As is well known two key features characterise prime numbers.

As discrete independent numbers, there is seemingly no discernible pattern connecting them.

However in terms of their overall relationship to the natural numbers an amazingly continuous pattern of interdependent regularity is in evidence.

Though this should immediately suggest both quantitative and qualitative aspects in the behaviour of prime numbers, due to the quantitative bias of Conventional Mathematics this is conveniently overlooked.

Even momentary reflection on the matter should suggest a problem.

From the linear (discrete) view, prime numbers are seen as the independent building blocks of the natural number system.

However from the corresponding holistic (continuous) view, prime numbers are seen as intimately dependent on the natural numbers for their general distribution.

So clearly, both prime and natural numbers are co-determined in a dynamic interactive manner.

To meaningfully refer to prime numbers implicitly requires the use of the natural numbers. The very listing of the prime numbers requires the ordinal use of the natural numbers 1, 2, 3, and 4 etc. And even here we have the relationship as between quantitative and qualitative (in the cardinal identification of prime numbers, together with the ordinal identification of natural numbers).

As is well-known the non-trivial zeros have a vital role to play with respect to the precise distribution of the prime numbers.

The trivial zeros also have an - unappreciated - qualitative significance with respect to their general distribution.

Because of the interdependence of quantitative and qualitative, all quantitative aspects of number have a corresponding qualitative meaning (and all qualitative a corresponding quantitative meaning).

However because of the merely quantitative bias of Conventional Mathematics, both in relation to individual prime numbers and their general distribution, only the quantitative aspect is formally recognised.

In quantitative terms, the simplest recognised expression for the general spread of the primes (i.e. average distance between successive primes) = log n.

Remarkably this in turn can be directly related to all the zeta values for 1, 2, 3,…

The Zeta Function for s = 1 generates the harmonic series. So where n is finite this will have a finite value and already approximates well to log n (for large n).

By successively subtracting and adding zeta values for s = 2, 3, 4,… (again taken over a finite range for n) and dividing by 2, 3, 4,.. we then progressively approximate closer to log n.

Now the Zeta Function for s = – 1, – 2, – 3, – 4,... provides the corresponding qualitative appreciation of this general distribution.

One advantage of such appreciation would lead to the realisation that prime numbers cannot be understood in a merely quantitative manner (but in fact entails a close intimate connection as between both quantitative and qualitative aspects).

The fruit of such understanding is the eventual realisation that the Riemann Hypothesis itself points to the ultimate condition necessary for the full reconciliation of both quantitative and qualitative aspects.

So the quantitative appreciation relates to rational type understanding, whereas the qualitative relates to directly intuitive type awareness (of the interdependence of primes).

Each trivial zero therefore can be seen as a further progression in the pure intuitive realisation of the interdependence of primes. The negative odd numbered integer dimensions (which we will look at more closely) then represent a disturbance of the existing intuitive realisation as not yet sufficient to fully incorporate the inherent mystery of the primes.

Ultimately however for very high numbered dimensions, contemplative type intuitive understanding would become so pure that discrete phenomenal elements would no longer be even distinguishable in experience (as rational understanding itself would attain an incredibly refined nature).

There is another remarkable qualitative connection here. We have already stated that the spread of the primes is quantitatively given as log n. This is based on the natural log (involving e).

Now with e (raised to any power x) both the differential and integral are the same in quantitative terms!

Likewise in dynamic holistic terms, with appropriate qualitative appreciation of the nature of primes, overall awareness becomes so refined that (differentiated) rational become indistinguishable from (integrated) intuitive aspects.

## Saturday, April 23, 2011

## Thursday, April 21, 2011

### The Trivial Zeros (7)

From a holistic perspective, every mathematical relationship has a dynamic application in both physical and psychological terms (which are complementary).

As we have seen the trivial zeros of the Riemann Zeta Function from a psychological perspective represent a series of ever more-refined spiritual energy states (through pure intuitive recognition).

This implies that these same zeros have a complementary explanation also as physical energy states.

Perhaps one can begin to see what is implied here through reference to the interaction of matter and anti-matter particles which annihilate each other in pure physical energy. So here we have the complementarity of opposite real poles (of matter) that corresponds directly to 2-dimensional reality (as holistically understood in mathematical terms).

However in practice it is very difficult to separate energy from associated phenomena of matter.

So if we could see - as it were - into the precise nature of physical energy (in itself) without dynamic association with matter particles, we would discover that it is strictly empty of form.

So the negative 2-dimensional application would correspond to this pure state of matter (as energy).

It is more difficult however to give expression to higher-dimensional states of physical energy.

A 4-dimensional state would imply the simultaneous annihilation of both (real) matter and anti-matter and also virtual matter and anti-matter particles. And then the negative state would imply the nature of such energy states (in themselves) without associated phenomena of matter.

Higher energy states would perhaps be harder to detect in “normal” physical reactions. However in more extreme temperatures situations, the clear implication is that correspondent physical energy states would also exist for all even dimensional numbers. And again with the negative sign of the dimension the inherent nature of such states would be revealed as 0 (i.e. qualitatively nothing with respect to phenomena of form).

Though we do not tend to look at physical reality in this manner, the clear implication is that, in dynamic terms, phenomena of matter - that can be quantitatively detected - continually interact with an invisible holistic ground. And the purer the physical energy state the closer it lies to this underlying holistic physical ground.

In corresponding psychological terms again our conscious recognition of phenomena (through sense and reason) likewise continually interact with an invisible holistic ground (which we recognise as the unconscious). And the purer a psychological energy state (as with pure contemplation of reality) the closer it lies to this invisible spiritual ground of reality.

The great limitation again with conventional scientific and mathematical appreciation is that - in formal terms - it attempts to ignore the role of the unconscious altogether.

This is turn leads to an approach where the holistic qualitative aspect of understanding (which is quite distinct) is thereby inevitably reduced to the quantitative.

There is another fascinating connection here!

We have seen in the Riemann Zeta Function for positive even values of s = 2, 4, 6,.. that a rational fraction is always associated with the corresponding expression for pi.

So for example when s = 10 the expression for pi will be raised to the power of 10 and multiplied by a rational fraction.

In this case the rational fraction is 1/93555

Now the denominator of this fraction will always represent various combinations of the products of all prime numbers from 3 to s + 1 and only those primes. Where s is a power of 2, the product will include all primes from 2 to s + 1 (and only those primes)

So for example 93555 = 3 * 5 * 5 * 7 * 11

As we have seen the spiritual contemplative life can be viewed as a journey through progressively higher dimensional states.

A key barrier in terms of such development is the intrusion of primitive projections from the unconscious in an involuntary manner.

Now qualitatively such primitive instincts bear a close relationship with the nature of prime numbers.

So the implication here is that through journeying to higher states, primitive instincts can be progressively mastered in an orderly fashion.

So the negative of these even dimensions represents in psycho-spiritual terms detachment from inordinate association with phenomena associated with such primitive instincts.

The earliest prime numbers are the most commonly used and frequently occurring. In similar fashion orderly progression through higher dimensional intuitive states involves detachment from the most frequently occurring primitive instincts. In this way involuntary promptings from the unconscious gradually cease.

So there is a whole world of qualitative interpretation out there associated with the primes still waiting to be uncovered!

As we have seen the trivial zeros of the Riemann Zeta Function from a psychological perspective represent a series of ever more-refined spiritual energy states (through pure intuitive recognition).

This implies that these same zeros have a complementary explanation also as physical energy states.

Perhaps one can begin to see what is implied here through reference to the interaction of matter and anti-matter particles which annihilate each other in pure physical energy. So here we have the complementarity of opposite real poles (of matter) that corresponds directly to 2-dimensional reality (as holistically understood in mathematical terms).

However in practice it is very difficult to separate energy from associated phenomena of matter.

So if we could see - as it were - into the precise nature of physical energy (in itself) without dynamic association with matter particles, we would discover that it is strictly empty of form.

So the negative 2-dimensional application would correspond to this pure state of matter (as energy).

It is more difficult however to give expression to higher-dimensional states of physical energy.

A 4-dimensional state would imply the simultaneous annihilation of both (real) matter and anti-matter and also virtual matter and anti-matter particles. And then the negative state would imply the nature of such energy states (in themselves) without associated phenomena of matter.

Higher energy states would perhaps be harder to detect in “normal” physical reactions. However in more extreme temperatures situations, the clear implication is that correspondent physical energy states would also exist for all even dimensional numbers. And again with the negative sign of the dimension the inherent nature of such states would be revealed as 0 (i.e. qualitatively nothing with respect to phenomena of form).

Though we do not tend to look at physical reality in this manner, the clear implication is that, in dynamic terms, phenomena of matter - that can be quantitatively detected - continually interact with an invisible holistic ground. And the purer the physical energy state the closer it lies to this underlying holistic physical ground.

In corresponding psychological terms again our conscious recognition of phenomena (through sense and reason) likewise continually interact with an invisible holistic ground (which we recognise as the unconscious). And the purer a psychological energy state (as with pure contemplation of reality) the closer it lies to this invisible spiritual ground of reality.

The great limitation again with conventional scientific and mathematical appreciation is that - in formal terms - it attempts to ignore the role of the unconscious altogether.

This is turn leads to an approach where the holistic qualitative aspect of understanding (which is quite distinct) is thereby inevitably reduced to the quantitative.

There is another fascinating connection here!

We have seen in the Riemann Zeta Function for positive even values of s = 2, 4, 6,.. that a rational fraction is always associated with the corresponding expression for pi.

So for example when s = 10 the expression for pi will be raised to the power of 10 and multiplied by a rational fraction.

In this case the rational fraction is 1/93555

Now the denominator of this fraction will always represent various combinations of the products of all prime numbers from 3 to s + 1 and only those primes. Where s is a power of 2, the product will include all primes from 2 to s + 1 (and only those primes)

So for example 93555 = 3 * 5 * 5 * 7 * 11

As we have seen the spiritual contemplative life can be viewed as a journey through progressively higher dimensional states.

A key barrier in terms of such development is the intrusion of primitive projections from the unconscious in an involuntary manner.

Now qualitatively such primitive instincts bear a close relationship with the nature of prime numbers.

So the implication here is that through journeying to higher states, primitive instincts can be progressively mastered in an orderly fashion.

So the negative of these even dimensions represents in psycho-spiritual terms detachment from inordinate association with phenomena associated with such primitive instincts.

The earliest prime numbers are the most commonly used and frequently occurring. In similar fashion orderly progression through higher dimensional intuitive states involves detachment from the most frequently occurring primitive instincts. In this way involuntary promptings from the unconscious gradually cease.

So there is a whole world of qualitative interpretation out there associated with the primes still waiting to be uncovered!

### The Trivial Zeros (6)

We have looked at the first of the trivial zeros in the Riemann Zeta Function for s = – 2 and explained its qualitative meaning (which is directly implied by the Function).

It now remains to explain why trivial zeros exist at all negative even values for s, i.e. – 2, – 4, – 6,…….

As we have seen the qualitative dimensional structure of a number is related to the corresponding root structure (when given a circular rather than linear interpretation).

So if we take the number 4 (as dimension) to illustrate, the corresponding root structure (i.e. the 4 roots) are + 1, – 1, + i, – i respectively. Now in quantitative terms these are interpreted in linear either/or fashion.

Therefore the corresponding (qualitative) dimensional structure entails the simultaneous appreciation of these as complementary opposites.

Now however as well as the two real opposites + 1 and – 1 that we have already encountered, we have in addition two imaginary opposites + i and – i.

Now the significance of these imaginary opposites in qualitative terms is that they relate to the manner in which the holistic unconscious projects its meaning indirectly in a conscious manner. Thus – as I have outlined in detail in my writings on the stages of development - such imaginary understanding becomes the very means through which circular meaning can be indirectly incorporated within the standard linear framework of Conventional Science.

However, once again though the use of the imaginary is now well established in both Mathematics and Science with respect to its (reduced) quantitative interpretation, little or no appreciation yet exists as to the enormous qualitative significance of the imaginary. For properly appreciated the imaginary points to the hidden holistic dimension of Mathematics (that receives no formal recognition in conventional interpretation).

Long before developing an interest in the Riemann Hypothesis, I had gone into considerable detail on the precise qualitative nature of 2, 4 and 8 dimensional understanding. Once again the positive sign of the dimension points to refined rational appreciation (of a circular paradoxical nature); in turn the negative sign of the dimension points to the erosion of any remaining linear rational elements of understanding in obtaining a purely intuitive appreciation (that is nothing in phenomenal terms).

Now in all cases, where even dimensional numbers are involved, a complementary structure as between the roots can be obtained.

Therefore we can look at the larger even dimensional numbers as representing ever more refined rational appreciation of the complementarity of opposites.

The corresponding negative sign of these dimensions in turn leads to ever purer intuitive recognition (of an empty kind).

So once again all of the trivial zeros relate to increasingly refined intuitive recognition states (which are nothing in phenomenal terms).

Therefore - quite literally - when one intuits the meaning of the Riemann Zeta Function for negative even dimensional values, it is empty of phenomenal rational interpretation.

So once again the 0 that is generated by the Function in such cases relates to a directly qualitative meaning (of a holistic kind).

Quite obviously this cannot be appreciated in conventional mathematical terms as it offers no formal recognition of such qualitative interpretation!

However once it is grasped that the values of the Riemann Zeta Function for s < 0 directly correspond to qualitative - rather than quantitative - meaning, then perhaps we can begin to appreciate the significant limitations of the standard approach.

It now remains to explain why trivial zeros exist at all negative even values for s, i.e. – 2, – 4, – 6,…….

As we have seen the qualitative dimensional structure of a number is related to the corresponding root structure (when given a circular rather than linear interpretation).

So if we take the number 4 (as dimension) to illustrate, the corresponding root structure (i.e. the 4 roots) are + 1, – 1, + i, – i respectively. Now in quantitative terms these are interpreted in linear either/or fashion.

Therefore the corresponding (qualitative) dimensional structure entails the simultaneous appreciation of these as complementary opposites.

Now however as well as the two real opposites + 1 and – 1 that we have already encountered, we have in addition two imaginary opposites + i and – i.

Now the significance of these imaginary opposites in qualitative terms is that they relate to the manner in which the holistic unconscious projects its meaning indirectly in a conscious manner. Thus – as I have outlined in detail in my writings on the stages of development - such imaginary understanding becomes the very means through which circular meaning can be indirectly incorporated within the standard linear framework of Conventional Science.

However, once again though the use of the imaginary is now well established in both Mathematics and Science with respect to its (reduced) quantitative interpretation, little or no appreciation yet exists as to the enormous qualitative significance of the imaginary. For properly appreciated the imaginary points to the hidden holistic dimension of Mathematics (that receives no formal recognition in conventional interpretation).

Long before developing an interest in the Riemann Hypothesis, I had gone into considerable detail on the precise qualitative nature of 2, 4 and 8 dimensional understanding. Once again the positive sign of the dimension points to refined rational appreciation (of a circular paradoxical nature); in turn the negative sign of the dimension points to the erosion of any remaining linear rational elements of understanding in obtaining a purely intuitive appreciation (that is nothing in phenomenal terms).

Now in all cases, where even dimensional numbers are involved, a complementary structure as between the roots can be obtained.

Therefore we can look at the larger even dimensional numbers as representing ever more refined rational appreciation of the complementarity of opposites.

The corresponding negative sign of these dimensions in turn leads to ever purer intuitive recognition (of an empty kind).

So once again all of the trivial zeros relate to increasingly refined intuitive recognition states (which are nothing in phenomenal terms).

Therefore - quite literally - when one intuits the meaning of the Riemann Zeta Function for negative even dimensional values, it is empty of phenomenal rational interpretation.

So once again the 0 that is generated by the Function in such cases relates to a directly qualitative meaning (of a holistic kind).

Quite obviously this cannot be appreciated in conventional mathematical terms as it offers no formal recognition of such qualitative interpretation!

However once it is grasped that the values of the Riemann Zeta Function for s < 0 directly correspond to qualitative - rather than quantitative - meaning, then perhaps we can begin to appreciate the significant limitations of the standard approach.

### The Trivial Zeros (5)

As we have seen in the case of the 1st trivial zero where s (the dimensional number)

= – 2, the value of the Riemann Zeta Function = 0.

However what is vital to appreciate is that this value does not relate to a standard quantitative notion of number but rather to its alternative qualitative aspect (which is unrecognised in conventional mathematical interpretation).

Once again the deeper implications of this finding go back to the very nature of mathematical experience which combines both rational and intuitive processes of understanding in dynamic interaction with each other. However though informally mathematicians may recognise the importance of intuitive insight (especially for creative work), in formal terms Conventional Mathematics is misleadingly presented as a solely rational body of truths. And as we have seen the type of rationality allowed here is confined purely to linear type understanding based on an unambiguous either/or logic.

So the qualitative notion of number arises when one attempts to reflect its holistic (intuitive) rather that (analytic) meaning.

Now to get a clue as to what this 1st trivial zero might mean, we need to enlarge our perspective to recognise that just as there is a wide spectrum for electromagnetic energy (of which natural light comprises one small band) likewise there is a wide spectrum of potential psychological energy states (in human development) of which the rational approach that informs conventional mathematical interpretation likewise comprises just one small band.

Traditionally the higher psychological energy states - that inform pure intuitive understanding - have been largely confined to a spiritual type treatment of development (devoid of any coherent mathematical implications).

As my own evolution in thinking owes a great deal to the Spanish mystic St. John of the Cross it is worth dealing briefly with his approach here.

St. John aimed at a very pure form of spiritually intuitive awareness (free of any lingering phenomenal attachment). This required putting a high degree of emphasis on the importance of negation of such attachments.

Initially St. John emphasises what he refers to “active nights” with respect to both sense and spirit. This could be referred to the context of mathematical understanding as the dynamic negation of linear perceptions and concepts (i.e. 1-dimensional understanding).

He later goes on to more deeply emphasise the importance of the “passive nights” with respect to both sense and spirit. Again in the context of mathematical understanding this would refer to the dynamic negation of circular type perceptions and concepts (of which 2-dimensional appreciation would be the earliest representative).

So these “passive nights” would therefore include - literally - the dynamic negation of 2-dimensional interpretation (to which the first of the trivial zeros relates).

Interestingly St. John himself frequently refers to the goal of this purgative process of negation as “nada” which means nothing.

When I was at school in Ireland the term zero was not used to refer to 0 but rather nought or nothing. And so “nada” as used by St. John in fact accurately refers to the qualitative meaning of 0 (which is actually implied by the trivial zeros).

What of course St. John means by this term is a purely intuitive experience of meaning (that is free of any secondary phenomenal attachment such as rational).

So we can have two extremes with respect to mathematical meaning;

1. rational quantitative interpretation (free of holistic intuitive connotations)

2. intuitive qualitative interpretation (free of partial rational considerations)

When properly appreciated the Riemann Zeta Function incorporates both extremes.

Where s > 1, values of the function correspond to rational quantitative interpretation (i.e. as conventionally understood).

Where s < 0, values of the function correspond to holistic qualitative interpretation (that is not formally recognised in present Mathematics).

Interestingly for values of s between 0 and 1 - to which the Riemann Hypothesis directly relates - both types of meaning are involved.

The huge problem with conventional mathematical interpretation is that it cannot give a coherent explanation for values of the Function where s < 1. Obviously it will be admitted that these values arise through the recognised process of analytic continuation and clearly have considerable importance. However it has no qualitative means of explaining such importance. Rather it attempts misleadingly to reduce all values generated - even though this leads to obvious contradictions - to mere quantitative interpretation.

= – 2, the value of the Riemann Zeta Function = 0.

However what is vital to appreciate is that this value does not relate to a standard quantitative notion of number but rather to its alternative qualitative aspect (which is unrecognised in conventional mathematical interpretation).

Once again the deeper implications of this finding go back to the very nature of mathematical experience which combines both rational and intuitive processes of understanding in dynamic interaction with each other. However though informally mathematicians may recognise the importance of intuitive insight (especially for creative work), in formal terms Conventional Mathematics is misleadingly presented as a solely rational body of truths. And as we have seen the type of rationality allowed here is confined purely to linear type understanding based on an unambiguous either/or logic.

So the qualitative notion of number arises when one attempts to reflect its holistic (intuitive) rather that (analytic) meaning.

Now to get a clue as to what this 1st trivial zero might mean, we need to enlarge our perspective to recognise that just as there is a wide spectrum for electromagnetic energy (of which natural light comprises one small band) likewise there is a wide spectrum of potential psychological energy states (in human development) of which the rational approach that informs conventional mathematical interpretation likewise comprises just one small band.

Traditionally the higher psychological energy states - that inform pure intuitive understanding - have been largely confined to a spiritual type treatment of development (devoid of any coherent mathematical implications).

As my own evolution in thinking owes a great deal to the Spanish mystic St. John of the Cross it is worth dealing briefly with his approach here.

St. John aimed at a very pure form of spiritually intuitive awareness (free of any lingering phenomenal attachment). This required putting a high degree of emphasis on the importance of negation of such attachments.

Initially St. John emphasises what he refers to “active nights” with respect to both sense and spirit. This could be referred to the context of mathematical understanding as the dynamic negation of linear perceptions and concepts (i.e. 1-dimensional understanding).

He later goes on to more deeply emphasise the importance of the “passive nights” with respect to both sense and spirit. Again in the context of mathematical understanding this would refer to the dynamic negation of circular type perceptions and concepts (of which 2-dimensional appreciation would be the earliest representative).

So these “passive nights” would therefore include - literally - the dynamic negation of 2-dimensional interpretation (to which the first of the trivial zeros relates).

Interestingly St. John himself frequently refers to the goal of this purgative process of negation as “nada” which means nothing.

When I was at school in Ireland the term zero was not used to refer to 0 but rather nought or nothing. And so “nada” as used by St. John in fact accurately refers to the qualitative meaning of 0 (which is actually implied by the trivial zeros).

What of course St. John means by this term is a purely intuitive experience of meaning (that is free of any secondary phenomenal attachment such as rational).

So we can have two extremes with respect to mathematical meaning;

1. rational quantitative interpretation (free of holistic intuitive connotations)

2. intuitive qualitative interpretation (free of partial rational considerations)

When properly appreciated the Riemann Zeta Function incorporates both extremes.

Where s > 1, values of the function correspond to rational quantitative interpretation (i.e. as conventionally understood).

Where s < 0, values of the function correspond to holistic qualitative interpretation (that is not formally recognised in present Mathematics).

Interestingly for values of s between 0 and 1 - to which the Riemann Hypothesis directly relates - both types of meaning are involved.

The huge problem with conventional mathematical interpretation is that it cannot give a coherent explanation for values of the Function where s < 1. Obviously it will be admitted that these values arise through the recognised process of analytic continuation and clearly have considerable importance. However it has no qualitative means of explaining such importance. Rather it attempts misleadingly to reduce all values generated - even though this leads to obvious contradictions - to mere quantitative interpretation.

## Wednesday, April 20, 2011

### The Trivial Zeros (4)

As is well known, Euler did great pioneering work on the Zeta Function (where it is defined for all real values of s > 1). s in this context representes the dimensional number (i.e. power or exponent) to which all the terms in the series are raised.

So for example when s = 2, we get

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

= 1 + 1/4 + 1/9 + 1/16 +……

Now Euler was able to show that the sum of these terms converged to (pi^2)/6.

This result is very revealing for not alone does the zeta function converge to a finite quantitative value (when defined in 2-dimensional terms) but the result is closely related to pi.

Now in quantitative terms pi represents the relationship between the circular circumference and its line diameter.

In corresponding qualitative fashion pi represents the relationship as between circular and linear type understanding.

And as we have seen the very attempt to rationally convey the nature of 2-dimensional interpretation requires both circular and linear type understanding!

When we consider the Euler Function for values of s < 1, it seemingly breaks down as the series diverges to infinity.

So for example when s = – 2, we get

1^2 + 2^2 + 3^2 + 4^2 +…..

= 1 + 4 + 9 + 16 +….. which clearly diverges in quantitative terms (to infinity).

However by defining s (the dimensional power) for zeta in complex terms, through a process of analytic continuation, Riemann was able to extend the domain of definition of the Riemann Hypothesis to all values of s (with the exception of 1).

So from this new extended perspective when s = – 2, the value of the function = 0.

Now on the face of it this result is hugely problematic in quantitative terms.

Remember from a standard linear interpretation, which defines quantitative values, when s = – 2, the value of the Zeta Function diverges to infinity.

However from this new perspective provided by Riemann, the value of the same Zeta Function = 0.

So the question arises - or at least should arise - as to how we can reconcile the two results.

The truly remarkable answer that unfolds is that whereas the divergent result for the Function corresponds to standard quantitative interpretation, the corresponding convergent result = 0 corresponds to the alternative unrecognised qualitative interpretation.

Put another way, all meaningful quantitative results are based qualitatively on linear interpretation whereby the values are reduced in default 1-dimensional terms.

Interestingly therefore the only value for which the Riemann Function is undefined is where s = 1.

As the quantitative result diverges here, the corresponding qualitative interpretation (which is identical in this one case to the quantitative) must also diverge.

However in all other cases, an alternative qualitative interpretation for the Function can be given in accordance with the dimension (power) of s.

And as both interpretations are paradoxical in terms of each other, this entails that if the Function diverges (from a standard quantitative perspective), it necessarily converges from the alternative qualitative perspective.

We have already given meaning to the negative sign of the 2nd dimension as corresponding to the purely intuitive appreciation of the complementarily of real opposite poles (such as external and internal).

So when we understand the Riemann Zeta Function from a qualitative perspective, with interpretation according to the negative sign of the 2nd dimension, this corresponds to pure intuitive recognition that is empty of form. Thus numerically this is represented by 0.

Therefore the remarkable finding we have found is that the first trivial zero - and by extension all trivial zeros - relate to a qualitative rather than quantitative interpretation of numerical values.

So for example when s = 2, we get

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

= 1 + 1/4 + 1/9 + 1/16 +……

Now Euler was able to show that the sum of these terms converged to (pi^2)/6.

This result is very revealing for not alone does the zeta function converge to a finite quantitative value (when defined in 2-dimensional terms) but the result is closely related to pi.

Now in quantitative terms pi represents the relationship between the circular circumference and its line diameter.

In corresponding qualitative fashion pi represents the relationship as between circular and linear type understanding.

And as we have seen the very attempt to rationally convey the nature of 2-dimensional interpretation requires both circular and linear type understanding!

When we consider the Euler Function for values of s < 1, it seemingly breaks down as the series diverges to infinity.

So for example when s = – 2, we get

1^2 + 2^2 + 3^2 + 4^2 +…..

= 1 + 4 + 9 + 16 +….. which clearly diverges in quantitative terms (to infinity).

However by defining s (the dimensional power) for zeta in complex terms, through a process of analytic continuation, Riemann was able to extend the domain of definition of the Riemann Hypothesis to all values of s (with the exception of 1).

So from this new extended perspective when s = – 2, the value of the function = 0.

Now on the face of it this result is hugely problematic in quantitative terms.

Remember from a standard linear interpretation, which defines quantitative values, when s = – 2, the value of the Zeta Function diverges to infinity.

However from this new perspective provided by Riemann, the value of the same Zeta Function = 0.

So the question arises - or at least should arise - as to how we can reconcile the two results.

The truly remarkable answer that unfolds is that whereas the divergent result for the Function corresponds to standard quantitative interpretation, the corresponding convergent result = 0 corresponds to the alternative unrecognised qualitative interpretation.

Put another way, all meaningful quantitative results are based qualitatively on linear interpretation whereby the values are reduced in default 1-dimensional terms.

Interestingly therefore the only value for which the Riemann Function is undefined is where s = 1.

As the quantitative result diverges here, the corresponding qualitative interpretation (which is identical in this one case to the quantitative) must also diverge.

However in all other cases, an alternative qualitative interpretation for the Function can be given in accordance with the dimension (power) of s.

And as both interpretations are paradoxical in terms of each other, this entails that if the Function diverges (from a standard quantitative perspective), it necessarily converges from the alternative qualitative perspective.

We have already given meaning to the negative sign of the 2nd dimension as corresponding to the purely intuitive appreciation of the complementarily of real opposite poles (such as external and internal).

So when we understand the Riemann Zeta Function from a qualitative perspective, with interpretation according to the negative sign of the 2nd dimension, this corresponds to pure intuitive recognition that is empty of form. Thus numerically this is represented by 0.

Therefore the remarkable finding we have found is that the first trivial zero - and by extension all trivial zeros - relate to a qualitative rather than quantitative interpretation of numerical values.

### The Trivial Zeros (3)

We have already contrasted 1-dimensional (linear) with 2-dimensional (circular) logic. Such circular logic is inherently empty (as intuitive awareness). However indirectly it can be expressed in a paradoxical rational manner as the complementarity of two opposite polarities.

So we have here in qualitative manner the binary digits of 1 and 0 (this time given a holistic meaning). Just as the binary digits 1 and 0 can be used in quantitative terms to encode all information, the same digits can be used in a qualitative manner to encode all transformation processes.

So all life processes physical and psychological combine both differentiation and integration (corresponding to the holistic digits 1 and 0).

In qualitative mathematical terms, + means to posit in a conscious manner.

We get a very clear example of this with conventional mathematical understanding. Though both external (objective) and internal (subjective) aspects are necessarily involved these are clearly separated in formal interpretation.

This is what gives such Mathematics its strong appeal in that its truths are believed to possess an absolute objective status (unaffected through psychological interaction).

This in fact once again represents the linear nature of understanding (i.e. with its one unambiguous positive direction of understanding).

By contrast in qualitative mathematical terms, – means to (dynamically) negate, what is conscious, in an unconscious manner. Though we may not have learnt to appreciate in this fashion, differentiation and integration always entail the continual positing and negating of phenomena of form. Through the positing aspect we are enabled to analytically differentiate such phenomena (as separate); through the corresponding negating aspect we are enabled to holistically integrate them again (as interdependent). And such holistic interdependence is strictly empty of form!

Just as we can have negative dimensions (or powers) in the standard quantitative mathematical sense, likewise we can have them also in a qualitative fashion.

As we have seen, in qualitative terms, the 2nd dimension relates to the logic of the complementarity of opposite poles (+ and – 1).

Now when we posit this dimension, in qualitative terms this implies an objective rational appreciation of this complementarity.

However such complementarity (and ultimate identity) of poles strictly is formless.

So just as the combination of + 1 and – 1 = 0 in quantitative terms, likewise the combination of + 1 and – 1 = 0 in corresponding qualitative fashion.

However the positive interpretation of the 2nd dimension gives it an indirect rational form.

Therefore to directly understand what is implied in an empty intuitive manner, one must negate in understanding the rational aspect (strictly any attachment to the rational aspect).

So purely intuitive appreciation, of what is implied by the complementarity of opposites, requires the negation of the 2nd dimension in qualitative terms and this is represented numerically as 0 (without form).

From a spiritual perspective, such empty intuitive appreciation (without phenomenal form) is identified with a pure contemplative state.

However it is important to appreciate that this state can be precisely represented in a holistic mathematical fashion (where symbols have a qualitative rather than quantitative meaning).

So we have identified the first of the purely intuitive contemplative states as representing understanding pertaining to the negation of the 2nd dimension.

And such interpretation is vital in properly appreciating the true nature of the very first of the trivial zeros in the Riemann Zeta Function!

We must remember that though in dynamic experiential terms all mathematical understanding necessarily combines both rational and intuitive processes, in formal interpretation, Conventional Mathematics is represented in merely reduced rational terms (where qualitative meaning is thereby reduced to quantitative).

So to properly represent both rational and intuitive understanding,the minimum that is required is the combination of both 1-dimensional (linear) and 2-dimensional (circular) interpretation.

It is also the minimum required in terms of appreciating the true nature of the Riemann Hypothesis!

So we have here in qualitative manner the binary digits of 1 and 0 (this time given a holistic meaning). Just as the binary digits 1 and 0 can be used in quantitative terms to encode all information, the same digits can be used in a qualitative manner to encode all transformation processes.

So all life processes physical and psychological combine both differentiation and integration (corresponding to the holistic digits 1 and 0).

In qualitative mathematical terms, + means to posit in a conscious manner.

We get a very clear example of this with conventional mathematical understanding. Though both external (objective) and internal (subjective) aspects are necessarily involved these are clearly separated in formal interpretation.

This is what gives such Mathematics its strong appeal in that its truths are believed to possess an absolute objective status (unaffected through psychological interaction).

This in fact once again represents the linear nature of understanding (i.e. with its one unambiguous positive direction of understanding).

By contrast in qualitative mathematical terms, – means to (dynamically) negate, what is conscious, in an unconscious manner. Though we may not have learnt to appreciate in this fashion, differentiation and integration always entail the continual positing and negating of phenomena of form. Through the positing aspect we are enabled to analytically differentiate such phenomena (as separate); through the corresponding negating aspect we are enabled to holistically integrate them again (as interdependent). And such holistic interdependence is strictly empty of form!

Just as we can have negative dimensions (or powers) in the standard quantitative mathematical sense, likewise we can have them also in a qualitative fashion.

As we have seen, in qualitative terms, the 2nd dimension relates to the logic of the complementarity of opposite poles (+ and – 1).

Now when we posit this dimension, in qualitative terms this implies an objective rational appreciation of this complementarity.

However such complementarity (and ultimate identity) of poles strictly is formless.

So just as the combination of + 1 and – 1 = 0 in quantitative terms, likewise the combination of + 1 and – 1 = 0 in corresponding qualitative fashion.

However the positive interpretation of the 2nd dimension gives it an indirect rational form.

Therefore to directly understand what is implied in an empty intuitive manner, one must negate in understanding the rational aspect (strictly any attachment to the rational aspect).

So purely intuitive appreciation, of what is implied by the complementarity of opposites, requires the negation of the 2nd dimension in qualitative terms and this is represented numerically as 0 (without form).

From a spiritual perspective, such empty intuitive appreciation (without phenomenal form) is identified with a pure contemplative state.

However it is important to appreciate that this state can be precisely represented in a holistic mathematical fashion (where symbols have a qualitative rather than quantitative meaning).

So we have identified the first of the purely intuitive contemplative states as representing understanding pertaining to the negation of the 2nd dimension.

And such interpretation is vital in properly appreciating the true nature of the very first of the trivial zeros in the Riemann Zeta Function!

We must remember that though in dynamic experiential terms all mathematical understanding necessarily combines both rational and intuitive processes, in formal interpretation, Conventional Mathematics is represented in merely reduced rational terms (where qualitative meaning is thereby reduced to quantitative).

So to properly represent both rational and intuitive understanding,the minimum that is required is the combination of both 1-dimensional (linear) and 2-dimensional (circular) interpretation.

It is also the minimum required in terms of appreciating the true nature of the Riemann Hypothesis!

### The Trivial Zeros (2)

I have mentioned before how at University a fresh source of disillusionment with respect to conventional mathematical procedures arose with respect to treatment of the infinite notion which once again is given a reduced meaning (robbing it of its true qualitative significance).

I then developed a growing interest in philosophy culminating with a strong interest especially in the Hegelian system.

When I then realised how key notions with respect to this system could be suitably expressed in a qualitative mathematical fashion, this led to the development of that missing aspect I was earlier searching for, in what I term Holistic Mathematics.

Hegelian logic - in contrast to standard Aristotelian logic - is circular and paradoxical in nature. It is commonly expressed in terms of the view that every thesis has a corresponding antithesis. Then in evolution the dynamic interaction of these opposites in experience (and in a wider context historical evolution) leads to a new synthesis, that in turn becomes the thesis for the next stage.

This circular paradoxical logic can also be expressed as the complementarity of opposite poles (such as objective and subjective). In contrast to standard linear logic which is based on clear either/or distinctions as between such opposites, this new circular logic is based on simultaneous both/and inclusion of both poles.

Both types of logic are necessarily involved in all experience enabling us to both analytically differentiate (in quantitative terms) while also holistically integrating (in a qualitative manner).

I have illustrated this repeatedly with a simple example. According to linear logic a turn on a straight road is either left or right. However according to circular logic, it is both left and right.

Now one can either travel up or down this straight road. So if we separate possible reference frames to identify movement as either up or down the road, then we can unambiguously identify a turn as left or right. However if we try to simultaneously combine reference frames, then a turn will be both left and right.

So linear reason - which is the direct product of conscious understanding - is suited to this sequential recognition (according to independent polar frames of reference). However circular reason - as the indirect expression of unconscious intuition - is suited by contrast to simultaneous recognition of interdependent polar frames.

Now remember that the square root of 1 has in quantitative terms two roots + 1 and – 1. The important inverse corollary of this is that the square of 1 (where we express the number with respect to the 2nd dimension) combines opposite polarities of form (in a qualitative both/and manner).

Therefore associated with the number 2 in qualitative dimensional terms is a uniquely distinctive (both/and) circular logic of understanding that is the indirect rational expression of intuitive type recognition.

By extension associated with every number (as dimension) is a uniquely distinctive logical system of interpretation.

So remarkably, in formal terms, Conventional Mathematics is conducted entirely through a default 1-dimensional logical mode. Though this does indeed constitute an enormously important special case, it is in the very nature of this system that the qualitative is reduced to mere quantitative interpretation.

However in every other potential logical system, mathematical reality is interpreted according a distinctive configuration of both quantitative and qualitative type meaning. And all of these systems have an unrecognised potential relevance with respect to mathematical interpretation!

So to properly understand the square root of 1 (in both quantitative and qualitative terms), we must combine both 1-dimensional and 2-dimensional interpretation.

With 1-dimensional interpretation opposite polarities are clearly separated. So according to this logic, the root is either + 1 or – 1 (in reduced quantitative terms).

However according to 2-dimensional interpretation, these opposite polarities are combined as complementary i.e. both + 1 and – 1 simultaneously. This actually points directly to holistic intuitive recognition (which is empty and thereby formless). Thus when we attempt to express this in a (reduced) 1-dimensional manner, the two poles that are inherently combined (in 2-dimensional terms) are clearly separated in an either/or manner.

Thus strictly speaking corresponding to every root (in quantitative terms) is a corresponding unique form of dimensional logic (from a qualitative inverse perspective). Now the structural form of both will be similar. However whereas the roots will be interpreted in reduced quantitative manner according to either/or logic as separate, in the corresponding qualitative appreciation they will be combined in an intuitive both/and manner as simultaneous.

So to sum up:

Every number - and by extension every mathematical symbol - can be given both a quantitative and qualitative interpretation.

Remarkably the qualitative interpretation of number corresponds to a unique dimension of logical interpretation (through which all mathematical symbols can in turn be expressed).

Therefore we have an inexhaustible number of possible logical interpretations of mathematical symbols and relationships (with the linear logic of Conventional Mathematics corresponding to the default dimensional interpretation of 1).

Needless to say however there is a truly enormous amount of meaning waiting to be uncovered through appreciation of the other dimensional interpretations (of which 2 is the most accessible).

I then developed a growing interest in philosophy culminating with a strong interest especially in the Hegelian system.

When I then realised how key notions with respect to this system could be suitably expressed in a qualitative mathematical fashion, this led to the development of that missing aspect I was earlier searching for, in what I term Holistic Mathematics.

Hegelian logic - in contrast to standard Aristotelian logic - is circular and paradoxical in nature. It is commonly expressed in terms of the view that every thesis has a corresponding antithesis. Then in evolution the dynamic interaction of these opposites in experience (and in a wider context historical evolution) leads to a new synthesis, that in turn becomes the thesis for the next stage.

This circular paradoxical logic can also be expressed as the complementarity of opposite poles (such as objective and subjective). In contrast to standard linear logic which is based on clear either/or distinctions as between such opposites, this new circular logic is based on simultaneous both/and inclusion of both poles.

Both types of logic are necessarily involved in all experience enabling us to both analytically differentiate (in quantitative terms) while also holistically integrating (in a qualitative manner).

I have illustrated this repeatedly with a simple example. According to linear logic a turn on a straight road is either left or right. However according to circular logic, it is both left and right.

Now one can either travel up or down this straight road. So if we separate possible reference frames to identify movement as either up or down the road, then we can unambiguously identify a turn as left or right. However if we try to simultaneously combine reference frames, then a turn will be both left and right.

So linear reason - which is the direct product of conscious understanding - is suited to this sequential recognition (according to independent polar frames of reference). However circular reason - as the indirect expression of unconscious intuition - is suited by contrast to simultaneous recognition of interdependent polar frames.

Now remember that the square root of 1 has in quantitative terms two roots + 1 and – 1. The important inverse corollary of this is that the square of 1 (where we express the number with respect to the 2nd dimension) combines opposite polarities of form (in a qualitative both/and manner).

Therefore associated with the number 2 in qualitative dimensional terms is a uniquely distinctive (both/and) circular logic of understanding that is the indirect rational expression of intuitive type recognition.

By extension associated with every number (as dimension) is a uniquely distinctive logical system of interpretation.

So remarkably, in formal terms, Conventional Mathematics is conducted entirely through a default 1-dimensional logical mode. Though this does indeed constitute an enormously important special case, it is in the very nature of this system that the qualitative is reduced to mere quantitative interpretation.

However in every other potential logical system, mathematical reality is interpreted according a distinctive configuration of both quantitative and qualitative type meaning. And all of these systems have an unrecognised potential relevance with respect to mathematical interpretation!

So to properly understand the square root of 1 (in both quantitative and qualitative terms), we must combine both 1-dimensional and 2-dimensional interpretation.

With 1-dimensional interpretation opposite polarities are clearly separated. So according to this logic, the root is either + 1 or – 1 (in reduced quantitative terms).

However according to 2-dimensional interpretation, these opposite polarities are combined as complementary i.e. both + 1 and – 1 simultaneously. This actually points directly to holistic intuitive recognition (which is empty and thereby formless). Thus when we attempt to express this in a (reduced) 1-dimensional manner, the two poles that are inherently combined (in 2-dimensional terms) are clearly separated in an either/or manner.

Thus strictly speaking corresponding to every root (in quantitative terms) is a corresponding unique form of dimensional logic (from a qualitative inverse perspective). Now the structural form of both will be similar. However whereas the roots will be interpreted in reduced quantitative manner according to either/or logic as separate, in the corresponding qualitative appreciation they will be combined in an intuitive both/and manner as simultaneous.

So to sum up:

Every number - and by extension every mathematical symbol - can be given both a quantitative and qualitative interpretation.

Remarkably the qualitative interpretation of number corresponds to a unique dimension of logical interpretation (through which all mathematical symbols can in turn be expressed).

Therefore we have an inexhaustible number of possible logical interpretations of mathematical symbols and relationships (with the linear logic of Conventional Mathematics corresponding to the default dimensional interpretation of 1).

Needless to say however there is a truly enormous amount of meaning waiting to be uncovered through appreciation of the other dimensional interpretations (of which 2 is the most accessible).

### The Trivial Zeros (1)

Precise knowledge of the nature of - even - the first of the trivial zeros in the Riemann Zeta Function is enough to appreciate the true nature of the Riemann Hypothesis (which establishes the key condition required for maintaining consistency as between both the quantitative and qualitative interpretation of mathematical symbols).

In terms of the gradual evolution of my own understanding in this regard the first seeds were set through a boyhood realisation of - what I saw - as a problem with mathematical interpretation.

At that time numbers were illustrated using the geometrical representation of the straight line. So 1 unit could be represented by a straight line segment (representing this magnitude).

So here we would represent the number - literally - in 1-dimensional terms (as a straight line).

If however we now went on to representing 1 in 2-dimensional terms we would get a square (with each of its four equal sides = 1 unit).

Though the answer in both cases, in quantitative terms is the same = 1, a significant qualitative difference is involved with the two answers.

So in the first case with the straight line we are referring to a 1-dimensional format of measurement, whereas in the second with the square we are referring to a qualitatively distinct 2-dimensional format.

Therefore though in reduced quantitative terms the answer remains the same = 1, clearly both answers in qualitative terms are quite distinct.

Again if we moved to measuring in 3 dimensions we could again get an answer = 1 by having all six sides of a cube representing squares (with side = 1 unit).

Conventional Mathematics is therefore - in a very precise manner - based on a linear rational approach. Thus whenever we multiply two numbers (or raise a number to a power other than 1) a qualitative - as well as quantitative - transformation in the numerical value takes place.

However the qualitative aspect of this transformation is then ignored with resulting value expressed in a merely reduced quantitative manner (in terms of the 1st dimension).

So for example in conventional terms 2 * 2 = 4. Though in qualitative terms this numerical operation entails a transformation (from the first to the second dimension), the result is expressed in a merely reduced 1-dimensional quantitative manner as 4 (i.e. 4 linear units).

It has often suggested that the mystery of Riemann Hypothesis relates to a fundamental connection as between addition and multiplication (that is not properly understood).

Well this fundamental connection is ultimately pointing to is the fact that all numbers (and indeed all mathematical symbols) have both quantitative and qualitative interpretations!

However Mathematics - as it is presently construed - is geared solely to the quantitative aspect. By its very nature it is not suited therefore to a proper appreciation – not alone resolution – of the true nature of the Hypothesis.

I became aware again at an early age of another related problem which I felt was glossed over in an unsatisfactory manner in Conventional Mathematics.

Once again - in standard conventional terms - when we square 1, the answer remains unchanged as 1. So the answer here is single valued and unambiguous. However when we now - in inverse terms - obtain the square root, two possible answers arise (in quantitative terms) i.e. + 1 and – 1. So we now - by contrast - have a highly paradoxical two valued solution (where each answer is the negative of the other).

This raises in fact a very deep qualitative issue, which is effectively ignored in Conventional Mathematics (precisely because of its lack of a qualitative dimension).

For example the fundamental notion of proof in Mathematics is based on obtaining a single valued qualitative result. In other words from this perspective a theorem is either true or not true (in unambiguous terms). Indeed the continued fascination with the Riemann Hypothesis is precisely to establish in this context whether it is true or not true. So the positive result i.e. that it is true would thereby - in terms of this logic - rule out the negative result i.e. that it is not true.

Yet in the very simple operation of obtaining the square root of 1 we find that both the positive and negative solutions i.e. + 1 and – 1 can both be true!

So this result, if it was properly appreciated in qualitative terms, would raise fundamental issues regarding the limitations of the standard (linear) logical approach used in Conventional Mathematics!

So to sum up! Even as a child (9- 10 years) two key mathematical issues troubled me.

1) I could vaguely see that every number had both a quantitative and qualitative meaning (with the qualitative meaning simply ignored in conventional terms).

2) I considered - though I would not have used these words at the time - that there was an unsatisfactory asymmetry as between the power of a number on the one hand and the corresponding inverse notion of its root on the other. Whereas again in conventional terms the square of 1, for example, has just one unambiguous result, the inverse operation of obtaining its square root yields two results (which are paradoxical in terms of each other).

However it took some considerable time before I found a way of resolving these issues to my own satisfaction (and realise that they in fact represented two sides of the same coin).

In terms of the gradual evolution of my own understanding in this regard the first seeds were set through a boyhood realisation of - what I saw - as a problem with mathematical interpretation.

At that time numbers were illustrated using the geometrical representation of the straight line. So 1 unit could be represented by a straight line segment (representing this magnitude).

So here we would represent the number - literally - in 1-dimensional terms (as a straight line).

If however we now went on to representing 1 in 2-dimensional terms we would get a square (with each of its four equal sides = 1 unit).

Though the answer in both cases, in quantitative terms is the same = 1, a significant qualitative difference is involved with the two answers.

So in the first case with the straight line we are referring to a 1-dimensional format of measurement, whereas in the second with the square we are referring to a qualitatively distinct 2-dimensional format.

Therefore though in reduced quantitative terms the answer remains the same = 1, clearly both answers in qualitative terms are quite distinct.

Again if we moved to measuring in 3 dimensions we could again get an answer = 1 by having all six sides of a cube representing squares (with side = 1 unit).

Conventional Mathematics is therefore - in a very precise manner - based on a linear rational approach. Thus whenever we multiply two numbers (or raise a number to a power other than 1) a qualitative - as well as quantitative - transformation in the numerical value takes place.

However the qualitative aspect of this transformation is then ignored with resulting value expressed in a merely reduced quantitative manner (in terms of the 1st dimension).

So for example in conventional terms 2 * 2 = 4. Though in qualitative terms this numerical operation entails a transformation (from the first to the second dimension), the result is expressed in a merely reduced 1-dimensional quantitative manner as 4 (i.e. 4 linear units).

It has often suggested that the mystery of Riemann Hypothesis relates to a fundamental connection as between addition and multiplication (that is not properly understood).

Well this fundamental connection is ultimately pointing to is the fact that all numbers (and indeed all mathematical symbols) have both quantitative and qualitative interpretations!

However Mathematics - as it is presently construed - is geared solely to the quantitative aspect. By its very nature it is not suited therefore to a proper appreciation – not alone resolution – of the true nature of the Hypothesis.

I became aware again at an early age of another related problem which I felt was glossed over in an unsatisfactory manner in Conventional Mathematics.

Once again - in standard conventional terms - when we square 1, the answer remains unchanged as 1. So the answer here is single valued and unambiguous. However when we now - in inverse terms - obtain the square root, two possible answers arise (in quantitative terms) i.e. + 1 and – 1. So we now - by contrast - have a highly paradoxical two valued solution (where each answer is the negative of the other).

This raises in fact a very deep qualitative issue, which is effectively ignored in Conventional Mathematics (precisely because of its lack of a qualitative dimension).

For example the fundamental notion of proof in Mathematics is based on obtaining a single valued qualitative result. In other words from this perspective a theorem is either true or not true (in unambiguous terms). Indeed the continued fascination with the Riemann Hypothesis is precisely to establish in this context whether it is true or not true. So the positive result i.e. that it is true would thereby - in terms of this logic - rule out the negative result i.e. that it is not true.

Yet in the very simple operation of obtaining the square root of 1 we find that both the positive and negative solutions i.e. + 1 and – 1 can both be true!

So this result, if it was properly appreciated in qualitative terms, would raise fundamental issues regarding the limitations of the standard (linear) logical approach used in Conventional Mathematics!

So to sum up! Even as a child (9- 10 years) two key mathematical issues troubled me.

1) I could vaguely see that every number had both a quantitative and qualitative meaning (with the qualitative meaning simply ignored in conventional terms).

2) I considered - though I would not have used these words at the time - that there was an unsatisfactory asymmetry as between the power of a number on the one hand and the corresponding inverse notion of its root on the other. Whereas again in conventional terms the square of 1, for example, has just one unambiguous result, the inverse operation of obtaining its square root yields two results (which are paradoxical in terms of each other).

However it took some considerable time before I found a way of resolving these issues to my own satisfaction (and realise that they in fact represented two sides of the same coin).

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