therefore in terms of a more comprehensive understanding, the proof that the square root of 2 is irrational, requires matching quantitative and qualitative aspects. [6]

In his “Elements” Euclid provides an ingenious proof demonstrating in quantitative terms how the square root of 2 is irrational and it is likely that the Pythagoreans would have used a similar approach.

This proof commences by assuming that the square root of 2 is a rational number a/b (with no common factors). Then by the process of squaring and use of a simple substitution, he establishes that both a and b are even numbers (that can be divided by 2). So the original assumption regarding a/b is thereby contradicted.

The clear implication of this proof is that a qualitative transformation in the nature of a number is entailed through converting from 1-dimensional to 2-dimensional format.

The task therefore of the qualitative aspect of proof is to show the precise nature of this transformation!

This latter aspect likewise proceeds through a process of negation (i.e. contradiction).

All experience is based on the interaction of fundamental polarities such as external and internal. The conscious (differentiated) aspect of experience requires that such polarities be clearly separated. In this way we have external objects that are considered independent of the internal interpreter. Conventional mathematics then represents an extreme example of this aspect whereby its objects are considered to have an absolute external identity (that is not altered through internal interpretation).

In qualitative terms this process represents the direct positing (+) of mathematical symbols.

However an integral aspect is also necessarily involved in experience whereby what was separate and independent (in terms of conscious understanding) is now - literally - seen as complementary and interdependent (in terms of holistic unconscious appreciation). In direct terms this relates to intuitive recognition which indirectly can be rationally expressed in terms of paradox.

As is well-known in physics, when matter and anti-matter particles are combined (in sub-atomic interactions) a fusion takes place in the form of physical energy.

Likewise in psychological terms when the opposite polarities of phenomenal understanding are combined, a qualitative fusion likewise takes place in the form of spiritual energy i.e. intuition.

Like the role of oil in a car engine it would not be possible for reason to function without intuitive recognition. However, though vitally important for all understanding it is given no formal recognition in Conventional Mathematics where it is reduced to mere rational interpretation.

So the very process through which intuition is generated in experience requires the dynamic negation (–) of rational type understanding. [7]

So in terms of rational understanding, holistic intuition is of a paradoxical nature.

Now the higher levels of understanding on the psychological spectrum are associated with the significant specialisation of a nondual intuitive type contemplative vision (that literally contradicts dualistic interpretation).

The first of these levels - which for convenience we can refer to as the psychic/subtle realm - corresponds closely with two-dimensional understanding in holistic mathematical terms.

So here we have the interaction of both a developed rational and intuitive type capacity. This leads qualitatively to “irrational” appreciation (which fully complements the nature of irrational quantities).

An irrational number such as the square root of 2 combines in its very nature both finite and infinite aspects. In finite terms it can be approximated as a discrete rational quantity (to any required degree of accuracy). So correct to 4 decimal places the square root of 2 is 1.4142. However it also contains a continuous infinite aspect in that its decimal sequence has no pattern and is without a finite limit.

In qualitative terms the appreciation of all symbols (including mathematical) at the 2-dimensional level of understanding is likewise “irrational”. [8]

Though one can still differentiate phenomena in a finite rational manner, they now likewise possess a transparent numinous quality as holistic archetypes of an infinite unseen reality.

The clear implication is that with the comprehensive mathematical approach (combining both quantitative and qualitative aspects) the irrational nature of the square root of 2 cannot be understood in merely 1-dimensional terms. [9]

To be properly appreciated in dynamic interactive terms, the quantitative nature of mathematical symbols must be related to the qualitative means through which they are interpreted.

Once again at the conventional 1-dimensional level this problem seemingly does not arise (because the qualitative aspect is reduced to quantitative interpretation).

So I have illustrated here with respect to the square root of 2 that - appropriately understood - a comprehensive proof requires the incorporation of both quantitative and qualitative aspects.

And this in principle can be applied to all proof. Therefore, as quantitative and qualitative are relative with respect to each other, an inevitable uncertainty principle applies to all mathematical proof.

Mathematical Proof and the Riemann Hypothesis

When appropriately understood, prime numbers represent the most fundamental manner in which both quantitative and qualitative interact in mathematical terms.

Because of the quantitative bias of Conventional Mathematics, prime numbers are generally looked on in reduced terms as the independent building blocks of the natural number system.

However prime numbers equally possess an amazing holistic capacity with respect to their general distribution among the natural numbers. The clear implication from this perspective is that the prime are fully interdependent with the natural numbers.

So there is a quantitative (independent) and qualitative (interdependent) aspect to prime numbers which cannot be given adequate recognition within Conventional Mathematics.

Once again the intrinsic nature of prime numbers is closely related to the behaviour of subatomic particles. Though we cannot know the precise location of a specific sub-atomic particle, yet their overall behaviour can be predicted with an amazing degree of accuracy. Likewise though we cannot know precisely the location of each individual prime number, yet an amazing regularity applies to their general behaviour.

It is now accepted in physics that particles can actually communicate with each other (which is the means through which they preserve their overall regularity).

The clear implication then is that the same communication capacity characterises prime numbers.

Though this might initially seem as a fanciful notion, the difficulty in its acceptance relates to a merely abstract notion of number that is divorced from living experience.

However when we seek to understand numbers in a more comprehensive manner, then we begin to appreciate how they necessarily embody inherent characteristics with respect to both physical and psychological processes. In this sense therefore prime numbers represent, through their behaviour, communication abilities which particles display at the sub-atomic level. [10]

So understood in this manner prime numbers can precisely articulate dynamic behavioural patterns in nature (with matching psychological correspondents in qualitative terms).

One clear implication of this realisation is that every mathematical symbol, theorem, relationship etc. in principle has - by definition - an intrinsic application to reality (in both physical and psychological terms).

The Riemann Hypothesis, which has an intimate bearing on the nature of prime numbers, is generally considered the most important unsolved problem in Mathematics.

In my article written for this web-site, I concluded that the Riemann Hypothesis was indeed truly fundamental and intimately related to the ultimate reconciliation of quantitative with qualitative meaning in mathematical terms (both of which are inherent in the nature of primes).

So, in the context of mathematical proof, the Riemann Hypothesis points to an ineffable state, where both its quantitative and qualitative aspects are finally unified.

One obvious consequence of this finding is that the Riemann Hypothesis cannot be proved (or disproved) in the accepted manner of Conventional Mathematics.

Put another way, the assumption to which the Riemann Hypothesis relates is already inherent within the axioms of Conventional Mathematics (while equally transcending any phenomenal attempt to grasp its ultimate nature).

So we have the lovely irony that in the truest and most profound sense the Riemann Hypothesis is therefore too simple to have a solution!

When asked once what was the most important problem in Mathematics - as claimed in Constance Reid's book - the great mathematician Hilbert replied! [11]

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

And to a very significant degree Hilbert was right.

For the Riemann Hypothesis (to which the zeros of the zeta function relate) serves as an equivalent mathematical statement of the famous Buddhist sutra,

“Form is not other than Emptiness
Emptiness is not other than Form”

In other words:

“The Quantitative is not other than the Qualitative
The Qualitative is not other than the Quantitative”

However if Hilbert was to awaken from his fabled long sleep, to see if the Riemann Hypothesis had been proved, he would be shocked by what he would find. [12]

His vision of a Mathematics where every problem in principle could be solved had already been undermined in his lifetime by Godel (who demonstrated that there would always be important mathematical problems that could not be proved within the accepted axioms).

Also, at a more fundamental level, an uncertainty principle necessarily applies to all mathematical proof (where both quantitative and qualitative aspects must be formally incorporated).

As for the Riemann Hypothesis, it simply is not capable of proof using existing axioms. It is in fact a key statement pertaining to the ultimate nature of this more comprehensive Mathematics!

Notes

1. The line in quantitative terms is 1-dimensional. Thus in a corresponding qualitative manner, linear rational understanding is also 1-dimensional. What this ultimately means is that mathematical interpretation is formally based on the clear separation of opposite polarities in experience. So truth is given one unambiguous direction with propositions for example assumed to have a merely objective validity.

The deeper implication is that numerical values are ultimately expressed - literally - in a 1-dimensional manner with respect to their quantitative aspect. Thus when we square - say - 2, a qualitative properly speaking a dimensional transformation takes place in the nature of units involved. However in linear interpretation the ultimate result (4) is expressed in a merely reduced quantitative fashion (with respect to the 1st dimension).

2. Though physicists have at one level been forced to deal with the paradoxical findings of quantum mechanics, the deeper implications of such findings for the conventional scientific paradigm have been avoided. Once again for a proper appreciation of quantum mechanics - and other key areas such as Relativity and String Theory - a more comprehensive model of physics is required that incorporates both quantitative and qualitative understanding.

(I hope to address this key issue in physics through further contributions!)

3. I can readily testify to this from my own experience as I found that initial immersion in Holistic Mathematics coincided with a marked erosion of conventional type mathematical ability.

Holistic Mathematics can in no way be approached through an extension of accepted mathematical ideas as it is based on a uniquely distinctive type of understanding (which is paradoxical in terms of accepted reason).

However ultimately the distinctive abilities required for conventional and holistic type appreciation can mutually be enhanced when used in close conjunction with each other.

4. Implicitly this is even true of Conventional Mathematics. Though it has no place in formal interpretation, informally the importance of intuition is readily accepted. Strictly speaking it would not be possible to confirm the truth of rational deductions without some holistic intuitive ability (to literally “see” what is implied by such deductions).

Indeed the reason why many people have such difficulty in following abstract type arguments is precisely because they lack the appropriate intuition to see what is implied!

Also, intuitive insight is vital for any truly creative work in Conventional Mathematics.

However though informally accepted, it is then excluded from formal interpretation.

So in practice conventional interpretation represents a significant misrepresentation of mathematical understanding.

5. The strong belief in the absolute type nature of mathematical proof is based on the fallacy that something can be objectively true (independent of its subjective means of interpretation).

Mathematical symbols have no meaning until they are interpreted in a certain qualitative manner and likewise mathematical proof has no meaning without such interpretation.

So the belief in the “objective truth” of mathematical propositions highlights once again the central issue of the reduction of qualitative to quantitative type meaning.

So when we interpret a proposition as “objectively true” we are simply reducing the qualitative means by which it is interpreted to its mere quantitative expression.

6. The proof that the square root of 2 is irrational is of a specific nature and thereby distinct from the general type of proof - such as the Pythagorean Theorem - that applies to all cases within its class.

However the same basic problem remains and the interaction as between finite and infinite notions of meaning is unavoidable. Thus with respect to number the general concept strictly has an infinite range applying potentially to all numbers, whereas specific number perceptions are actual and finite in nature.

So for example, once we let x be a rational number, we are faced with this inevitable interaction as between finite and infinite meaning.

7. Even in normal language the use of the word “unconscious” implies such a negation. So someone who is unconscious is literally not-conscious. In a more refined manner the development of unconscious states of awareness (as with spiritual contemplation) requires the dynamic negation of conscious type understanding. Such awareness is thereby of a nondual nature (in contrast to the dualistic manner of recognising distinct phenomena of form).

8. The square root of 2 can be expressed as 2 raised to the power (i.e. dimension) of ½.

This then has an intimate connection with the Riemann Hypothesis where all the non-trivial zeros of the zeta function are postulated to lie on the line with real part = ½ (where ½ refers to a dimensional number).

There is an inverse relationship as between the qualitative interpretation of a dimensional number and the quantitative root form structure to which it relates.

In pure qualitative terms we need only concern ourselves with the dimensional numbers to which the number 1 is raised and the corresponding roots of 1.

So the dimension 2 (in qualitative terms) is associated with its corresponding structure by raising 1 to the power of ½ (as quantitatively understood).

So here in quantitative terms we get either + 1 or – 1 (expressed in linear logic). In corresponding qualitative terms we get both + 1 and – 1 (expressed in circular logic). In other words from a qualitative perspective the circular logical structure of 2 as a dimension entails the complementarity of (real) opposite polarities of experience.

In this way we can define a unique circular logical system for each number (as dimension) with reference to its corresponding root structure (where 1 is raised to the inverse of that number). And the incorporation of this circular logical structure for 2 is the key to appreciating the true nature of the Riemann Hypothesis!

What this entails in practice is that each dimension (as a qualitative means of interpretation) represents a unique interaction with respect to both unconscious and unconscious in the combination of both rational (quantitative) and intuitive (qualitative) meaning. Once again in the default case of 1 (which defines conventional mathematical interpretation) both quantitative and qualitative meaning coincide (as 1 = the inverse of 1). So here we can see in a very precise manner how this number dimension of 1 when used as a means of interpretation necessarily reduces qualitative to quantitative meaning (representing a special limiting case with respect to a much more comprehensive understanding).

9. I have demonstrated here in qualitative terms with respect to the simplest possible case the nature of irrational (algebraic) - as opposed to rational - interpretation.

Then by extension, associated with each number type e.g. prime, natural, rational, irrational (algebraic), transcendental, imaginary, complex is a unique qualitative interpretation.

In particular as it is now recognised in quantitative terms that the number system is complex (with real and imaginary parts), likewise appropriate interpretation (in qualitative terms) needs to be complex (with real and imaginary aspects).

However - again in qualitative terms - Conventional Mathematics confines itself to merely real (rational) means of interpretation.

The clear implication is that the imaginary aspect is entirely missing!

Such imaginary interpretation simply represents the indirect rational means of incorporating qualitative holistic notions in understanding (though circular logical means).

So in the context of mathematical proof, comprehensive interpretation needs to be complex combing both real (quantitative) and imaginary (qualitative) aspects.

Through illustrating here just a few key mathematical concepts, it can perhaps be appreciated that for every mathematical notion (as defined in conventional quantitative terms) a corresponding definition can be given in Holistic Mathematics (defined in a corresponding qualitative manner).

10. Recent attempts to prove the Riemann Hypothesis are in fact based on the assumption that a direct connection does in fact exist as between the non-trivial zeros of the Riemann Function and the energy states of a certain quantum chaotic system.

This finding came as no surprise to me, as I had already assumed that some quantum physical relationship with the Riemann zeros necessarily existed.

Indeed I would go considerably further in maintaining that the Riemann zeros equally have an intimate qualitative connection - what one might term a qualtum relationship - with certain advanced spiritual energy states (which I outlined in the Riemann article).

Ultimately a direct correspondence exists as between the (quantitative) physical and the (qualitative) psychological energy states. In other words true quantitative significance of the Riemann zeros cannot be divorced from the appropriate qualitative means by which they are interpreted. And when appropriately understood, this indeed is the key implication of the Riemann Hypothesis!

However, belief that the physical connection - though important in its own right - can lead to a possible proof is misplaced. In fact the quantum link that already has been established should suggest that a classical type of proof of the Hypothesis is not in fact possible.

11. P.92 of her autobiography “Hilbert”.

12. Hilbert is reputed to have stated "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"