This latest contribution in many ways entails a summary of the points that I have made in recent blog entries and in discussion with Anthony Judge in relation to “Transforming the Art of Conversation - conversing as the transformative science of development” at his hugely impressive “Laetus in Praesens” site.

I have been long fascinated by the fact that the two binary digits (1 and 0)

when used in a quantitative manner can potentially encode all

information processes.

I am therefore of the opinion that the same two digits when used in an

appropriate qualitative manner can likewise potentially encode all transformation processes.

So transformation itself (in all its manifestations) is basically encoded in number when appreciated in a qualitative manner.

Now as geometrical symbols, 1 can be identified with the straight line and 0 with a circular circumference. So the relationship of 1 and 0 in qualitative terms implies the relationship between (rational) linear and (intuitive) circular understanding. (In this context circular refers to the indirect rational attempt through paradox to portray the nature of intuitive understanding).

From a physical perspective this would imply that all transformation processes entail the interaction of a visible phenomenal aspect together with an equally important invisible holistic dimension.

At a deeper level this circular aspect relates to the manner in which the

fundamental polarities - which necessarily underlie all phenomenal

relationships - are configured.

For convenience, I would see that two key sets here are essential to all dynamic relationships i.e. external and internal and whole and part. In dynamic terms, external always implies internal (and internal external). Likewise wholes imply parts (and parts wholes). All phenomenal creation necessarily entails the two-way interaction of both sets of polarities.

Conventional Science and Mathematics are decidedly linear (i.e. 1-dimensional) in the manner that these polarities are treated with isolated independent reference frames employed. So the external (objective) is abstracted from the internal (subjective) aspect; likewise wholes are typically viewed as composed of parts in a mere quantitative manner. Not surprisingly this leads to a highly reduced interpretation of truth!

However an unlimited number of higher dimensional perspectives are possible which all entail an authentic dynamic interaction as between polarities.

The nature of each number, as qualitative dimension, is structurally related to the corresponding notion of quantitative roots of unity.

So the nature of 2-dimensional understanding bears a close relationship

therefore with the two roots of 1, i.e. + 1 and - 1. However whereas with

standard quantitative appreciation, these two values are separated, in

holistic qualitative terms they are interdependent. Thus 2-dimensional

understanding can be therefore expressed as the complementarity of (real)

opposites in the dynamic interaction of poles which are positive and

negative with respect to each other.

These dimensions can be given a geometrical representation (though we must remember that the interpretation is now of a holistic nature).

For example 3-dimensional understanding can be geometrically represented in terms of the well-known Mercedes-Benz logo (which equally is a geometrical representation of the 3 roots of 1).

So in short, each number as dimension, relates to a unique manner of

configuring the dynamic interaction of the two fundamental sets of

polarities. So rather like with a compass with four starting coordinates, we

can obtain ever more detailed notions of direction, likewise starting with

the two fundamental polarity sets we can give ever more refined expression

to the dynamic interaction between opposite coordinates through moving to

higher dimensional numbers! So once again each number in this qualitative

sense represents a unique manner of configuring the dynamic interaction as

between the essential polarities that necessarily underlie phenomenal experience.

This key issue is avoided completely in conventional scientific (and

mathematical) terms through sole concentration on the special limiting case

where understanding in formal terms is 1-dimensional.

Now, I believe that this qualitative holistic notion of dimension intimately

applies to the true nature of space and time. So if we were to map

space-time reality, we could validly say that it is truly multi-dimensional

where the ceaseless interweaving of these qualitative numbers are involved. Going even further, the distinctive qualitative features that phenomena possess, thereby represent multi-dimensional configurations with respect to space and time that are ultimately rooted in the qualitative notion of number.

I would go even further. In dynamic relative terms, phenomena represent but appearances (in continual transformation) of an ultimate reality that is ineffable.

In fact, from this perspective, we can say that such phenomena (which

possess no ultimate substance) fundamentally represent but the dynamic

configurations of number (with respect to both their quantitative and

qualitative aspects).

From a geometrical perspective the quantitative shape of all phenomena can

be understood in terms of the interplay of both linear and circular

properties in varying dimensions.

The corresponding qualitative "shape" of these phenomena in their uniquely

distinctive features can likewise be understood in terms of both specific

and holistic features again with respect to the combined interplay of

multiple dimensional numbers (which again represent a distinctive manner in which the fundamental polarities are dynamically configured).

In a direct sense I would see the quantitative aspect of understanding as

relating to form, with the qualitative relating to the mysterious

transformation of this form.

So if we are to isolate what is common to all patterns of transformation, it

is the intersection of this holistic qualitative aspect with established

quantitative notions of form.

However when one accepts that the very nature of the standard paradigm of

science and mathematics is to attempt to reduce this interaction in a merely quantitative manner, then one can perhaps appreciate why it is inimical to transformation.

It is not that science as such is opposed to such transformation, but rather the present limited version that is wrongly accepted as solely synonymous with valid scientific interpretation!

Now there is much greater freedom for both the development and expression of the qualitative aspect within the arts.

So in the quest to transform present conversation - even scientific conversation - it would be helpful to informally dialogue with artistic metaphors.

Of course acceptance of the (neglected) qualitative aspect of science (and mathematics) would eventually pave the way towards better integration with the arts (with both seen as complementary expressions of the same truth).

My key point again is this!

There is not just one Mathematics (that is qualitatively 1-dimensional in nature) but potentially an infinite set, with each interpretation as the complex expression of a number dimension possessing a partial relative validity. And as phenomenal reality can be expressed as the dynamic interplay of all these dimensional systems in complex space and time (with quantitative and qualitative aspects), ultimately it is vital that we abandon the present total adherence to just one! Even with the best intentions, it therefore continually leads to a reduced form of understanding that eventually can serve as the enemy of true transformation.

## Saturday, September 22, 2012

## Friday, September 21, 2012

### Connections to Taoism

I have always felt a special affinity to Taoism where the basic nature of reality is explained in a manner that readily lends itself to holistic mathematical understanding.

So the Tao represents the ineffable undivided unity (which equally is a nothingness in phenomenal terms).

Then phenomenal reality arises from the splitting of this unity into polar opposites that are understood as separate from each other. However a deeper understanding of the nature of these opposites leads to the realisation that they are complementary (and ultimately identical in nondual terms) as yin and yang. So it is this latter realisation that enables the process of harmonising phenomenal reality with the original absolute nature of Tao.

Last night I was briefly reading the section on Taoism in that wonderful little book on "Mysticism" by F.C. Happold. There, I saw the seeds of an even closer relationship in its thought to my recent notions expressed in these blogs on the all important role of number.

For example on P. 152 we have this statement

"As soon as Tao creates order, it becomes nameable."

Now the very basis of ordering is number, both in its recognised quantitative, and also in its much less recognised qualitative manner.

So, number itself freely arises from the absolute ineffable nature of reality, which then becomes the very means of identification of phenomena.

Just a few sentences later on the same page we have a more graphic statement on the fundamental nature of number!

"Tao produced Unity; Unity produced Duality: Duality produced Trinity; and Trinity produced all existing objects.

These myriad objects leave darkness behind them and embrace the light, being harmonised by contact with the Vital Force."

So Unity, Duality and Trinity simply represent the qualitative holistic notions of 1, 2 and 3 respectively from which all phenomenal objects arise. Now one might validly query the sole emphasis on the holistic notion of number here! However the key point is the direct connection then made as between number and the manifest identity of phenomenal objects!

So these multiple objects then leave darkness behind. What is implied here is that in the original state of Tao, where - by definition - no differentiation (or integration) has yet taken place, such objects would enjoy a mere potential for existence. So in becoming differentiated as separate objects, evolution can begin the process of gradual actualisation of Tao. And it is in the recognition of the ultimate nondual nature of phenomena (through the complementary yin and yang aspects of nature) that the integrated state of all phenomena thereby arises (which is inseparable from their absolute identity in Tao).

My simple purpose in all these blogs is to understand the true nature of Mathematics and Science as fully consistent with the accumulated great wisdom of the various mystical traditions.

And when one looks carefully, the seeds of such reconciliation are already evident in these traditions (as illustrated here in an emphatic manner in Taoist literature).

So the Tao represents the ineffable undivided unity (which equally is a nothingness in phenomenal terms).

Then phenomenal reality arises from the splitting of this unity into polar opposites that are understood as separate from each other. However a deeper understanding of the nature of these opposites leads to the realisation that they are complementary (and ultimately identical in nondual terms) as yin and yang. So it is this latter realisation that enables the process of harmonising phenomenal reality with the original absolute nature of Tao.

Last night I was briefly reading the section on Taoism in that wonderful little book on "Mysticism" by F.C. Happold. There, I saw the seeds of an even closer relationship in its thought to my recent notions expressed in these blogs on the all important role of number.

For example on P. 152 we have this statement

"As soon as Tao creates order, it becomes nameable."

Now the very basis of ordering is number, both in its recognised quantitative, and also in its much less recognised qualitative manner.

So, number itself freely arises from the absolute ineffable nature of reality, which then becomes the very means of identification of phenomena.

Just a few sentences later on the same page we have a more graphic statement on the fundamental nature of number!

"Tao produced Unity; Unity produced Duality: Duality produced Trinity; and Trinity produced all existing objects.

These myriad objects leave darkness behind them and embrace the light, being harmonised by contact with the Vital Force."

So Unity, Duality and Trinity simply represent the qualitative holistic notions of 1, 2 and 3 respectively from which all phenomenal objects arise. Now one might validly query the sole emphasis on the holistic notion of number here! However the key point is the direct connection then made as between number and the manifest identity of phenomenal objects!

So these multiple objects then leave darkness behind. What is implied here is that in the original state of Tao, where - by definition - no differentiation (or integration) has yet taken place, such objects would enjoy a mere potential for existence. So in becoming differentiated as separate objects, evolution can begin the process of gradual actualisation of Tao. And it is in the recognition of the ultimate nondual nature of phenomena (through the complementary yin and yang aspects of nature) that the integrated state of all phenomena thereby arises (which is inseparable from their absolute identity in Tao).

My simple purpose in all these blogs is to understand the true nature of Mathematics and Science as fully consistent with the accumulated great wisdom of the various mystical traditions.

And when one looks carefully, the seeds of such reconciliation are already evident in these traditions (as illustrated here in an emphatic manner in Taoist literature).

## Saturday, September 8, 2012

### What is Number?

We have to be careful here. It is very hard in practice to distinguish numbers from the symbols used for their representation.

And the very nature of such representation is that we thereby give a distinct phenomenal identity to number (as represented by its symbol).

So when I use the symbol "1" to represent the notion of one, it thereby assumes this phenomenal identity.

Furthermore because understanding of number in our culture is dominated by its quantitative aspect, numbers thereby become misleadingly identified as abstract phenomenal objects (with an absolute identity).

However in truth the meaning of number is much more elusive.

As I have been at pains to illustrate, every number has both a qualitative as well as recognised quantitative aspect. Basically, the quantitative aspect relates to the notion of number as independent (i.e. where phenomenal poles such as external and internal are separated). The corresponding qualitative aspect relates to the corresponding notion of number as interdependent (where these same poles are understood as inherently complementary and ultimately identical).

We can easily illustrate this with respect to 1.

In conventional terms 1 is given a mere quantitative meaning i.e. as a separate number object. This notion is indeed extremely important and serves as the fundamental basis for discrimination of any phenomenal object. Therefore in order to recognise an object phenomenon as a distinctive unit, the quantitative notion of 1 must necessarily be already implicit in such understanding.

However 1 can equally be given a qualitative holistic meaning as "oneness". The best example of this relates to the ultimate experience of spiritual oneness (where the explicit notion of an object as a separate phenomenon no longer arises).

So the very notion of 1 in this alternative qualitative sense pertains to the notion of pure interdependent relatedness (based on the identity of opposite poles).

Put another way, the quantitative notion of number is based on either/or linear logic, where the positive poles excludes the negative..

Therefore in the expression where 1 - 1 = 0, 1 ≠ 0.

However the qualitative notion of number is based by contrast on both/and circular logic, where the positive pole includes the negative.

Therefore from this perspective where 1 - 1 = 0, 1 (as oneness now defined in this complementary manner) = 0 (as nothingness).

However before we can understand the (common) interdependence of opposite poles, we must recognise their (separate) independence (and vice versa).

So properly understood, both the quantitative and qualitative notions of number are inextricably linked in all experience.

Thus, the ultimate notion of number (though necessarily implicit in all phenomenal observation) is of an ineffable nature where both quantitative and qualitative aspects coincide.

In this sense, though we must necessarily represent numbers in phenomenal terms with symbols, they cannot be confused with physical phenomena (where number is already inherent in their recognition).

Put another way, physical phenomena themselves represent a certain rigid confusion with respect to the quantitative and qualitative aspects of number. In other words, we can only recognise such phenomena, through maintaining a certain imbalance with respect to the quantitative and qualitative aspects of number.

Once we recognise a physical object for example, we thereby associate number with its merely quantitative aspect.

In this sense the very quest for ultimate spiritual unity is the corresponding desire to reconcile both the quantitative and qualitative aspects of number in their original ineffable state.

So 1, in the unity of all form (through circular understanding) as pure interdependence is inseparable from 0 (as the emptiness or nothingness with respect to separate phenomena).

Thus once again, 1 - 1 = 0.

However when we switch to linear (quantitative) logic, both poles are now positive

So we have 1 + 1 = 2.

Thus duality (as the qualitative meaning of 2) arises from application of the alternative logic.

In dynamic terms, all phenomenal reality in its forms and transformations represents the dynamic interaction of both types of logic (representing the quantitative and qualitative aspects of number).

So from this perspective, we could say that the very goal of all evolution is to ultimately realise the true original state of number (where quantitative and qualitative aspects are indistinguishable).

And all the fundamental mathematical operations can be validly seen as an extension as to what is implied through the notion of number.

This thereby gives an extraordinary significance to the role of a more comprehensive mathematical understanding (where both its quantitative and qualitative aspects are explicitly recognised).

And the very nature of such representation is that we thereby give a distinct phenomenal identity to number (as represented by its symbol).

So when I use the symbol "1" to represent the notion of one, it thereby assumes this phenomenal identity.

Furthermore because understanding of number in our culture is dominated by its quantitative aspect, numbers thereby become misleadingly identified as abstract phenomenal objects (with an absolute identity).

However in truth the meaning of number is much more elusive.

As I have been at pains to illustrate, every number has both a qualitative as well as recognised quantitative aspect. Basically, the quantitative aspect relates to the notion of number as independent (i.e. where phenomenal poles such as external and internal are separated). The corresponding qualitative aspect relates to the corresponding notion of number as interdependent (where these same poles are understood as inherently complementary and ultimately identical).

We can easily illustrate this with respect to 1.

In conventional terms 1 is given a mere quantitative meaning i.e. as a separate number object. This notion is indeed extremely important and serves as the fundamental basis for discrimination of any phenomenal object. Therefore in order to recognise an object phenomenon as a distinctive unit, the quantitative notion of 1 must necessarily be already implicit in such understanding.

However 1 can equally be given a qualitative holistic meaning as "oneness". The best example of this relates to the ultimate experience of spiritual oneness (where the explicit notion of an object as a separate phenomenon no longer arises).

So the very notion of 1 in this alternative qualitative sense pertains to the notion of pure interdependent relatedness (based on the identity of opposite poles).

Put another way, the quantitative notion of number is based on either/or linear logic, where the positive poles excludes the negative..

Therefore in the expression where 1 - 1 = 0, 1 ≠ 0.

However the qualitative notion of number is based by contrast on both/and circular logic, where the positive pole includes the negative.

Therefore from this perspective where 1 - 1 = 0, 1 (as oneness now defined in this complementary manner) = 0 (as nothingness).

However before we can understand the (common) interdependence of opposite poles, we must recognise their (separate) independence (and vice versa).

So properly understood, both the quantitative and qualitative notions of number are inextricably linked in all experience.

Thus, the ultimate notion of number (though necessarily implicit in all phenomenal observation) is of an ineffable nature where both quantitative and qualitative aspects coincide.

In this sense, though we must necessarily represent numbers in phenomenal terms with symbols, they cannot be confused with physical phenomena (where number is already inherent in their recognition).

Put another way, physical phenomena themselves represent a certain rigid confusion with respect to the quantitative and qualitative aspects of number. In other words, we can only recognise such phenomena, through maintaining a certain imbalance with respect to the quantitative and qualitative aspects of number.

Once we recognise a physical object for example, we thereby associate number with its merely quantitative aspect.

In this sense the very quest for ultimate spiritual unity is the corresponding desire to reconcile both the quantitative and qualitative aspects of number in their original ineffable state.

So 1, in the unity of all form (through circular understanding) as pure interdependence is inseparable from 0 (as the emptiness or nothingness with respect to separate phenomena).

Thus once again, 1 - 1 = 0.

However when we switch to linear (quantitative) logic, both poles are now positive

So we have 1 + 1 = 2.

Thus duality (as the qualitative meaning of 2) arises from application of the alternative logic.

In dynamic terms, all phenomenal reality in its forms and transformations represents the dynamic interaction of both types of logic (representing the quantitative and qualitative aspects of number).

So from this perspective, we could say that the very goal of all evolution is to ultimately realise the true original state of number (where quantitative and qualitative aspects are indistinguishable).

And all the fundamental mathematical operations can be validly seen as an extension as to what is implied through the notion of number.

This thereby gives an extraordinary significance to the role of a more comprehensive mathematical understanding (where both its quantitative and qualitative aspects are explicitly recognised).

## Friday, September 7, 2012

### Binary Wonder

We are already well aware of the great significance of number from the conventional quantitative perspective. However what we have not yet recognised yet is the equal significance of number from the greatly neglected qualitative dimensional perspective.

And when we combine these two aspects of number in interactive terms, then it is but a short leap to the recognition that - at its most fundamental - phenomenal reality is but the dynamic representation of number configurations (in both quantitative and qualitative terms).

Now a widespread view in contemporary physics is that reality is fundamentally composed of tiny 1-dimensional strings, the unique vibrations of which give rise to all the particles from which more conventional material forms are composed.

However the notion of "physical strings" in any meaningful philosophical sense is but a fiction arising from the reductionist quantitative viewpoint that matter must ultimately be composed of smaller constituent parts of matter with the "strings" thereby representing the most fundamental parts.

However from a more correct dynamic perspective all such matter particles must necessarily be of a merely relative nature, arising from dynamic interaction of quantitative and qualitative aspects. And as neither quantitative or qualitative aspects have meaning in the absence of such interaction we can thereby never ultimately isolate the basic constituents of matter in merely quantitative terms.

This then leads to the even more extraordinary realisation that - what we recognise as matter - represents but the dynamic configuration of what in phenomenal terms are recognised as numbers.

Now a number has no physical identity in either quantitative or qualitative terms (in isolation). However it is the unique vibration of both the quantitative and qualitative aspects of number that give rise to the rigid physical appearances in nature that we recognise as matter.

The two most important numbers - what I refer to as the original numbers - are 1 and 0.

We are now discovering in this digital age the great significance of these two numbers as a potential means of encoding all information.

However again what is not equally recognised is the qualitative counterpart of this digital revolution, whereby the same two digits can be seen as a potential means of encoding all transformation processes.

It has always impressed me how these two numbers play such a big role in representing the mystical experience of reality.

In the Western religious traditions - based more heavily on linear notions of form - the peak experience of transformation is commonly expressed in terms of union (or oneness) which is the holistic qualitative notion of 1.

In the Eastern traditions - based more on circular cyclical notions - the peak experience is by contrast expressed in terms of a void or emptiness (i.e. nothingness) which is the holistic qualitative notion of 0.

And as phenomenal reality itself is an information system undergoing continual transformation, we can perhaps recognise that such reality represents but the dynamic interaction of a digital system (based on the numbers 1 and 0) with twin aspects that are quantitative and qualitative respectively.

So the numbers 1 and 0 in this sense are sufficient to encode all reality with respect to (quantitative) information and (qualitative) transformation.

However before this system can operate in phenomenal terms, a precise requirement is necessary relating to the prime number code.

Therefore though prime numbers can be represented in a binary manner, with respect to both quantitative and qualitative characteristics, only one configuration is possible (in the generation of the prime numbers) as - literally - the prime condition for the subsequent unfolding of the phenomenal universe.

And when we combine these two aspects of number in interactive terms, then it is but a short leap to the recognition that - at its most fundamental - phenomenal reality is but the dynamic representation of number configurations (in both quantitative and qualitative terms).

Now a widespread view in contemporary physics is that reality is fundamentally composed of tiny 1-dimensional strings, the unique vibrations of which give rise to all the particles from which more conventional material forms are composed.

However the notion of "physical strings" in any meaningful philosophical sense is but a fiction arising from the reductionist quantitative viewpoint that matter must ultimately be composed of smaller constituent parts of matter with the "strings" thereby representing the most fundamental parts.

However from a more correct dynamic perspective all such matter particles must necessarily be of a merely relative nature, arising from dynamic interaction of quantitative and qualitative aspects. And as neither quantitative or qualitative aspects have meaning in the absence of such interaction we can thereby never ultimately isolate the basic constituents of matter in merely quantitative terms.

This then leads to the even more extraordinary realisation that - what we recognise as matter - represents but the dynamic configuration of what in phenomenal terms are recognised as numbers.

Now a number has no physical identity in either quantitative or qualitative terms (in isolation). However it is the unique vibration of both the quantitative and qualitative aspects of number that give rise to the rigid physical appearances in nature that we recognise as matter.

The two most important numbers - what I refer to as the original numbers - are 1 and 0.

We are now discovering in this digital age the great significance of these two numbers as a potential means of encoding all information.

However again what is not equally recognised is the qualitative counterpart of this digital revolution, whereby the same two digits can be seen as a potential means of encoding all transformation processes.

It has always impressed me how these two numbers play such a big role in representing the mystical experience of reality.

In the Western religious traditions - based more heavily on linear notions of form - the peak experience of transformation is commonly expressed in terms of union (or oneness) which is the holistic qualitative notion of 1.

In the Eastern traditions - based more on circular cyclical notions - the peak experience is by contrast expressed in terms of a void or emptiness (i.e. nothingness) which is the holistic qualitative notion of 0.

And as phenomenal reality itself is an information system undergoing continual transformation, we can perhaps recognise that such reality represents but the dynamic interaction of a digital system (based on the numbers 1 and 0) with twin aspects that are quantitative and qualitative respectively.

So the numbers 1 and 0 in this sense are sufficient to encode all reality with respect to (quantitative) information and (qualitative) transformation.

However before this system can operate in phenomenal terms, a precise requirement is necessary relating to the prime number code.

Therefore though prime numbers can be represented in a binary manner, with respect to both quantitative and qualitative characteristics, only one configuration is possible (in the generation of the prime numbers) as - literally - the prime condition for the subsequent unfolding of the phenomenal universe.

## Wednesday, September 5, 2012

### Prime Movers

We now come back to highlighting the significance of the prime numbers.

Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system.

What this again implies is that all numbers (as dimensions) are built up from prime number constituents.

Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms).

Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature.

Thus, looked at from these two distinct perspectives (in isolation) the prime numbers can be thereby seen as the basis for all natural characteristics (quantitative and qualitative) .

But we now come back to a familiar paradox. What seems unambiguous within isolated reference frames, becomes deeply paradoxical when these frames are treated as interdependent.

So one once again, when I walk up a road (understood in isolation) movement takes place positively in space and time.

Then when I walk down the same road (in isolation) movement likewise takes place positively in space and time.

However when we understood these two reference frames (i.e. "up" and "down") as interdependent, movement takes place - relatively - in both positive and negative directions in space and time.

It is the same with respect to the prime numbers.

When we consider both the quantitative and qualitative aspects in isolation, the prime appear unambiguously as the building blocks of the natural numbers.

However when we consider both quantitative and qualitative in dynamic relationship to each other (as interdependent) then this comforting picture breaks down with both prime and natural numbers simultaneously giving rise to each other.

What this means in effect is that the mysterious connection, that links primes and natural numbers in such a synchronous manner, is of an ineffable nature (and already inherent in number processes when they phenomenally arise).

And once again it is this mysterious connection to which the Riemann Hypothesis directly applies!

So, properly understood, the Riemann Hypothesis is a mathematical statement of the condition required to reconcile both the quantitative and qualitative behaviour of the primes.

And as Conventional Mathematics is formally defined by a merely quantitative interpretation of its symbols, the Riemann Hypothesis not only cannot be proved (or disproved) in this manner; it cannot even be properly understood from this perspective.

However far from being a defeat, a proper realisation of this fact would then open the way for a much more comprehensive appreciation of the true nature of Mathematics.

Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system.

What this again implies is that all numbers (as dimensions) are built up from prime number constituents.

Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms).

Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature.

Thus, looked at from these two distinct perspectives (in isolation) the prime numbers can be thereby seen as the basis for all natural characteristics (quantitative and qualitative) .

But we now come back to a familiar paradox. What seems unambiguous within isolated reference frames, becomes deeply paradoxical when these frames are treated as interdependent.

So one once again, when I walk up a road (understood in isolation) movement takes place positively in space and time.

Then when I walk down the same road (in isolation) movement likewise takes place positively in space and time.

However when we understood these two reference frames (i.e. "up" and "down") as interdependent, movement takes place - relatively - in both positive and negative directions in space and time.

It is the same with respect to the prime numbers.

When we consider both the quantitative and qualitative aspects in isolation, the prime appear unambiguously as the building blocks of the natural numbers.

However when we consider both quantitative and qualitative in dynamic relationship to each other (as interdependent) then this comforting picture breaks down with both prime and natural numbers simultaneously giving rise to each other.

What this means in effect is that the mysterious connection, that links primes and natural numbers in such a synchronous manner, is of an ineffable nature (and already inherent in number processes when they phenomenally arise).

And once again it is this mysterious connection to which the Riemann Hypothesis directly applies!

So, properly understood, the Riemann Hypothesis is a mathematical statement of the condition required to reconcile both the quantitative and qualitative behaviour of the primes.

And as Conventional Mathematics is formally defined by a merely quantitative interpretation of its symbols, the Riemann Hypothesis not only cannot be proved (or disproved) in this manner; it cannot even be properly understood from this perspective.

However far from being a defeat, a proper realisation of this fact would then open the way for a much more comprehensive appreciation of the true nature of Mathematics.

## Tuesday, September 4, 2012

### Reality as Number

We now come back to highlighting the significance of the prime numbers.

Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system.

What this again implies is that all numbers (as dimensions) are built up from prime number constituents.

Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms).

Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature.

Thus, looked at from these two distinct perspectives (in isolation) the prime numbers can be thereby seen as the basis for all natural characteristics (quantitative and qualitative) .

But we now come back to a familiar paradox. What seems unambiguous within isolated reference frames, becomes deeply paradoxical when these frames are treated as interdependent.

So one once again, when I walk up a road (understood in isolation) movement takes place positively in space and time.

Then when I walk down the same road (in isolation) movement likewise takes place positively in space and time.

However when we understood these two reference frames (i.e. "up" and "down") as interdependent, movement takes place - relatively - in both positive and negative directions in space and time.

It is the same with respect to the prime numbers.

When we consider both the quantitative and qualitative aspects in isolation, the prime appear unambiguously as the building blocks of the natural numbers.

However when we consider both quantitative and qualitative in dynamic relationship to each other (as interdependent) then this comforting picture breaks down with both prime and natural numbers simultaneously giving rise to each other.

What this means in effect is that the mysterious connection, that links primes and natural numbers in such a synchronous manner, is of an ineffable nature (and already inherent in number processes when they phenomenally arise).

And once again it is this mysterious connection to which the Riemann Hypothesis directly applies!

So, properly understood, the Riemann Hypothesis is a mathematical statement of the condition required to reconcile both the quantitative and qualitative behaviour of the primes.

And as Conventional Mathematics is formally defined by a merely quantitative interpretation of its symbols, the Riemann Hypothesis not only cannot be proved (or disproved) in this manner; it cannot even be properly understood from this perspective.

However far from being a defeat, a proper realisation of this fact would then open the way for a much more comprehensive appreciation of the true nature of Mathematics.

Just as the prime numbers are recognised in quantitative terms as the building blocks of the natural number system, likewise the prime numbers - though conventionally unrecognised - are equally the building blocks in qualitative terms of the natural number system.

What this again implies is that all numbers (as dimensions) are built up from prime number constituents.

Then, as the number dimensions directly relate to the dimensions of space and time (physically and psychologically) these likewise are built from prime numbers (in qualitative terms).

Furthermore, as the qualitative characteristics that are inherent in natural phenomena are but manifestations of such space and time configurations, the prime numbers can then be clearly seen - in literal terms - as the fundamental basis of all qualitative characteristics in nature.

Thus, looked at from these two distinct perspectives (in isolation) the prime numbers can be thereby seen as the basis for all natural characteristics (quantitative and qualitative) .

But we now come back to a familiar paradox. What seems unambiguous within isolated reference frames, becomes deeply paradoxical when these frames are treated as interdependent.

So one once again, when I walk up a road (understood in isolation) movement takes place positively in space and time.

Then when I walk down the same road (in isolation) movement likewise takes place positively in space and time.

However when we understood these two reference frames (i.e. "up" and "down") as interdependent, movement takes place - relatively - in both positive and negative directions in space and time.

It is the same with respect to the prime numbers.

When we consider both the quantitative and qualitative aspects in isolation, the prime appear unambiguously as the building blocks of the natural numbers.

However when we consider both quantitative and qualitative in dynamic relationship to each other (as interdependent) then this comforting picture breaks down with both prime and natural numbers simultaneously giving rise to each other.

What this means in effect is that the mysterious connection, that links primes and natural numbers in such a synchronous manner, is of an ineffable nature (and already inherent in number processes when they phenomenally arise).

And once again it is this mysterious connection to which the Riemann Hypothesis directly applies!

So, properly understood, the Riemann Hypothesis is a mathematical statement of the condition required to reconcile both the quantitative and qualitative behaviour of the primes.

And as Conventional Mathematics is formally defined by a merely quantitative interpretation of its symbols, the Riemann Hypothesis not only cannot be proved (or disproved) in this manner; it cannot even be properly understood from this perspective.

However far from being a defeat, a proper realisation of this fact would then open the way for a much more comprehensive appreciation of the true nature of Mathematics.

## Monday, September 3, 2012

### Multidimensional Nature of Time and Space (20)

Yesterday, I briefly attempted to explain the qualitative significance of the Euler Identity which essentially represents a holistic mathematical description of the precise nature of spiritual transformation where emptiness and form (and form and emptiness) are united. Here the contemplative journey - literally - comes full circle with the transcendent goal (beyond all phenomenal form) finally revealed as identical with its immanent source (as already inherent within such form).

Last week I was looking at a fascinating programme on the mapping of the Universe. We are of course accustomed to the mapping of planet Earth and to a lesser extent our planetary system. But ambitious attempts have already been made to provide a map of the entire Milky Way galaxy. And even beyond that considerable progress has been made with respect to the mapping of the visible universe (made up of countless billions of galaxies). Even some are already attempting to know what lies beyond the visible universe, with the current view that it is infinite in extent (or as I would prefer to put finitely unlimited with respect to its size).

Also it is quite apparent that on the grand scale that we have to move well beyond conventional notions of matter and energy. Though only dimly understood the prevailing view now is that most of the Universe is comprised of dark matter and dark energy. Now this seems to complement my own finding that when we look at Mathematics on a grand scale that the unconscious aspect of interpretation thereby assumes great significance!

So there are strong parallels here to my attempts to map both the mathematical and scientific Universe with the prevailing paradigm akin to the mapping of planet Earth. However there is so, so much more out there waiting to be discovered that we have not yet even begun to consider.

It perhaps will provide more perspective on my approach by giving a little more information on the "map of development" that I have now been using for the past few years.

I break this down into seven major bands (with each band comprising three major stages of development).

The first band is well recognised in Developmental Psychology and is concerned with the gradual differentiation of conscious type abilities (in what might be referred as the archaic, magical and mythic stages).

The second band then is concerned with the specialisation of consciousness through (linear) reason. As we have seen such thinking defines the prevailing paradigm and has become long established in - what we know as - Mathematics and Science. This is the gross realm containing both conop (concrete operational), formop (formal operational) and vision logic understanding. The third stage refers to reason that is infused with intuition, and is especially important for creative work of an original kind!

Now the second band - correctly understood - relates merely to specialisation with respect to the quantitative aspect of mathematical (and scientific) understanding i.e Type 1.

The third band is then concerned - from a scientific perspective - with the unfolding of higher dimensional understanding that is directly intuitive in nature, though indirectly can be logically expressed in a circular rational manner. The stages of this band are sometimes referred to as the psychic, subtle and causal realms and we have already looked at the nature of the higher dimensional numbers associated with these realms.

The fourth band (nondual reality) relates to both an extremely refined intuitive and rational type understanding and is concerned with the consolidation of this circular type appreciation.

So the fourth band - correctly interpreted - is concerned with complementary specialisation of the qualitative aspect of both mathematical (and scientific) understanding.

And putting it bluntly this aspect (Type 2) still remains completely unrecognised from the conventional mathematical (and scientific) perspectives.

However I now recognise three further bands on the full Spectrum of Development which I customarily refer to as radial development.

These would entail the growing mature interpenetration of reason (linear and circular) with contemplation.

In relation to both Science and Mathematics it would lead to the increasing dynamic interplay of both the quantitative and qualitative aspects of understanding. Band 5 would relate to its initial unfolding whereas with Band 6, full specialised understanding would commence. In relation to all the number types therefore, one would now seek the combined understanding of both quantitative and qualitative appreciation in what I refer to as Type 3 Mathematics.

This would likewise be associated with a final advance with respect to the qualitative aspect of dimensional understanding.

We have already looked at the distinction as between the real and imaginary notions of number with respect to dimension. However until both transcendent and immanent directions are fully harmonised in experience, these tend to be used separately.

However, with the unfolding of the radial stages, they can now increasingly be used in a simultaneous manner. So this means in effect that the qualitative aspect of dimensional numbers as complex can now be appropriately reflected in experience. This likewise entails a corresponding dynamic appreciation of the nature of time (and space) in likewise complex terms that once again applies in complementary physical and psychological terms.

Last week I was looking at a fascinating programme on the mapping of the Universe. We are of course accustomed to the mapping of planet Earth and to a lesser extent our planetary system. But ambitious attempts have already been made to provide a map of the entire Milky Way galaxy. And even beyond that considerable progress has been made with respect to the mapping of the visible universe (made up of countless billions of galaxies). Even some are already attempting to know what lies beyond the visible universe, with the current view that it is infinite in extent (or as I would prefer to put finitely unlimited with respect to its size).

Also it is quite apparent that on the grand scale that we have to move well beyond conventional notions of matter and energy. Though only dimly understood the prevailing view now is that most of the Universe is comprised of dark matter and dark energy. Now this seems to complement my own finding that when we look at Mathematics on a grand scale that the unconscious aspect of interpretation thereby assumes great significance!

So there are strong parallels here to my attempts to map both the mathematical and scientific Universe with the prevailing paradigm akin to the mapping of planet Earth. However there is so, so much more out there waiting to be discovered that we have not yet even begun to consider.

It perhaps will provide more perspective on my approach by giving a little more information on the "map of development" that I have now been using for the past few years.

I break this down into seven major bands (with each band comprising three major stages of development).

The first band is well recognised in Developmental Psychology and is concerned with the gradual differentiation of conscious type abilities (in what might be referred as the archaic, magical and mythic stages).

The second band then is concerned with the specialisation of consciousness through (linear) reason. As we have seen such thinking defines the prevailing paradigm and has become long established in - what we know as - Mathematics and Science. This is the gross realm containing both conop (concrete operational), formop (formal operational) and vision logic understanding. The third stage refers to reason that is infused with intuition, and is especially important for creative work of an original kind!

Now the second band - correctly understood - relates merely to specialisation with respect to the quantitative aspect of mathematical (and scientific) understanding i.e Type 1.

The third band is then concerned - from a scientific perspective - with the unfolding of higher dimensional understanding that is directly intuitive in nature, though indirectly can be logically expressed in a circular rational manner. The stages of this band are sometimes referred to as the psychic, subtle and causal realms and we have already looked at the nature of the higher dimensional numbers associated with these realms.

The fourth band (nondual reality) relates to both an extremely refined intuitive and rational type understanding and is concerned with the consolidation of this circular type appreciation.

So the fourth band - correctly interpreted - is concerned with complementary specialisation of the qualitative aspect of both mathematical (and scientific) understanding.

And putting it bluntly this aspect (Type 2) still remains completely unrecognised from the conventional mathematical (and scientific) perspectives.

However I now recognise three further bands on the full Spectrum of Development which I customarily refer to as radial development.

These would entail the growing mature interpenetration of reason (linear and circular) with contemplation.

In relation to both Science and Mathematics it would lead to the increasing dynamic interplay of both the quantitative and qualitative aspects of understanding. Band 5 would relate to its initial unfolding whereas with Band 6, full specialised understanding would commence. In relation to all the number types therefore, one would now seek the combined understanding of both quantitative and qualitative appreciation in what I refer to as Type 3 Mathematics.

This would likewise be associated with a final advance with respect to the qualitative aspect of dimensional understanding.

We have already looked at the distinction as between the real and imaginary notions of number with respect to dimension. However until both transcendent and immanent directions are fully harmonised in experience, these tend to be used separately.

However, with the unfolding of the radial stages, they can now increasingly be used in a simultaneous manner. So this means in effect that the qualitative aspect of dimensional numbers as complex can now be appropriately reflected in experience. This likewise entails a corresponding dynamic appreciation of the nature of time (and space) in likewise complex terms that once again applies in complementary physical and psychological terms.

## Saturday, September 1, 2012

### Multidimensional Nature of Time and Space (19)

Yesterday we considered how number (as qualitative dimension) can be given an imaginary (as well as real) meaning and that this thereby also applies to time (and space) in both physical and psychological terms.

Basically what this entails is that development can take two complementary directions that are transcendent and immanent with respect to each other. Therefore if we associate real numbers (as dimensions) with the transcendent aspect, then the corresponding imaginary numbers are then - relatively - associated with the immanent aspect.

Though all these numbers (as dimensions) are implicit in actual human experience, remarkably little progress has yet been made with respect to any coherent explicit appreciation. And as I have stated repeatedly the conventional paradigm of Science and Mathematics as we know is merely of a 1-dimensional nature (in qualitative terms).

Now, when appropriately interpreted, the other dimensional numbers do unfold in varying degrees through the process of (authentic) contemplative development.

However as practitioners in the past were rarely directly concerned with the mathematical implications of their newly acquired spiritual vision, the important scientific consequences were never made (except in the most general terms).

So my own special concern from the start has been to marry the contemplative vision with rational understanding through exploration of the amazing new mathematical (and associated scientific landscapes) that thereby emerge.

One significant clue as to the nature of the imaginary number (as dimension) can be given through raising 1 to the power of i.

Now as we have seen when we raise 1 to a rational number (such as 1/3) we generate a new number in the circular number system (i.e. on the circle of unit radius in the complex plane).

However when we now raise 1 to the imaginary number i, we generate a new number in the linear number system. So we can see here how real and imaginary numbers (as qualitative dimensions) are associated with circular and linear quantitative results respectively.

So a key task with healthy contemplative development is the successful balancing of both transcendent and immanent directions. This implies likewise the successful balancing of appreciation of number (as qualitative dimension) in a - relatively - real and imaginary manner.

As we saw yesterday transcendental type understanding (in qualitative terms) is of the most refined manner possible in the phenomenal realm.

So before moving directly into the subject of today's entry, I will briefly summarise on the various types of transcendental dimensions.

Real transcendental dimensions are of the most refined type whereby one understands all phenomenal relations - not in terms of (separate) linear or (circular) holistic notions - but rather as the relationship between both.

The positive refer to the subtle rational appreciation of this relationship; the negative then relate to direct intuitive realisation.

However the final step in the phenomenal realm is making the understanding associated with corresponding imaginary directions explicit.

So the imaginary transcendental dimensions relate to understanding of projections (in the indirect conscious expression of unconscious meaning) as the relationship between both (separate) linear and (holistic) circular understanding.

Indeed it is precisely in successfully being able to understand projections in terms of this necessary relationship of conscious and unconscious that the involuntary nature of such projection ceases. So involuntary projections always arises due to a certain failure in properly relating the unconscious desire for meaning (embodied in such projections) with the conscious phenomenal circumstances through which they are expressed.

Once again the positive expression of such imaginary transcendental dimensions relates to a highly refined form of rational understanding of their nature; the negative expression again relates to the more direct intuitive realisation of this nature.

Now, I have already likened the contemplative journey to a steep mountain climb. The real transcendent aspect of this journey - notice the close association here with the qualitative mathematical meaning of transcendental - relates to the ascent (that ultimately leads to a spiritual experience beyond all phenomena of form).

The corresponding descent then relates to the immanent aspect of the journey resulting in a spiritual experience that is inherent within all phenomena.

Just as the Riemann Hypothesis is generally considered the most important unsolved problem in Mathematics, the Euler Identity is likewise considered its most remarkable equation (formula, relationship).

Now because every quantitative relationship equally has a qualitative significance (that is formally unrecognised in conventional mathematical terms), this suggests that an extremely important qualitative meaning is associated with the Euler Identity (that can only be decoded in the appropriate manner).

Now conventionally the Euler Identity is expressed by the equation,

e^(πi) + 1 = 0;

Therefore e^(πi) = - 1.

Then by squaring both sides

e^(2πi) = 1.

We now have e (which is itself a transcendental number) raised to a dimensional expression (that is of an imaginary transcendental nature).

Notice how when we raised a rational number to a rational number the result was irrational; then when we raised an irrational number to an irrational number the result was transcendental. So we have continued to move in the direction of increasing transcendence (from a qualitative perspective).

However now in this special case where the base transcendental number is e, we raise it to a special case of a transcendental number where the dimension is 2πi, we obtain the simplest of all rational numbers (which in qualitative terms is thereby of the most immanent nature).

So putting it simply, the Euler Identity, when understood in an appropriate qualitative manner, points to the mysterious transformation in contemplative development where both form and emptiness are perfectly reconciled.

We saw earlier how development entails both differentiation and integration (through the odd and even numbered dimensions respectively. So the ultimate task of transformation is to reach a state where differentiation and integration both coincide.

Now e is the perfect numerical symbol of such transformation as both the differential and integral of e^x are uniquely the same!

Now in the real unit circle, the circular circumference is 2π. However if we now consider the radius as imaginary - rather than real - then the circular circumference is 2πi. However imaginary in this qualitative sense combines both positive and negative directions. So the imaginary circle is better represented as a non-dimensional point. Here both line and circle are perfectly reconciled from a quantitative perspective; likewise both linear and circular understanding are perfectly reconciled in qualitative terms.

So e^(2πi) in qualitative terms is inseparable from e^0 in quantitative terms!

Therefore in qualitative terms - when finally experience becomes of a pure formless nature as transcendence - it thereby equally becomes immanent as inherent in all form (represented qualitatively as 1).

In the various mystical traditions extensive attention has been given to the nature of this key transformation.

Perhaps the most famous expression is given in the Buddhist sutra:

"Form is nor other than Emptiness

Emptiness is not other than Form."

Well in a precise mathematical manner (where symbols are appropriately understand in the qualitative manner) the Euler Identity describes the same transformation.

However even this is not the end of the mathematical story of number

Basically what this entails is that development can take two complementary directions that are transcendent and immanent with respect to each other. Therefore if we associate real numbers (as dimensions) with the transcendent aspect, then the corresponding imaginary numbers are then - relatively - associated with the immanent aspect.

Though all these numbers (as dimensions) are implicit in actual human experience, remarkably little progress has yet been made with respect to any coherent explicit appreciation. And as I have stated repeatedly the conventional paradigm of Science and Mathematics as we know is merely of a 1-dimensional nature (in qualitative terms).

Now, when appropriately interpreted, the other dimensional numbers do unfold in varying degrees through the process of (authentic) contemplative development.

However as practitioners in the past were rarely directly concerned with the mathematical implications of their newly acquired spiritual vision, the important scientific consequences were never made (except in the most general terms).

So my own special concern from the start has been to marry the contemplative vision with rational understanding through exploration of the amazing new mathematical (and associated scientific landscapes) that thereby emerge.

One significant clue as to the nature of the imaginary number (as dimension) can be given through raising 1 to the power of i.

Now as we have seen when we raise 1 to a rational number (such as 1/3) we generate a new number in the circular number system (i.e. on the circle of unit radius in the complex plane).

However when we now raise 1 to the imaginary number i, we generate a new number in the linear number system. So we can see here how real and imaginary numbers (as qualitative dimensions) are associated with circular and linear quantitative results respectively.

So a key task with healthy contemplative development is the successful balancing of both transcendent and immanent directions. This implies likewise the successful balancing of appreciation of number (as qualitative dimension) in a - relatively - real and imaginary manner.

As we saw yesterday transcendental type understanding (in qualitative terms) is of the most refined manner possible in the phenomenal realm.

So before moving directly into the subject of today's entry, I will briefly summarise on the various types of transcendental dimensions.

Real transcendental dimensions are of the most refined type whereby one understands all phenomenal relations - not in terms of (separate) linear or (circular) holistic notions - but rather as the relationship between both.

The positive refer to the subtle rational appreciation of this relationship; the negative then relate to direct intuitive realisation.

However the final step in the phenomenal realm is making the understanding associated with corresponding imaginary directions explicit.

So the imaginary transcendental dimensions relate to understanding of projections (in the indirect conscious expression of unconscious meaning) as the relationship between both (separate) linear and (holistic) circular understanding.

Indeed it is precisely in successfully being able to understand projections in terms of this necessary relationship of conscious and unconscious that the involuntary nature of such projection ceases. So involuntary projections always arises due to a certain failure in properly relating the unconscious desire for meaning (embodied in such projections) with the conscious phenomenal circumstances through which they are expressed.

Once again the positive expression of such imaginary transcendental dimensions relates to a highly refined form of rational understanding of their nature; the negative expression again relates to the more direct intuitive realisation of this nature.

Now, I have already likened the contemplative journey to a steep mountain climb. The real transcendent aspect of this journey - notice the close association here with the qualitative mathematical meaning of transcendental - relates to the ascent (that ultimately leads to a spiritual experience beyond all phenomena of form).

The corresponding descent then relates to the immanent aspect of the journey resulting in a spiritual experience that is inherent within all phenomena.

Just as the Riemann Hypothesis is generally considered the most important unsolved problem in Mathematics, the Euler Identity is likewise considered its most remarkable equation (formula, relationship).

Now because every quantitative relationship equally has a qualitative significance (that is formally unrecognised in conventional mathematical terms), this suggests that an extremely important qualitative meaning is associated with the Euler Identity (that can only be decoded in the appropriate manner).

Now conventionally the Euler Identity is expressed by the equation,

e^(πi) + 1 = 0;

Therefore e^(πi) = - 1.

Then by squaring both sides

e^(2πi) = 1.

We now have e (which is itself a transcendental number) raised to a dimensional expression (that is of an imaginary transcendental nature).

Notice how when we raised a rational number to a rational number the result was irrational; then when we raised an irrational number to an irrational number the result was transcendental. So we have continued to move in the direction of increasing transcendence (from a qualitative perspective).

However now in this special case where the base transcendental number is e, we raise it to a special case of a transcendental number where the dimension is 2πi, we obtain the simplest of all rational numbers (which in qualitative terms is thereby of the most immanent nature).

So putting it simply, the Euler Identity, when understood in an appropriate qualitative manner, points to the mysterious transformation in contemplative development where both form and emptiness are perfectly reconciled.

We saw earlier how development entails both differentiation and integration (through the odd and even numbered dimensions respectively. So the ultimate task of transformation is to reach a state where differentiation and integration both coincide.

Now e is the perfect numerical symbol of such transformation as both the differential and integral of e^x are uniquely the same!

Now in the real unit circle, the circular circumference is 2π. However if we now consider the radius as imaginary - rather than real - then the circular circumference is 2πi. However imaginary in this qualitative sense combines both positive and negative directions. So the imaginary circle is better represented as a non-dimensional point. Here both line and circle are perfectly reconciled from a quantitative perspective; likewise both linear and circular understanding are perfectly reconciled in qualitative terms.

So e^(2πi) in qualitative terms is inseparable from e^0 in quantitative terms!

Therefore in qualitative terms - when finally experience becomes of a pure formless nature as transcendence - it thereby equally becomes immanent as inherent in all form (represented qualitatively as 1).

In the various mystical traditions extensive attention has been given to the nature of this key transformation.

Perhaps the most famous expression is given in the Buddhist sutra:

"Form is nor other than Emptiness

Emptiness is not other than Form."

Well in a precise mathematical manner (where symbols are appropriately understand in the qualitative manner) the Euler Identity describes the same transformation.

However even this is not the end of the mathematical story of number

## Thursday, August 30, 2012

### Multidimensional Nature of Time and Space (18)

We looked briefly at the qualitative nature of a transcendental number yesterday.

Once again it requires the explicit recognition of both linear (discrete) and circular (continuous) notions, with the transcendental aspect relating directly to the necessary (irreducible) relationship as between both.

Therefore to stress an important point, if we wish to avoid gross reductionism, we cannot deal with the nature of a transcendental number such as π or e in a merely rational manner!

And of course Conventional Mathematics is defined by such reductionism!

Thus the value of π properly relates therefore to a mysterious conjunction as between (finite) discrete and (infinite) continuous notions which - literally - transcends the linear interpretation of reason.

So the transcendental notion of time (and space) arises from this explicit recognition of the dynamic relationship as between analytic (rational) and holistic (intuitive) type aspects. In the most accurate sense, it reflects therefore an understanding of dimension that serves as the relationship of both finite and infinite meaning.

Now as the very recognition of any phenomenon requires a certain degree of linear separation in experience, the implication is that the purest form of transcendental understanding ultimately is so refined that (separate) phenomena can no longer be explicitly recognised.

However as actual experience represents but an approximation to this state, refined phenomenal recognition necessarily arises.

So the positive aspect of qualitative transcendental recognition is in in the refined rational understanding of its dynamic nature. The negative aspect then relates to its direct intuitive recognition. So as positive and negative aspects interact in experience, clearly phenomena that arise become of an ever transparent nature (as relative expressions of the continual present nature of reality that is absolute).

However there is even one more step to take here.

As we know the importance of imaginary quantities is now well recognised. This implies therefore that this notion of imaginary has an equally important meaning in the qualitative sense of dimension.

Just as the dimensions can be given real numbers (with a corresponding interpretation of the nature of space and time), equally they can be given an imaginary interpretation.

So to what do these imaginary dimensional numbers precisely relate?

Basically I would explain it like this!

Progression with respect to the real numbers as dimensions, relates directly to an increasing transcendent experience of reality. Here - literally - its ultimate spiritual nature (at the ever present moment continually renewed) is gradually seen to transcend all its more limited phenomenal expressions. And as we have just demonstrated, if one has reached contemplative experience (corresponding to these transcendental numbers) these phenomenal expressions are necessarily of a highly refined transparent nature.

However there is an equally important immanent aspect to development, whereby the ultimate nature of reality (as the ever present spiritual moment) is understand to be already inherent in every phenomenal form that arises.

So to use an analogy, that may be of some assistance! The transcendent aspect of development is akin to the ascent in reaching the summit of the mountain. However having reached the summit, one is faced with the opposite problem of achieving the successful descent and getting back on familiar ground once more.

Thus if contemplative development is to be properly grounded as it were, both the immanent and transcendent aspects must be equally emphasised.

And the key role of the imaginary numbers as dimensions is that - when appropriately understood - these are directly tied up with theses corresponding immanent dimension.

The basic idea is not too difficult to express! Basically what is imaginary in qualitative terms, relates to the unconscious. Now, as we have seen with the Olympics this Summer, many athletes at a young age form a dream of one day reaching the summit with respect to their own particular event in becoming the Olympic champion.

Thus this dream thereby represents the potential for transcendence, in going completely beyond all obstacles standing in the way of fulfilling one's goal.

However though this dream is very important, it is not sufficient in itself. So if for successfully realisation, it must become grounded in actual life, through all the practice and training required. So when the gold medal is eventually won, the dream thereby now becomes the reality (with both transcendent and immanent aspects successfully united).

So the actual attempt to realise the dream, consists in transferring this great drive and energy emanating from the unconscious back into the conscious domain through long dedicated preparation. So in this very process of transference, the unconscious is gradually made conscious, and the imaginary becomes real.

In fact when properly understood, this is related directly to the imaginary dimensions of time (and space).

So the real dimensions lead to an increasing intensification in depth with respect to the unconscious (through transcendence); the imaginary dimensions lead to the transference of this unconscious energy back into the conscious domain of everyday life.

So if we are to look at the most advanced development possible in the qualitative (contemplative) domain, it would involve transcendental structures of an imaginary kind!

In the next blog entry, we will see the truly remarkable culmination of such understanding with respect to the famed Euler Identity (where its inherent qualitative significance can be made manifest).

Once again it requires the explicit recognition of both linear (discrete) and circular (continuous) notions, with the transcendental aspect relating directly to the necessary (irreducible) relationship as between both.

Therefore to stress an important point, if we wish to avoid gross reductionism, we cannot deal with the nature of a transcendental number such as π or e in a merely rational manner!

And of course Conventional Mathematics is defined by such reductionism!

Thus the value of π properly relates therefore to a mysterious conjunction as between (finite) discrete and (infinite) continuous notions which - literally - transcends the linear interpretation of reason.

So the transcendental notion of time (and space) arises from this explicit recognition of the dynamic relationship as between analytic (rational) and holistic (intuitive) type aspects. In the most accurate sense, it reflects therefore an understanding of dimension that serves as the relationship of both finite and infinite meaning.

Now as the very recognition of any phenomenon requires a certain degree of linear separation in experience, the implication is that the purest form of transcendental understanding ultimately is so refined that (separate) phenomena can no longer be explicitly recognised.

However as actual experience represents but an approximation to this state, refined phenomenal recognition necessarily arises.

So the positive aspect of qualitative transcendental recognition is in in the refined rational understanding of its dynamic nature. The negative aspect then relates to its direct intuitive recognition. So as positive and negative aspects interact in experience, clearly phenomena that arise become of an ever transparent nature (as relative expressions of the continual present nature of reality that is absolute).

However there is even one more step to take here.

As we know the importance of imaginary quantities is now well recognised. This implies therefore that this notion of imaginary has an equally important meaning in the qualitative sense of dimension.

Just as the dimensions can be given real numbers (with a corresponding interpretation of the nature of space and time), equally they can be given an imaginary interpretation.

So to what do these imaginary dimensional numbers precisely relate?

Basically I would explain it like this!

Progression with respect to the real numbers as dimensions, relates directly to an increasing transcendent experience of reality. Here - literally - its ultimate spiritual nature (at the ever present moment continually renewed) is gradually seen to transcend all its more limited phenomenal expressions. And as we have just demonstrated, if one has reached contemplative experience (corresponding to these transcendental numbers) these phenomenal expressions are necessarily of a highly refined transparent nature.

However there is an equally important immanent aspect to development, whereby the ultimate nature of reality (as the ever present spiritual moment) is understand to be already inherent in every phenomenal form that arises.

So to use an analogy, that may be of some assistance! The transcendent aspect of development is akin to the ascent in reaching the summit of the mountain. However having reached the summit, one is faced with the opposite problem of achieving the successful descent and getting back on familiar ground once more.

Thus if contemplative development is to be properly grounded as it were, both the immanent and transcendent aspects must be equally emphasised.

And the key role of the imaginary numbers as dimensions is that - when appropriately understood - these are directly tied up with theses corresponding immanent dimension.

The basic idea is not too difficult to express! Basically what is imaginary in qualitative terms, relates to the unconscious. Now, as we have seen with the Olympics this Summer, many athletes at a young age form a dream of one day reaching the summit with respect to their own particular event in becoming the Olympic champion.

Thus this dream thereby represents the potential for transcendence, in going completely beyond all obstacles standing in the way of fulfilling one's goal.

However though this dream is very important, it is not sufficient in itself. So if for successfully realisation, it must become grounded in actual life, through all the practice and training required. So when the gold medal is eventually won, the dream thereby now becomes the reality (with both transcendent and immanent aspects successfully united).

So the actual attempt to realise the dream, consists in transferring this great drive and energy emanating from the unconscious back into the conscious domain through long dedicated preparation. So in this very process of transference, the unconscious is gradually made conscious, and the imaginary becomes real.

In fact when properly understood, this is related directly to the imaginary dimensions of time (and space).

So the real dimensions lead to an increasing intensification in depth with respect to the unconscious (through transcendence); the imaginary dimensions lead to the transference of this unconscious energy back into the conscious domain of everyday life.

So if we are to look at the most advanced development possible in the qualitative (contemplative) domain, it would involve transcendental structures of an imaginary kind!

In the next blog entry, we will see the truly remarkable culmination of such understanding with respect to the famed Euler Identity (where its inherent qualitative significance can be made manifest).

## Wednesday, August 29, 2012

### Multidimensional Nature of Time and Space (17)

Yesterday we looked briefly at the qualitative nature of time (and space) from an (algebraic) irrational perspective.

Now an (algebraic) irrational number arises as the solution to a polynomial equation with rational coefficients. The famed square root of 2 - which is the best known example of an irrational number - arises from the simple polynomial expression x^2 = 2!

What this implies with respect to the nature of time (and space) is that a hybrid dynamic mix of the two logical systems (linear and circular) are involved, whereby relative notions are continually reduced in somewhat absolute terms and - in reverse - absolute notions quickly transformed in a relative manner.

Once again, we see this clearly in nature at the sub-atomic level where energy is continually reduced in terms of mass and mass once more transformed into energy.

So in holistic mathematical terms, such interactions properly take place in an environment characterised by irrational notions of time (and space).

In corresponding psychological terms, understanding of these dimensions typically unfolds through authentic contemplative development, where nondual notions of reality are continually reduced in a dualistic manner and then likewise such dualistic notions continually transformed again in a nondual manner.

And this leads to a a more refined dynamic type of understanding whereby reason and intuition continually interact in experience.

We have already looked at the differentiated nature of experience that corresponds with the odd integers and the corresponding integrated nature of the even dimensions. So in a very accurate sense, irrational understanding arises when both odd and even dimensions are combined. So once the first two dimensions unfold in experience, then irrational type understanding (in this strict mathematical sense) will then arise through the process of attempting to relate both dimensions.

As with rational, all irrational numbers can be given both a positive and negative identity.

So this then raises the question as to what is implied by the nature of time (and space) in negative irrational dimensions.

Now, perhaps initially this can be more easily approached from the psychological perspective. We have already seen how with the odd dimensions (as positive) i.e. where one attempts to understand in a direct rational manner, that the main problem relates to unrecognised projections (of an unconscious intuitive kind).

By contrast with the even dimensions (as positive) i.e. where one attempts to understand in a directly intuitive manner, the main problem arising is that of (unrecognised) rational attachments.

Therefore negation with respect to the irrational number dimensions, implies the dual attempt to erode unwanted attachments of both an unconscious and conscious nature.

When successful therefore, this leads to both a purer rational and intuitive appreciation of the dynamic nature of reality involved.

However an important limitation attaches to (algebraic) irrational understanding in that the (imaginary) unconscious nature of personality initially remains comparatively undeveloped. This then sets limitations to the extent to which dynamic negation can be successful in eroding all unnecessary confusion.

This is where we come to another remarkable holistic mathematical finding.

When Hilbert in his famous address named his 23 important - and yet unsolved -mathematical problems one of these related to the status of a number such as 2^(square root of 2). It was believed to be of a transcendental nature but this had not yet been proved. Indeed Hilbert mistakenly believed that this problem would take longer to solve than the Riemann Hypothesis!

In fact it was proved within Hilbert's lifetime. However it demonstrates once more how the the very nature of a number is transformed in quantitative terms through relating a base expression to a dimensional number (as power).

So we saw yesterday with respect to a^b, when both a and b are rational (with b a fraction, an (algebraic) irrational number arises.

We can take this one step further by showing how when b is now irrational (and a either rational or irrational) that a transcendental number arises.

In conventional mathematical terms, a transcendental number is expressed as an irrational number that cannot arise as a solution to a polynomial equation with rational coefficients. The most famous examples of such numbers are π and e.

However as always we can give such a number a qualitative as well as quantitative meaning.

As we have seen the earlier stages with respect to authentic contemplative development (in what is sometimes is referred to as the subtle realm) imply the irrational interpretation of dimensions (from the holistic qualitative perspective).

Typically one's perceptions of reality are much more fluid where both dual (rational) and nondual (intuitive) aspects increasingly interpenetrate. Later in development more deep rooted concepts likewise unfold with again dual and nondual aspects interpenetrating.

However, as one now begins to increasingly match both perceptions and concepts of this nature a further important transformation in development is required, whereby experience now becomes transcendental (in a precise qualitative mathematical manner)

Now it would be valuable to probe more closely here what such transcendental experience entails.

Putting in bluntly, at the earlier irrational (subtle) stages, a certain mismatch of conscious and unconscious is in evidence. From one perspective, one is still too ready to reduce what is properly unconscious (and nondual) in rational terms. Likewise from the other perspective one is equally too ready to elevate what is properly conscious and nondual in an intuitive manner.

However because rational and intuitive aspects are complementary in nature, the proper balancing of both irrational perceptions and concepts requires that both conscious and unconscious aspects come into equal balance.

Thus when the new transcendental type knowledge unfolds (in what - again - is sometimes in Eastern terms referred to as the causal realm) it is of a new even more refined nature. So with respect to the nature of reality, one does not emphasise either dual or nondual aspects (as separate) but rather as the relationship between both dual and nondual aspects. Attaining such a position requires that attention focus more directly on the harmonising nature of the will (as a means of reconciling both conscious and unconscious).

This also provides a fascinating qualitative insight into why a transcendental number cannot be the solution of a polynomial equation.

Such a solution always entails a reduction of a higher dimensional value in 1-dimensional terms.

So once again if we have x^2 = 2, the higher dimensional value here (corresponding to 2 as dimension) = 2. Then we obtain x = the square root of 2, it thereby corresponds to the reduced 1-dimensional value.

However the very nature of transcendental is that reality essentially represents the relationship as between dual and nondual. Therefore we cannot attempt to either reduce or elevate one in terms of the other.

So the transcendental nature of time (and space) is now of an extremely subtle variety as representing the essential relationship as between actual (finite) and potential (infinite) aspects of understanding. This corresponds in experience with a highly refined and dynamic understanding where both reason and intuition interpenetrate in pretty equal measure.

In fact perhaps the best representation of the nature of such understanding is with respect to most famous transcendental number π.

Now π in quantitative terms represents the (perfect) relationship as between the circular circumference and its line diameter.

Equally in qualitative terms, π represents the (perfect) relationship as between both circular and linear type understanding. And what is common to both is the point at the centre of the circle which equally is at the centre of the line diameter.

So pure transcendental understanding, therefore can be expressed as the ineffable midpoint (or singularity) where linear or circular understanding of a separate nature no longer remains.

Just one further observation is worth making here!

I have mentioned before how from a higher dimensional perspective the very nature of mathematical proof is inherently subject to an Uncertainty Principle.

What this entails is that - properly understood - such proof represents an inevitable dynamic interaction as between two aspects which are quantitative and qualitative with respect to each other.

As we have seen elsewhere, implicitly the Pythagoreans were searching for this type of proof. From a quantitative perspective they were indeed able to show why the square root of 2 is irrational! However what really troubled them was that they were unable to provide a satisfactory qualitative rationale as why this was the case!

So now we have yet another example with respect to the nature of a transcendental number. From a quantitative perspective, it can be proved why any rational (or irrational number) other than 1 raised to an irrational dimensional power will result in a transcendental number.

And from my own holistic mathematical investigation of the nature of the stages of contemplative development, I have been able to provide a corresponding qualitative explanation as why to this behaviour occurs.

Therefore, a comprehensive understanding of this relationship entails both quantitative and qualitative aspects (with an inevitable uncertainty attaching to both). Thus current mathematical proof with respect to the merely quantitative nature of transcendental numbers, reveals a subtle confusion (for in quantitative terms we can never precisely determine the value of any transcendental number).

So for example the wide held belief that π is a constant, strictly has no meaning in - merely - quantitative terms (as we can never precisely determine its value).

When we look at Mathematics in a more comprehensive manner we realise that the quantitative is always balanced by a corresponding (holistic) qualitative aspect.

So a rational number therefore (in quantitative terms) corresponds to rational type understanding (from a holistic qualitative perspective).

Equally however a transcendental number (in quantitative terms) corresponds to transcendental type understanding (from a holistic qualitative perspective) And as we have see transcendental in this qualitative sense relates to the the highly refined understanding where both linear (rational) and circular (intuitive) type understanding are both explicitly recognised and kept in a certain balance to each other. (And as we have seen with the purest development of such understanding they are kept in perfect balance!)

Therefore one cannot properly attempt - without gross reductionism - a rational proof (in qualitative terms) of what is transcendental (from a quantitative perspective).

So once again for example we cannot give a - merely - rational meaning to the notion that π is a constant because it is not a rational number (with it's value strictly indeterminate from a merely quantitative perspective!)

Now an (algebraic) irrational number arises as the solution to a polynomial equation with rational coefficients. The famed square root of 2 - which is the best known example of an irrational number - arises from the simple polynomial expression x^2 = 2!

What this implies with respect to the nature of time (and space) is that a hybrid dynamic mix of the two logical systems (linear and circular) are involved, whereby relative notions are continually reduced in somewhat absolute terms and - in reverse - absolute notions quickly transformed in a relative manner.

Once again, we see this clearly in nature at the sub-atomic level where energy is continually reduced in terms of mass and mass once more transformed into energy.

So in holistic mathematical terms, such interactions properly take place in an environment characterised by irrational notions of time (and space).

In corresponding psychological terms, understanding of these dimensions typically unfolds through authentic contemplative development, where nondual notions of reality are continually reduced in a dualistic manner and then likewise such dualistic notions continually transformed again in a nondual manner.

And this leads to a a more refined dynamic type of understanding whereby reason and intuition continually interact in experience.

We have already looked at the differentiated nature of experience that corresponds with the odd integers and the corresponding integrated nature of the even dimensions. So in a very accurate sense, irrational understanding arises when both odd and even dimensions are combined. So once the first two dimensions unfold in experience, then irrational type understanding (in this strict mathematical sense) will then arise through the process of attempting to relate both dimensions.

As with rational, all irrational numbers can be given both a positive and negative identity.

So this then raises the question as to what is implied by the nature of time (and space) in negative irrational dimensions.

Now, perhaps initially this can be more easily approached from the psychological perspective. We have already seen how with the odd dimensions (as positive) i.e. where one attempts to understand in a direct rational manner, that the main problem relates to unrecognised projections (of an unconscious intuitive kind).

By contrast with the even dimensions (as positive) i.e. where one attempts to understand in a directly intuitive manner, the main problem arising is that of (unrecognised) rational attachments.

Therefore negation with respect to the irrational number dimensions, implies the dual attempt to erode unwanted attachments of both an unconscious and conscious nature.

When successful therefore, this leads to both a purer rational and intuitive appreciation of the dynamic nature of reality involved.

However an important limitation attaches to (algebraic) irrational understanding in that the (imaginary) unconscious nature of personality initially remains comparatively undeveloped. This then sets limitations to the extent to which dynamic negation can be successful in eroding all unnecessary confusion.

This is where we come to another remarkable holistic mathematical finding.

When Hilbert in his famous address named his 23 important - and yet unsolved -mathematical problems one of these related to the status of a number such as 2^(square root of 2). It was believed to be of a transcendental nature but this had not yet been proved. Indeed Hilbert mistakenly believed that this problem would take longer to solve than the Riemann Hypothesis!

In fact it was proved within Hilbert's lifetime. However it demonstrates once more how the the very nature of a number is transformed in quantitative terms through relating a base expression to a dimensional number (as power).

So we saw yesterday with respect to a^b, when both a and b are rational (with b a fraction, an (algebraic) irrational number arises.

We can take this one step further by showing how when b is now irrational (and a either rational or irrational) that a transcendental number arises.

In conventional mathematical terms, a transcendental number is expressed as an irrational number that cannot arise as a solution to a polynomial equation with rational coefficients. The most famous examples of such numbers are π and e.

However as always we can give such a number a qualitative as well as quantitative meaning.

As we have seen the earlier stages with respect to authentic contemplative development (in what is sometimes is referred to as the subtle realm) imply the irrational interpretation of dimensions (from the holistic qualitative perspective).

Typically one's perceptions of reality are much more fluid where both dual (rational) and nondual (intuitive) aspects increasingly interpenetrate. Later in development more deep rooted concepts likewise unfold with again dual and nondual aspects interpenetrating.

However, as one now begins to increasingly match both perceptions and concepts of this nature a further important transformation in development is required, whereby experience now becomes transcendental (in a precise qualitative mathematical manner)

Now it would be valuable to probe more closely here what such transcendental experience entails.

Putting in bluntly, at the earlier irrational (subtle) stages, a certain mismatch of conscious and unconscious is in evidence. From one perspective, one is still too ready to reduce what is properly unconscious (and nondual) in rational terms. Likewise from the other perspective one is equally too ready to elevate what is properly conscious and nondual in an intuitive manner.

However because rational and intuitive aspects are complementary in nature, the proper balancing of both irrational perceptions and concepts requires that both conscious and unconscious aspects come into equal balance.

Thus when the new transcendental type knowledge unfolds (in what - again - is sometimes in Eastern terms referred to as the causal realm) it is of a new even more refined nature. So with respect to the nature of reality, one does not emphasise either dual or nondual aspects (as separate) but rather as the relationship between both dual and nondual aspects. Attaining such a position requires that attention focus more directly on the harmonising nature of the will (as a means of reconciling both conscious and unconscious).

This also provides a fascinating qualitative insight into why a transcendental number cannot be the solution of a polynomial equation.

Such a solution always entails a reduction of a higher dimensional value in 1-dimensional terms.

So once again if we have x^2 = 2, the higher dimensional value here (corresponding to 2 as dimension) = 2. Then we obtain x = the square root of 2, it thereby corresponds to the reduced 1-dimensional value.

However the very nature of transcendental is that reality essentially represents the relationship as between dual and nondual. Therefore we cannot attempt to either reduce or elevate one in terms of the other.

So the transcendental nature of time (and space) is now of an extremely subtle variety as representing the essential relationship as between actual (finite) and potential (infinite) aspects of understanding. This corresponds in experience with a highly refined and dynamic understanding where both reason and intuition interpenetrate in pretty equal measure.

In fact perhaps the best representation of the nature of such understanding is with respect to most famous transcendental number π.

Now π in quantitative terms represents the (perfect) relationship as between the circular circumference and its line diameter.

Equally in qualitative terms, π represents the (perfect) relationship as between both circular and linear type understanding. And what is common to both is the point at the centre of the circle which equally is at the centre of the line diameter.

So pure transcendental understanding, therefore can be expressed as the ineffable midpoint (or singularity) where linear or circular understanding of a separate nature no longer remains.

Just one further observation is worth making here!

I have mentioned before how from a higher dimensional perspective the very nature of mathematical proof is inherently subject to an Uncertainty Principle.

What this entails is that - properly understood - such proof represents an inevitable dynamic interaction as between two aspects which are quantitative and qualitative with respect to each other.

As we have seen elsewhere, implicitly the Pythagoreans were searching for this type of proof. From a quantitative perspective they were indeed able to show why the square root of 2 is irrational! However what really troubled them was that they were unable to provide a satisfactory qualitative rationale as why this was the case!

So now we have yet another example with respect to the nature of a transcendental number. From a quantitative perspective, it can be proved why any rational (or irrational number) other than 1 raised to an irrational dimensional power will result in a transcendental number.

And from my own holistic mathematical investigation of the nature of the stages of contemplative development, I have been able to provide a corresponding qualitative explanation as why to this behaviour occurs.

Therefore, a comprehensive understanding of this relationship entails both quantitative and qualitative aspects (with an inevitable uncertainty attaching to both). Thus current mathematical proof with respect to the merely quantitative nature of transcendental numbers, reveals a subtle confusion (for in quantitative terms we can never precisely determine the value of any transcendental number).

So for example the wide held belief that π is a constant, strictly has no meaning in - merely - quantitative terms (as we can never precisely determine its value).

When we look at Mathematics in a more comprehensive manner we realise that the quantitative is always balanced by a corresponding (holistic) qualitative aspect.

So a rational number therefore (in quantitative terms) corresponds to rational type understanding (from a holistic qualitative perspective).

Equally however a transcendental number (in quantitative terms) corresponds to transcendental type understanding (from a holistic qualitative perspective) And as we have see transcendental in this qualitative sense relates to the the highly refined understanding where both linear (rational) and circular (intuitive) type understanding are both explicitly recognised and kept in a certain balance to each other. (And as we have seen with the purest development of such understanding they are kept in perfect balance!)

Therefore one cannot properly attempt - without gross reductionism - a rational proof (in qualitative terms) of what is transcendental (from a quantitative perspective).

So once again for example we cannot give a - merely - rational meaning to the notion that π is a constant because it is not a rational number (with it's value strictly indeterminate from a merely quantitative perspective!)

## Tuesday, August 28, 2012

### Multidimensional Nature of Time and Space (16)

As stated so often when properly understood as the very nature of experience, Mathematics has both quantitative and qualitative aspects in dynamic interaction with each other. So from this perspective one does not understand symbols in static terms as absolute forms, but rather in dynamic interactive terms as symbols of transformation!

I will now attempt to illustrate one extremely important example of this new understanding (with intimate parallels to the nature of psychological development).

As befits the dynamic approach, in a number expression such as a^b, if we designate the base number a in quantitative terms - the dimensional number b is - relatively of a qualitative nature.

And it is this interaction as between quantitative and qualitative aspects that can then be used to explain how the nature of number itself evolves to "higher" forms.

So for example if we start with the simplest of prime numbers 2 and then raise this to 2 (i.e. 2^2), the result is a natural number integer (which is not prime).

So we can se how the very nature of the number has now changed.

Now to obtain the appropriate corresponding situation in psychological terms, we must remember that all experience necessarily entails the dynamic interaction of perceptions and concepts which are - relatively - quantitative and qualitative with respect to each other.

Therefore if we designate the perceptions as quantitative (which is the standard approach in Conventional Science) then corresponding concepts - are relatively - of a qualitative nature. (Of course in formal scientific and mathematical interpretation, concepts are misleadingly also treated as quantitative leading to a merely reduced interpretation of subsequent dynamics).

So in other words an infant at the primitive stage of development initially will develop primitive perceptions and later primitive concepts (both of a transient nature) . It is then in the successful fusion of such perceptions and concepts that development reaches the next natural stage (i.e. where natural objects with a greater degree of constancy emerge in experience).

So in this sense we see how psychological development - in line with the nature of number - evolves from a prime to a natural stage. So what we are really showing here is how number possesses both a qualitative and quantitative relationship to order (with the qualitative aspect of number directly relevant to ordering the various stages of development).

Now switching back to the quantitative nature of natural numbers, the next development is to recognise a negative as well as positive identity in the generation of all the integers.

Then when we raise - as for example the number 2 to - 1 i.e. 2^(- 1) a remarkable number transformation takes place whereby we generate a new type of rational number (i.e. a fraction).

Now again there are direct correspondents on the psychological side. The negative integers here refer to the increasing ability of the child to hold objects in memory even when temporarily absent (giving them a greater absolute constancy).

This in turn enables the child to experience concepts in negative terms i.e. where they can be held in memory as a basis for organising experience when dealing with corresponding perceptions.

And this is the very basis of rational ability whereby both object perceptions and concepts can be repeatedly subdivided in analytical terms and rearranged into new aggregate wholes.

And once again Conventional Science (and Mathematics) is defined by the specialisation to the nth degree of such ability.

However now we come to the interesting part!

If we take again the simple number 2 and now raise it to its reciprocal fraction 1/2, we generate a new type of number that is (algebraically) irrational in nature. Indeed this is the famed square root of 2 that caused so much difficulty for the Pythagoreans many years ago!.

The psychological correspondent implies that if we now try to dynamically relate rational perceptions with rational concepts, which is the very nature of scientific and mathematical experience, we should generate a new (qualitative) form of irrational understanding in holistic terms!

The obvious question then arises as to why this does not typically happen and the answer is - as we have seen - due to the misleading manner in which both perceptions and concepts are interpreted in formal terms with respect merely to their quantitative aspect. Therefore qualitative understanding, in the form of supporting intuition, remains of a merely implicit nature that is quickly reduced in rational terms.

So here we are giving a demonstration using the simplest of numbers to highlight an extremely important limitation of standard scientific and mathematical practice.

Thus even from the quantitative perspective, we cannot properly understood the nature of an irrational number without likewise also explicitly recognising a qualitative aspect! Now the Pythagoreans recognised this and their consternation arose from the inability to properly explain this qualitative aspect. However such appreciation has subsequently become lost through a greatly reduced - merely quantitative - interpretation of mathematical symbols in Western culture.

Therefore once again, because the (qualitative) dimensional nature of number relates holistically to the nature of both the physical and psychological dimensions, we must thereby recognise that time (and space) can necessarily be given an (algebraic) irrational meaning.

Now the square root of 2 has two irrational roots i.e. that are positive and negative with respect to each other.

I have attempted before to explain the nature of the corresponding experience of time and space with respect to the appreciation of a flower such as a rose.

Now formerly one would have largely experienced this object as largely separate and discrete in experience. Initially this would be of a linear (1-dimensional) nature where the rose is viewed as a separate object in space and time. Then later with 2-dimensional understanding, greater subtlety would pertain with an appreciation of both mental and object perception of the rose as - relatively - external and internal with respect to each other.

These two directions would equally apply with this new irrational appreciation. However in relation to both the external and internal directions, a mixture of rational and intuitive appreciation would now be involved. Thus from the rational perspective, one would still appreciate the rose as a finite discrete object; however from the holistic intuitive perspective, one would recognise its continuity with all creation (as an archetype) whereby it radiates an infinite quality.

So quite simply the irrational nature of time and space arises when both discrete (finite) and continuous (infinite) aspects are so intertwined.

This category of irrational dimensions likewise has deep implications for the true nature of sub-atomic particles, where again total independence (from other particles) does not strictly pertain, but rather a hybrid existence combining both discrete and continuous aspects.

I will now attempt to illustrate one extremely important example of this new understanding (with intimate parallels to the nature of psychological development).

As befits the dynamic approach, in a number expression such as a^b, if we designate the base number a in quantitative terms - the dimensional number b is - relatively of a qualitative nature.

And it is this interaction as between quantitative and qualitative aspects that can then be used to explain how the nature of number itself evolves to "higher" forms.

So for example if we start with the simplest of prime numbers 2 and then raise this to 2 (i.e. 2^2), the result is a natural number integer (which is not prime).

So we can se how the very nature of the number has now changed.

Now to obtain the appropriate corresponding situation in psychological terms, we must remember that all experience necessarily entails the dynamic interaction of perceptions and concepts which are - relatively - quantitative and qualitative with respect to each other.

Therefore if we designate the perceptions as quantitative (which is the standard approach in Conventional Science) then corresponding concepts - are relatively - of a qualitative nature. (Of course in formal scientific and mathematical interpretation, concepts are misleadingly also treated as quantitative leading to a merely reduced interpretation of subsequent dynamics).

So in other words an infant at the primitive stage of development initially will develop primitive perceptions and later primitive concepts (both of a transient nature) . It is then in the successful fusion of such perceptions and concepts that development reaches the next natural stage (i.e. where natural objects with a greater degree of constancy emerge in experience).

So in this sense we see how psychological development - in line with the nature of number - evolves from a prime to a natural stage. So what we are really showing here is how number possesses both a qualitative and quantitative relationship to order (with the qualitative aspect of number directly relevant to ordering the various stages of development).

Now switching back to the quantitative nature of natural numbers, the next development is to recognise a negative as well as positive identity in the generation of all the integers.

Then when we raise - as for example the number 2 to - 1 i.e. 2^(- 1) a remarkable number transformation takes place whereby we generate a new type of rational number (i.e. a fraction).

Now again there are direct correspondents on the psychological side. The negative integers here refer to the increasing ability of the child to hold objects in memory even when temporarily absent (giving them a greater absolute constancy).

This in turn enables the child to experience concepts in negative terms i.e. where they can be held in memory as a basis for organising experience when dealing with corresponding perceptions.

And this is the very basis of rational ability whereby both object perceptions and concepts can be repeatedly subdivided in analytical terms and rearranged into new aggregate wholes.

And once again Conventional Science (and Mathematics) is defined by the specialisation to the nth degree of such ability.

However now we come to the interesting part!

If we take again the simple number 2 and now raise it to its reciprocal fraction 1/2, we generate a new type of number that is (algebraically) irrational in nature. Indeed this is the famed square root of 2 that caused so much difficulty for the Pythagoreans many years ago!.

The psychological correspondent implies that if we now try to dynamically relate rational perceptions with rational concepts, which is the very nature of scientific and mathematical experience, we should generate a new (qualitative) form of irrational understanding in holistic terms!

The obvious question then arises as to why this does not typically happen and the answer is - as we have seen - due to the misleading manner in which both perceptions and concepts are interpreted in formal terms with respect merely to their quantitative aspect. Therefore qualitative understanding, in the form of supporting intuition, remains of a merely implicit nature that is quickly reduced in rational terms.

So here we are giving a demonstration using the simplest of numbers to highlight an extremely important limitation of standard scientific and mathematical practice.

Thus even from the quantitative perspective, we cannot properly understood the nature of an irrational number without likewise also explicitly recognising a qualitative aspect! Now the Pythagoreans recognised this and their consternation arose from the inability to properly explain this qualitative aspect. However such appreciation has subsequently become lost through a greatly reduced - merely quantitative - interpretation of mathematical symbols in Western culture.

Therefore once again, because the (qualitative) dimensional nature of number relates holistically to the nature of both the physical and psychological dimensions, we must thereby recognise that time (and space) can necessarily be given an (algebraic) irrational meaning.

Now the square root of 2 has two irrational roots i.e. that are positive and negative with respect to each other.

I have attempted before to explain the nature of the corresponding experience of time and space with respect to the appreciation of a flower such as a rose.

Now formerly one would have largely experienced this object as largely separate and discrete in experience. Initially this would be of a linear (1-dimensional) nature where the rose is viewed as a separate object in space and time. Then later with 2-dimensional understanding, greater subtlety would pertain with an appreciation of both mental and object perception of the rose as - relatively - external and internal with respect to each other.

These two directions would equally apply with this new irrational appreciation. However in relation to both the external and internal directions, a mixture of rational and intuitive appreciation would now be involved. Thus from the rational perspective, one would still appreciate the rose as a finite discrete object; however from the holistic intuitive perspective, one would recognise its continuity with all creation (as an archetype) whereby it radiates an infinite quality.

So quite simply the irrational nature of time and space arises when both discrete (finite) and continuous (infinite) aspects are so intertwined.

This category of irrational dimensions likewise has deep implications for the true nature of sub-atomic particles, where again total independence (from other particles) does not strictly pertain, but rather a hybrid existence combining both discrete and continuous aspects.

## Monday, August 27, 2012

### Multidimensional Nature of Time and Space (15)

I have commented before on - what I refer to as - the Pythagorean Dilemma.

In other words the significance of the discovery that the square root of 2 is an (algebraic) irrational number, was as much of a qualitative as a quantitative nature.

As I have stated, the Pythagoreans recognised an important qualitative significance to number. Prior to their discovery of the irrational nature of 2, they had assumed that all number quantities were of a rational nature. Happily this complemented well the scientific paradigm they used to interpret this reality which qualitatively was also of a rational nature.

So the true significance of the irrational nature of 2, is that the Pythagoreans lacked the qualitative holistic means to explain how it could arise, thus shattering the harmonious balance they sougth to preserve with respect to mathematical activity.

The rational paradigm which still dominates present scientific and mathematical thinking is basically suited to interpretation of meaning that is of a finite discrete nature.

However an irrational number by its very nature combines both finite (discrete) and infinite (continuous) aspects. Thus its quantitative value can be approximated as a rational number to any required degree of accuracy. However its qualitative nature leads to a continuous unending decimal sequence (with no fixed pattern).

Therefore though the very nature of an irrational number - literally - transcends the mere rational perspective, Conventional Mathematics can only attempt to deal with such a number in a reduced quantitative manner.

Now once again the (linear) rational paradigm is 1-dimensional in nature (where all number quantities are ultimately expressed in 1-dimensional terms).

Over the past 14 blog entries however I have been at pains to point out that a complementary (circular) rational paradigm exists where every dimensional number is defined in items of the same base quantity of 1. And as we have seen these dimensions then bear an important inverse relationship with their corresponding roots (in quantitative terms).

So in these contributions, I have shown how all rational numbers (positive and negative) possess a unique qualitative significance that intimately applies to the nature of time and space (in both physical and psychological terms).

However just as an irrational number properly combines both finite (discrete) and infinite (continuous) aspects in its very nature, the same equally applies to an irrational number when given its appropriate qualitative interpretation.

So the upshot of this is that from both the quantitative and qualitative perspectives, irrational numbers are of a hybrid nature truly requiring both Type 1 (analytic) and Type 2 (holistic) mathematical interpretation.

And when this is done we can then give meaning to the irrational nature of time and space (in physical and psychological terms).

In other words the significance of the discovery that the square root of 2 is an (algebraic) irrational number, was as much of a qualitative as a quantitative nature.

As I have stated, the Pythagoreans recognised an important qualitative significance to number. Prior to their discovery of the irrational nature of 2, they had assumed that all number quantities were of a rational nature. Happily this complemented well the scientific paradigm they used to interpret this reality which qualitatively was also of a rational nature.

So the true significance of the irrational nature of 2, is that the Pythagoreans lacked the qualitative holistic means to explain how it could arise, thus shattering the harmonious balance they sougth to preserve with respect to mathematical activity.

The rational paradigm which still dominates present scientific and mathematical thinking is basically suited to interpretation of meaning that is of a finite discrete nature.

However an irrational number by its very nature combines both finite (discrete) and infinite (continuous) aspects. Thus its quantitative value can be approximated as a rational number to any required degree of accuracy. However its qualitative nature leads to a continuous unending decimal sequence (with no fixed pattern).

Therefore though the very nature of an irrational number - literally - transcends the mere rational perspective, Conventional Mathematics can only attempt to deal with such a number in a reduced quantitative manner.

Now once again the (linear) rational paradigm is 1-dimensional in nature (where all number quantities are ultimately expressed in 1-dimensional terms).

Over the past 14 blog entries however I have been at pains to point out that a complementary (circular) rational paradigm exists where every dimensional number is defined in items of the same base quantity of 1. And as we have seen these dimensions then bear an important inverse relationship with their corresponding roots (in quantitative terms).

So in these contributions, I have shown how all rational numbers (positive and negative) possess a unique qualitative significance that intimately applies to the nature of time and space (in both physical and psychological terms).

However just as an irrational number properly combines both finite (discrete) and infinite (continuous) aspects in its very nature, the same equally applies to an irrational number when given its appropriate qualitative interpretation.

So the upshot of this is that from both the quantitative and qualitative perspectives, irrational numbers are of a hybrid nature truly requiring both Type 1 (analytic) and Type 2 (holistic) mathematical interpretation.

And when this is done we can then give meaning to the irrational nature of time and space (in physical and psychological terms).

## Sunday, August 26, 2012

### Multidimensional Nature of Time and Space (14)

To follow the next section requires even subtler understanding of psychological and complementary physical dynamics.

My basic starting point with respect to the dynamic understanding of number, is that in any context the base quantity and dimensional number are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.

Thus in the simple expression 1^2, the base number here (1) is understood in quantitative, whereas the corresponding dimensional number (2) is understood - relatively - in a qualitative manner.

As we have seen Conventional Mathematics is interpreted in terms of the (default) dimensional number of 1 (as qualitative) whereby qualitative is necessarily reduced to quantitative meaning.

Therefore if we take the expression 2^3 to illustrate, the result will be expressed, from this perspective, in reduced quantitative terms as 8 (i.e. 8^1).

Now to explore the qualitative nature of mathematical symbols in isolation, we then reverse interpretation, whereby every mathematical expression is defined in terms of a default base quantity of 1!

And in our exploration of the nature of time (and space) we have illustrated the varying configurations that arise through changing the dimensional numbers as powers with respect to the fixed quantity of 1 (which have an inverse quantitative interpretation as corresponding roots of 1). Thus the first 2 dimensions (where only 2 are involved) - which intimately apply to the dynamic nature of time and space (+ 1 and - 1) - bear an inverse relationship to the corresponding 2 roots of 1 (in quantitative terms).

However we could equally adopt as our starting point the position whereby the base number is now understood in qualitative terms and the corresponding dimensional number - relatively - as quantity.

And in the actual dynamics of psychological experience (and the complementary physical reality corresponding to such experience) continual switching takes place whereby both base and dimensional numbers keep alternating as between quantitative and qualitative interpretation. With respect to psychological understanding this simply means that both perceptions and concepts likewise continually alternate between actual and potential meanings resulting in a continual transformation of experience.

And once again the actual aspect (with respect to both perceptions and concepts) is directly associated with (conscious) reason whereas the corresponding potential aspect is directly associated with (unconscious) intuition.

This likewise means that with respect to the fractional nature of time (and space) that we briefly explored in the last blog entry, that understanding likewise continually alternates as between qualitative and quantitative interpretation.

This represents a generalisation, with respect to the nature of space and time, of what we take for granted on a more mundane level.

For example if a cake is divided into 4 slices one will naturally be able to view each slice as unit whole and also as a fractional part of the whole cake. Likewise one will be able to appreciate the cake itself as a whole unit that is composed of multiple unit parts. Implicitly the dynamics of such understanding requires that we are able to view both parts and wholes (in quantitative and qualitative terms) in order to make these connections. However the qualitative aspect remains merely implicit in customary understanding, with the results interpreted in reduced quantitative terms!

So for this reverse understanding with respect to the nature of dimensions, whereas the emphasis is now explicitly on qualitative type appreciation, implicitly it equally requires the ability to view these dimensions in quantitative terms.

Using the more spiritualised language, that customarily is associated with respect to multidimensional understanding, when the dimensional number is seen as qualitative - relative to a base number as quantitative - this will lead to a more transcendent appreciation of the nature of reality (where its holistic nature is gradually understood as beyond all form).

However when the dimension is now seen as quantitative - relative to a base number as qualitative - it will lead to a more immanent appreciation of the nature of reality (where its holistic nature is gradually seen as inherent within all form).

Again for truly balanced appreciation of reality both aspects are required.

My basic starting point with respect to the dynamic understanding of number, is that in any context the base quantity and dimensional number are quantitative as to qualitative (and qualitative as to quantitative) with respect to each other.

Thus in the simple expression 1^2, the base number here (1) is understood in quantitative, whereas the corresponding dimensional number (2) is understood - relatively - in a qualitative manner.

As we have seen Conventional Mathematics is interpreted in terms of the (default) dimensional number of 1 (as qualitative) whereby qualitative is necessarily reduced to quantitative meaning.

Therefore if we take the expression 2^3 to illustrate, the result will be expressed, from this perspective, in reduced quantitative terms as 8 (i.e. 8^1).

Now to explore the qualitative nature of mathematical symbols in isolation, we then reverse interpretation, whereby every mathematical expression is defined in terms of a default base quantity of 1!

And in our exploration of the nature of time (and space) we have illustrated the varying configurations that arise through changing the dimensional numbers as powers with respect to the fixed quantity of 1 (which have an inverse quantitative interpretation as corresponding roots of 1). Thus the first 2 dimensions (where only 2 are involved) - which intimately apply to the dynamic nature of time and space (+ 1 and - 1) - bear an inverse relationship to the corresponding 2 roots of 1 (in quantitative terms).

However we could equally adopt as our starting point the position whereby the base number is now understood in qualitative terms and the corresponding dimensional number - relatively - as quantity.

And in the actual dynamics of psychological experience (and the complementary physical reality corresponding to such experience) continual switching takes place whereby both base and dimensional numbers keep alternating as between quantitative and qualitative interpretation. With respect to psychological understanding this simply means that both perceptions and concepts likewise continually alternate between actual and potential meanings resulting in a continual transformation of experience.

And once again the actual aspect (with respect to both perceptions and concepts) is directly associated with (conscious) reason whereas the corresponding potential aspect is directly associated with (unconscious) intuition.

This likewise means that with respect to the fractional nature of time (and space) that we briefly explored in the last blog entry, that understanding likewise continually alternates as between qualitative and quantitative interpretation.

This represents a generalisation, with respect to the nature of space and time, of what we take for granted on a more mundane level.

For example if a cake is divided into 4 slices one will naturally be able to view each slice as unit whole and also as a fractional part of the whole cake. Likewise one will be able to appreciate the cake itself as a whole unit that is composed of multiple unit parts. Implicitly the dynamics of such understanding requires that we are able to view both parts and wholes (in quantitative and qualitative terms) in order to make these connections. However the qualitative aspect remains merely implicit in customary understanding, with the results interpreted in reduced quantitative terms!

So for this reverse understanding with respect to the nature of dimensions, whereas the emphasis is now explicitly on qualitative type appreciation, implicitly it equally requires the ability to view these dimensions in quantitative terms.

Using the more spiritualised language, that customarily is associated with respect to multidimensional understanding, when the dimensional number is seen as qualitative - relative to a base number as quantitative - this will lead to a more transcendent appreciation of the nature of reality (where its holistic nature is gradually understood as beyond all form).

However when the dimension is now seen as quantitative - relative to a base number as qualitative - it will lead to a more immanent appreciation of the nature of reality (where its holistic nature is gradually seen as inherent within all form).

Again for truly balanced appreciation of reality both aspects are required.

## Saturday, August 25, 2012

### Multidimensional Nature of Time and Space (13)

As we know from a quantitative perspective rational numbers exist that are not integers i.e. fractions.

This applies therefore that from a qualitative perspective, we equally can give meaning to rational numbers as fractions.

And as the very nature of time (and space) when appropriately understood is intimately related to the qualitative dimensional notion of number, this likewise applies that we can give meaning to the fractional nature of time (and space) from both complementary physical and psychological perspectives.

It perhaps will be easiest in this respect to start with the number 2 (as ordinal dimension). As we have seen this ordinal dimension (from a qualitative perspective) is intimately connected with its corresponding root (in quantitative terms).

Thus the 2nd root of 1 can be written as 1^(1/2) = - 1 and in quantitative terms this result matches the corresponding 2nd dimension i.e. 1^2 = - 1 (which here relates to a qualitative interpretation).

Thus as we have seen the 2nd dimension in qualitative terms relates to the - relative - negative direction of the nature of time (and space) in switching as between polar opposites in experience. And we already have seen how this dimension is implicitly involved in all scientific interpretation (though explicitly completely ignored).

However because each whole number (as dimension) is intimately linked with its reciprocal (in quantitative terms) this implies that we can now give a meaning to 1/2 with respect to the nature of time (in quantitative terms).

What this simply means is that because now one explicitly recognises the existence of two dimensions with respect to the qualitative nature of time (that are positive and negative with respect to each other) then if we isolate just one of these directions (in absolute terms) it thereby represents 1/2 of the total number of dimensions.

Once again let us illustrate with the simple example of a straight road. So starting from a given point, I can move up or down the road. Now if I separate the two reference frames (considering movement with either "up" or "down" as independent), movement along the road will take place positively in space and time. So clearly because there are two possible directions, one of these in isolation represents 1/2 (of the total number of possible directions).

However if I now consider the two directions as interdependent (befitting the qualitative approach) movement is of a merely relative nature. So positive movement up the road, thereby - relatively - implies negative movement with respect to the corresponding "down" direction. And it is this relatively negative movement that the 2nd dimension directly implies (in qualitative terms).

So an integer number (in qualitative terms) is necessarily associated with a corresponding fraction (from a quantitative perspective). So in this restricted quantitative sense, we can thereby give a fractional meaning to time (and space).

Let us further illustrate with respect to the especially important case of 4 dimensions. The 4 dimensions of 1 correspond in turn to the 4 roots of 1^1, 1^2, 1^3 and 1^4 respectively.

Therefore the 4 roots of 1 in quantitative terms are 1^(1/4), 1^(2/4) = 1^(1/2), 1^(3/4) and 1^(4/4) = 1^1.

The corresponding dimensions in qualitative terms are provided through the reciprocals of these powers i.e. 4, 2, 4/3 and 1.

Now three of these are integral dimensions relating to the 1st, 2nd and 4th dimensions respectively.

However the 3rd dimension (in this context of 4 dimensions) is already expressed as a fractional number. And in this case the fractional number has a qualitative rather than quantitative meaning!

Underlying this is a very deep issue indeed with enormous consequences for the very nature of Mathematics which seems to me entirely overlooked in conventional understanding.

Putting it simply, an unavoidable ambiguity attaches to the ordinal notion of number.

For example we might consider that 3 is an unambiguous number. However 3 can be given both a cardinal and ordinal meaning.

And when we look at 3 in an ordinal sense its meaning is entirely relative. In other words the 3rd of a group of 4 items is quite distinct from the 3rd of a group of 5.

Equally the 3rd dimension (as the 3rd of 4) is quite distinct from the 3rd (as the 3rd of 5).

In other words, properly understood, the ordinal nature of number is merely relative. And as the ordinal itself is inseparably linked with its corresponding cardinal meaning, this implies that the cardinal notion of number - when properly understood - is likewise of a merely relative nature.

This is just another way of recognising that the number system itself represents - when appropriately understood - a dynamic interaction as between quantitative and qualitative aspects (which are - relatively - cardinal and ordinal with respect to each other).

Furthermore, Riemann's finding that a harmonic system of wavelike numbers (the non-trivial zeros) underlines the number system is simply evidence - again when correctly understood in dynamic terms - of the dual relative nature of number.

Therefore in considering higher numbered dimensions (in ordinal terms) we are inevitably led to the generation of fractional numbers for most of these dimensions. And the integer dimensions represent but a special case of these fractional dimensions.

In other words, if we limit ourselves to n members (in cardinal terms) the nth member (as ordinal) can be given an unambiguous interpretation. Thus the 4th dimension (of 4 dimensions) can be written with the integer 4 (in qualitative terms). However the 4th member of any higher number of dimensions will be represented as a fraction (in qualitative terms).

Thus from this perspective, the dimensions of time (and space) can be given meaning in terms of rational fractions both directly in qualitative and indirectly in quantitative terms.

This applies therefore that from a qualitative perspective, we equally can give meaning to rational numbers as fractions.

And as the very nature of time (and space) when appropriately understood is intimately related to the qualitative dimensional notion of number, this likewise applies that we can give meaning to the fractional nature of time (and space) from both complementary physical and psychological perspectives.

It perhaps will be easiest in this respect to start with the number 2 (as ordinal dimension). As we have seen this ordinal dimension (from a qualitative perspective) is intimately connected with its corresponding root (in quantitative terms).

Thus the 2nd root of 1 can be written as 1^(1/2) = - 1 and in quantitative terms this result matches the corresponding 2nd dimension i.e. 1^2 = - 1 (which here relates to a qualitative interpretation).

Thus as we have seen the 2nd dimension in qualitative terms relates to the - relative - negative direction of the nature of time (and space) in switching as between polar opposites in experience. And we already have seen how this dimension is implicitly involved in all scientific interpretation (though explicitly completely ignored).

However because each whole number (as dimension) is intimately linked with its reciprocal (in quantitative terms) this implies that we can now give a meaning to 1/2 with respect to the nature of time (in quantitative terms).

What this simply means is that because now one explicitly recognises the existence of two dimensions with respect to the qualitative nature of time (that are positive and negative with respect to each other) then if we isolate just one of these directions (in absolute terms) it thereby represents 1/2 of the total number of dimensions.

Once again let us illustrate with the simple example of a straight road. So starting from a given point, I can move up or down the road. Now if I separate the two reference frames (considering movement with either "up" or "down" as independent), movement along the road will take place positively in space and time. So clearly because there are two possible directions, one of these in isolation represents 1/2 (of the total number of possible directions).

However if I now consider the two directions as interdependent (befitting the qualitative approach) movement is of a merely relative nature. So positive movement up the road, thereby - relatively - implies negative movement with respect to the corresponding "down" direction. And it is this relatively negative movement that the 2nd dimension directly implies (in qualitative terms).

So an integer number (in qualitative terms) is necessarily associated with a corresponding fraction (from a quantitative perspective). So in this restricted quantitative sense, we can thereby give a fractional meaning to time (and space).

Let us further illustrate with respect to the especially important case of 4 dimensions. The 4 dimensions of 1 correspond in turn to the 4 roots of 1^1, 1^2, 1^3 and 1^4 respectively.

Therefore the 4 roots of 1 in quantitative terms are 1^(1/4), 1^(2/4) = 1^(1/2), 1^(3/4) and 1^(4/4) = 1^1.

The corresponding dimensions in qualitative terms are provided through the reciprocals of these powers i.e. 4, 2, 4/3 and 1.

Now three of these are integral dimensions relating to the 1st, 2nd and 4th dimensions respectively.

However the 3rd dimension (in this context of 4 dimensions) is already expressed as a fractional number. And in this case the fractional number has a qualitative rather than quantitative meaning!

Underlying this is a very deep issue indeed with enormous consequences for the very nature of Mathematics which seems to me entirely overlooked in conventional understanding.

Putting it simply, an unavoidable ambiguity attaches to the ordinal notion of number.

For example we might consider that 3 is an unambiguous number. However 3 can be given both a cardinal and ordinal meaning.

And when we look at 3 in an ordinal sense its meaning is entirely relative. In other words the 3rd of a group of 4 items is quite distinct from the 3rd of a group of 5.

Equally the 3rd dimension (as the 3rd of 4) is quite distinct from the 3rd (as the 3rd of 5).

In other words, properly understood, the ordinal nature of number is merely relative. And as the ordinal itself is inseparably linked with its corresponding cardinal meaning, this implies that the cardinal notion of number - when properly understood - is likewise of a merely relative nature.

This is just another way of recognising that the number system itself represents - when appropriately understood - a dynamic interaction as between quantitative and qualitative aspects (which are - relatively - cardinal and ordinal with respect to each other).

Furthermore, Riemann's finding that a harmonic system of wavelike numbers (the non-trivial zeros) underlines the number system is simply evidence - again when correctly understood in dynamic terms - of the dual relative nature of number.

Therefore in considering higher numbered dimensions (in ordinal terms) we are inevitably led to the generation of fractional numbers for most of these dimensions. And the integer dimensions represent but a special case of these fractional dimensions.

In other words, if we limit ourselves to n members (in cardinal terms) the nth member (as ordinal) can be given an unambiguous interpretation. Thus the 4th dimension (of 4 dimensions) can be written with the integer 4 (in qualitative terms). However the 4th member of any higher number of dimensions will be represented as a fraction (in qualitative terms).

Thus from this perspective, the dimensions of time (and space) can be given meaning in terms of rational fractions both directly in qualitative and indirectly in quantitative terms.

## Friday, August 24, 2012

### Multidimensional Nature of Time and Space (12)

We will now consider directly the nature of time (and space) associated with the negative integers (as qualitative dimensions).

The even integer dimensions (- 2, - 4, - 6, - 8,...) are easier to explain, for here in all cases - from the psychological perspective - time has no phenomenal meaning with experience relating directly a present moment continually renewed. This in turn would be consistent with a pure contemplative state.

Of course, because in actual experience, all the varying dimensions co-exist (at least with the potential to exist) we cannot completely isolate the experience of any one dimension. However having said this, at any moment one or more can be especially prominent.

So therefore for example if the experience of the negative 2nd dimension is predominant then indeed one will have little consciousness of time (or space) but rather the spiritual awareness of the (absolute) present moment. Once again such experience is of a purely intuitive nature resulting from the successful negation of any secondary rational attachment to the notion of reality as merely relative (based on the the complementarity of real opposite poles).

So the positive 2-dimensional experience relates to refined rational understanding of the relativity of opposite real poles, in which case one understands time (and space) as having both positive and negative linear directions.

The negative 2-dimensional experience then relates to the direct intuitive realisation of the relativity of these same poles (which results in immediate experience of a present reality).

And again because physical and psychological aspects are themselves complementary, this equally implies that a physical correspondent exists for all negative even dimensions. Again the positive 2-dimensional structure would relate to the dynamic nature of matter particles resulting - relatively - from both positive and negative aspects (of time (and space) i.e. real matter and anti-matter particles. The negative 2-dimensional state would relate directly to the energy fusion (arising from the interaction) which again would exist in an immediate present moment.

Once again, the negative odd dimensions are more difficult to describe, especially in relation to the dynamic nature of time and space involved.

In one important respect they represent the reverse of what is involved with the even number dimensions.

Perhaps again it may be initially easier to appreciate this from a psychological contemplative perspective (where explicit experiential knowledge of such dimensions unfolds).

Now all the mystical traditions speak of the dangers of possessive attachment (which essentially relate to a confused interpretation of the nature of reality). One such attachment is where rational understanding tends to dominate purer intuitive realisation of what is appropriate to the situation leading to an unfortunate form of reductionism. As we have seen such reductionism is endemic with respect to conventional (formal) interpretation in both Science and Mathematics.

Therefore from the psychological perspective though (discriminating) reason and (holistic) intuition necessarily interact in all such understanding, formal interpretation completely excludes intuition - and thereby distorts - its true nature; equally from the complementary physical perspective though phenomenal reality entails the dynamic interaction of visible finite particles with an invisible holistic ground, again experience of reality formally is based on the direct reduction of the infinite aspect in finite terms.

Thus one important task in the development of an authentic contemplative vision is the erosion of (rigid) conscious attachments (of sense and reason) thereby freeing the intuitive light, with which is equally associated a new refined appreciation of the nature of physical reality.

And essentially, it is this type of passive purgation (i.e. negation) that applies with the even numbered dimensions which - when successful - leads to the experience of ever more refined intuitive states.

However, there an equally important type of active detachment required (which to my mind is not sufficiently emphasised in the mystical literature). Thus, higher stages of development are not just concerned with the attainment of ever more refined intuitive states. Equally they are concerned with with the attainment of ever more refined rational structures, whereby one is enabled to think about reality in a much clearer manner (which should be one important goal of Science).

So, whereas purer intuition is developed through cleansing of the confusion of unnecessary rational attachment, purer reason, is developed - by contrast - through cleansing the confusion associated with unnecessary intuitive clutter.

Such clutter typically arises through unrecognised projections of an unconscious nature that then can considerably interfere with conscious type activity.

So much as we might profess the neutrality of science, based merely on objective rational assessment of truth, in practice this is but an illusion with scientists' judgement at every turn subtly - and sometimes not so subtly - influenced by all sorts of unconscious prejudices (of which they generally are not aware).

As I have stated before, the higher odd dimensions are always associated with the pursuit of linear activity. However the higher the dimension involved, the more aware one becomes of new unconscious projections that interfere with direct rational activity.

Putting it bluntly, all scientists and mathematicians inevitably fall victim to unconscious projections and prejudices that interfere with the neutral pursuit of rational truth. However at the 1-dimensional level, they are likely to remain largely unaware of these projections, whereas at the higher odd dimensions, there will be a growing realisation of their nature (and how they interfere with conscious reason).

Thus the negative odd dimensions are, in psychological terms, associated with the gradual erosion of involuntary projections. So when successful traversed, a purer form of rational activity results (largely free of involuntary projections).

Once again there is a remarkable correspondent of this with the Riemann Zeta Function, whereby for all odd integers a rational number results.

So therefore with respect to time and space, the negative odd dimensions are associated with a purer experience of their linear nature (where both move in a forward direction). And this linear nature corresponds with the ability to actively involve oneself in a conscious rational manner (free of interference from involuntary projections).

So strictly when one is the victim of phenomenal projections, all sorts of confusion arise. One may still interpret that one is operating within a framework of linear time (and space) but in truth this will be mingled also with unrecognised rigid forms of imaginary time (and space).

Most of my attention over the years has been to provide a truly scientific rationale (of a holistic mathematical nature) with respect to the higher stages of development. Though such stages have indeed been successfully traversed by the spiritual superstars of the varying mystical traditions, accounts are generally couched in the language of the various religions they represent.

Unfortunately such accounts do not lend themselves readily to qualitative mathematical interpretation. So in some ways I would see myself as in the process of attempting to develop a new mathematical language that is consistent with the transformed understanding that unfolds with each of these stages.

And of paramount significance in this context is the holistic mathematical notion of dimension. So the stages of higher level development literally entail journeying through these varying dimensions (in their positive and negative form).

And in this quest I would emphasise the importance of balance.

1. Higher level rational understanding of reality must be counterbalanced equally by higher level intuitive realisation, for in dynamic terms both mutually serve each other. Traditionally there has been far too much emphasis on mere reason within the scientific community and then too much emphasis on mere intuition within the esoteric mystical traditions. This has resulted in a considerable division as between the religious and scientific quests though in truth they should be seen as mutually complementary.

2. Even numbered dimensional stages are directly concerned with the integration of reality and the ultimate attainment of pure intuitive states as negative dimensions (which have an indirect rational interpretation as positive).

Odd numbered stages - by contrast - are directly concerned with the differentiation of reality and the ultimate attainment of pure rational structures as negative dimensions (which have an indirect intuitive interpretation as positive).

3. For proper balance both odd and even numbered dimensions need to be emphasised; equally both positive and negative aspects likewise need to be emphasised.

So contemplation (intuition) and reason are designed to mutually serve each other; Likewise differentiation (in active engagement with reality) and integration (in passive withdrawal) are equally complementary and likewise need to be kept in balance.

Thus the contemplative quest is not designed just as a means of going beyond reality (as transcendent); equally it is designed as a means of more fully engaging with the world (as immanent). And the intuitive nature of both of these aspects needs to be always finely balanced with the complementary use of reason.

The even integer dimensions (- 2, - 4, - 6, - 8,...) are easier to explain, for here in all cases - from the psychological perspective - time has no phenomenal meaning with experience relating directly a present moment continually renewed. This in turn would be consistent with a pure contemplative state.

Of course, because in actual experience, all the varying dimensions co-exist (at least with the potential to exist) we cannot completely isolate the experience of any one dimension. However having said this, at any moment one or more can be especially prominent.

So therefore for example if the experience of the negative 2nd dimension is predominant then indeed one will have little consciousness of time (or space) but rather the spiritual awareness of the (absolute) present moment. Once again such experience is of a purely intuitive nature resulting from the successful negation of any secondary rational attachment to the notion of reality as merely relative (based on the the complementarity of real opposite poles).

So the positive 2-dimensional experience relates to refined rational understanding of the relativity of opposite real poles, in which case one understands time (and space) as having both positive and negative linear directions.

The negative 2-dimensional experience then relates to the direct intuitive realisation of the relativity of these same poles (which results in immediate experience of a present reality).

And again because physical and psychological aspects are themselves complementary, this equally implies that a physical correspondent exists for all negative even dimensions. Again the positive 2-dimensional structure would relate to the dynamic nature of matter particles resulting - relatively - from both positive and negative aspects (of time (and space) i.e. real matter and anti-matter particles. The negative 2-dimensional state would relate directly to the energy fusion (arising from the interaction) which again would exist in an immediate present moment.

Once again, the negative odd dimensions are more difficult to describe, especially in relation to the dynamic nature of time and space involved.

In one important respect they represent the reverse of what is involved with the even number dimensions.

Perhaps again it may be initially easier to appreciate this from a psychological contemplative perspective (where explicit experiential knowledge of such dimensions unfolds).

Now all the mystical traditions speak of the dangers of possessive attachment (which essentially relate to a confused interpretation of the nature of reality). One such attachment is where rational understanding tends to dominate purer intuitive realisation of what is appropriate to the situation leading to an unfortunate form of reductionism. As we have seen such reductionism is endemic with respect to conventional (formal) interpretation in both Science and Mathematics.

Therefore from the psychological perspective though (discriminating) reason and (holistic) intuition necessarily interact in all such understanding, formal interpretation completely excludes intuition - and thereby distorts - its true nature; equally from the complementary physical perspective though phenomenal reality entails the dynamic interaction of visible finite particles with an invisible holistic ground, again experience of reality formally is based on the direct reduction of the infinite aspect in finite terms.

Thus one important task in the development of an authentic contemplative vision is the erosion of (rigid) conscious attachments (of sense and reason) thereby freeing the intuitive light, with which is equally associated a new refined appreciation of the nature of physical reality.

And essentially, it is this type of passive purgation (i.e. negation) that applies with the even numbered dimensions which - when successful - leads to the experience of ever more refined intuitive states.

However, there an equally important type of active detachment required (which to my mind is not sufficiently emphasised in the mystical literature). Thus, higher stages of development are not just concerned with the attainment of ever more refined intuitive states. Equally they are concerned with with the attainment of ever more refined rational structures, whereby one is enabled to think about reality in a much clearer manner (which should be one important goal of Science).

So, whereas purer intuition is developed through cleansing of the confusion of unnecessary rational attachment, purer reason, is developed - by contrast - through cleansing the confusion associated with unnecessary intuitive clutter.

Such clutter typically arises through unrecognised projections of an unconscious nature that then can considerably interfere with conscious type activity.

So much as we might profess the neutrality of science, based merely on objective rational assessment of truth, in practice this is but an illusion with scientists' judgement at every turn subtly - and sometimes not so subtly - influenced by all sorts of unconscious prejudices (of which they generally are not aware).

As I have stated before, the higher odd dimensions are always associated with the pursuit of linear activity. However the higher the dimension involved, the more aware one becomes of new unconscious projections that interfere with direct rational activity.

Putting it bluntly, all scientists and mathematicians inevitably fall victim to unconscious projections and prejudices that interfere with the neutral pursuit of rational truth. However at the 1-dimensional level, they are likely to remain largely unaware of these projections, whereas at the higher odd dimensions, there will be a growing realisation of their nature (and how they interfere with conscious reason).

Thus the negative odd dimensions are, in psychological terms, associated with the gradual erosion of involuntary projections. So when successful traversed, a purer form of rational activity results (largely free of involuntary projections).

Once again there is a remarkable correspondent of this with the Riemann Zeta Function, whereby for all odd integers a rational number results.

So therefore with respect to time and space, the negative odd dimensions are associated with a purer experience of their linear nature (where both move in a forward direction). And this linear nature corresponds with the ability to actively involve oneself in a conscious rational manner (free of interference from involuntary projections).

So strictly when one is the victim of phenomenal projections, all sorts of confusion arise. One may still interpret that one is operating within a framework of linear time (and space) but in truth this will be mingled also with unrecognised rigid forms of imaginary time (and space).

Most of my attention over the years has been to provide a truly scientific rationale (of a holistic mathematical nature) with respect to the higher stages of development. Though such stages have indeed been successfully traversed by the spiritual superstars of the varying mystical traditions, accounts are generally couched in the language of the various religions they represent.

Unfortunately such accounts do not lend themselves readily to qualitative mathematical interpretation. So in some ways I would see myself as in the process of attempting to develop a new mathematical language that is consistent with the transformed understanding that unfolds with each of these stages.

And of paramount significance in this context is the holistic mathematical notion of dimension. So the stages of higher level development literally entail journeying through these varying dimensions (in their positive and negative form).

And in this quest I would emphasise the importance of balance.

1. Higher level rational understanding of reality must be counterbalanced equally by higher level intuitive realisation, for in dynamic terms both mutually serve each other. Traditionally there has been far too much emphasis on mere reason within the scientific community and then too much emphasis on mere intuition within the esoteric mystical traditions. This has resulted in a considerable division as between the religious and scientific quests though in truth they should be seen as mutually complementary.

2. Even numbered dimensional stages are directly concerned with the integration of reality and the ultimate attainment of pure intuitive states as negative dimensions (which have an indirect rational interpretation as positive).

Odd numbered stages - by contrast - are directly concerned with the differentiation of reality and the ultimate attainment of pure rational structures as negative dimensions (which have an indirect intuitive interpretation as positive).

3. For proper balance both odd and even numbered dimensions need to be emphasised; equally both positive and negative aspects likewise need to be emphasised.

So contemplation (intuition) and reason are designed to mutually serve each other; Likewise differentiation (in active engagement with reality) and integration (in passive withdrawal) are equally complementary and likewise need to be kept in balance.

Thus the contemplative quest is not designed just as a means of going beyond reality (as transcendent); equally it is designed as a means of more fully engaging with the world (as immanent). And the intuitive nature of both of these aspects needs to be always finely balanced with the complementary use of reason.

## Thursday, August 23, 2012

### Multidimensional Nature of Time and Space (11)

We made the distinction yesterday as between implicit qualitative recognition of the 1st dimension as negative (where it remains completely ignored in formal mathematical interpretation), and full explicit recognition which inevitably leads to a redefinition of the nature of Mathematics (whereby both quantitative and qualitative aspects are recognised).

So once again, a mathematician may well recognise the important role of intuition with respect to important new discoveries. And this inherently requires to a degree - sometimes marked - the temporary negation of customary rational understanding. This then allows deeper holistic insight to incubate in the unconscious which is essential in enabling an important new breakthrough. But unfortunately such a mathematician will then formally interpret this new finding in a merely reduced rational manner (with the 1st dimension as positive solely recognised).

As I live in Dublin I can identify with the inscription on Brougham Bridge in honour of William Rowan Hamilton.

"Here as he walked by on the 16th of October, 1843, Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication

i^2 = j^2 = k^2 = ijk = - 1"

So this inscription indicates well how the "discovery" essentially relates to a sudden illumination (releasing holistic intuition into consciousness). Notice how this does not happen in the normal sequential manner of successive rational linkages spread out in linear time! Rather it represents the present moment thrust as it were into linear time (where the relationship of all aspects of the problem to each other are understood simultaneously). Indeed so fearful was Hamilton at losing such inspiration that he felt compelled to carve the equation immediately into the stone at the bridge (though alas no record of this now remains).

However in formal terms, Mathematics has nothing to say about the role of intuition in understanding, or its important dynamic interaction with rational type understanding.

So, in the most accurate sense, conventional mathematical interpretation thereby offers but a reduced and ultimately quite distorted account of the nature of mathematical truth.

In other words, in the qualitative mathematical manner that I now use these terms, Conventional Mathematics is entirely defined within a merely (positive) 1-dimensional framework of interpretation, where qualitative is reduced to quantitative meaning. However proper incorporation of quantitative with qualitative requires recognition that all other numbers (as dimensions) have an important potential role to play in mathematical interpretation.

So once the negative 1st dimension - which remains merely implicit in conventional mathematical interpretation - is then explicitly recognised, the very nature of Mathematics changes from an absolute fixed to a relative dynamic approach, which necessarily entails the interaction of both quantitative and qualitative aspects.

Now, we have already looked at the nature of 2 as a dimensional number, which immediately arises through explicit dynamic recognition of the negative aspect of linear understanding. So 2-dimensional interpretation contains both positive and negative aspects, in dynamic relationship with each other (as the complementarity of real opposites).

There is a remarkable evidence of this provided - when appropriately interpreted - by the Riemann Functional Equation. So s, representing a dimensional number (i.e. power) of the Function on the RHS, can be given a complementary expression on the LHS, now expressed with respect to the dimensional number 1 - s.

This suggest therefore that there are intimate connections as between 2 as dimensional number and - 1 (on opposite sides of the equation). What this means in effect is that we must keep switching as between quantitative and qualitative (and qualitative and quantitative) type understanding with respect to interpreting both sides of the equation.

Therefore when we explicitly recognise the holistic intuitive significance of the result for the Function, with - 1 as dimension on the LHS, this immediately leads to a corresponding recognition of the rational nature of the result for 2 (as dimension) on the RHS. In other words whereas the numerical result (π^2)/6 makes sense from a rational linear perspective on the RHS, this is not so with respect to the corresponding result (- 1/12) on the LHS! And the reason for this is that the LHS result does not conform directly to a linear quantitative, but rather a circular qualitative interpretation (of a holistic kind).

The deeper implication of this is that proper interpretation of the nature of the Riemann Zeta Function cannot be carried out from within the conventional mathematical perspective. As the real secret of the primes relates to this fundamental relationship as between its quantitative and qualitative aspects, clearly this is completely missed from a mere 1-dimensional perspective (where qualitative meaning is inevitably reduced in quantitative terms).

So just as the Riemann Zeta Function is uniquely undefined in quantitative terms where s = 1, equally it remains uniquely undefined in qualitative terms (in terms of overall interpretation) likewise where s = 1.

We can now suggest a further important connection with the Riemann Zeta Function.

We have already defined 2 (as dimensional number) as the rational interpretation of the complementarity of opposite real poles.

However the very nature of reason is to separate poles. So we are attempting therefore to express with the number 2 (as positive dimension) what properly relates to the true nature of interdependence in an indirect rational manner (which tends to give it a somewhat independent identity).

Therefore to move to the true intuitive meaning of what is implied by 2 (as dimension) we must negate such rational interpretation.

Then when we successfully negate any lingering independent element we are left with the intuitive recognition of true interdependence (which is nothing in phenomenal terms).

Now this is deeply illuminating as the value of the Riemann Zeta Function (the first trivial zero) for which s is - 2, = 0.

This strongly suggests that this numerical value corresponds directly to the holistic qualitative - rather than specific quantitative - meaning of 0. So once again, whereas we can interpret values on the RHS of the Functional Equation (> 1) in quantitative terms, corresponding values on the left are - relatively - of a qualitative nature.

In short whereas the positive sign with respect to any dimensional number, represents its rational interpretation, the corresponding negative sign represents its direct intuitive recognition (through negation of independent rational elements).

This explains therefore in qualitative terms, why the Riemann Zeta Function = 0 for the trivial zeros (i.e. negative even integer values of s). In all cases, these represent the complementarity of opposites where pure interdependence arises. And such interdependence is directly grasped through intuitive recognition (which implies negation of indirect rational understanding provided through the positive even number dimensions). And this recognition = 0 in phenomenal terms.

My own route to this understanding was based on a deep resonance with the work of St. John of the Cross, who deals very well with the negative dimensions (from a mystical contemplative perspective).

So the "dark nights" or purgations are directly concerned with the experience of negative dimensions (in qualitative mathematical terms).

The active purgations relate to the odd numbered dimensions (especially 1). The passive purgations relate to the even numbered dimensions. And the direct goal of such passive purgation in St. John's terms is "nada" i.e. nothing (= 0 in qualitative terms).

Finally he talks of "nights of sense" and "nights of spirit". The former would relate in scientific terms to empirical perceptions whereas the latter would relate to more deep rooted theories and concepts. And we will later demonstrate a further startling holistic mathematical result that arises through the dynamic interaction of perceptions (as parts) and concepts (as wholes) respectively!

Thus once again we can see in the process of discovery of the greatest scientists and mathematicians (e.g. recently with Andrew Wiles) long periods spent in the intellectual wilderness. These implicitly in a mathematical context, constituted active nights of sense and spirit i.e. where attachment to former customary perceptions and concepts required considerable erosion before essential new insights could successfully develop.

Just to complete this section we return to the fact that what is true in psychological terms has - by definition - a complementary meaning from a physical perspective.

Now just as interdependence in psychological terms leads to the generation of spiritual energy (in the form of holistic intuition) equally interdependence in physical terms leads to the generation of physical energy. However the mysterious feature of such energy as with light, is that it has no phenomenal existence in itself, but rather only indirectly through interaction with other phenomenal processes.

So if you look at the world through contemplative eyes, you will realise that because mass represents just another form of energy, that phenomena essentially do not exist! Rather what we term "physical reality" relates to arbitrary appearances of a merely relative nature that have no ultimate substance.

However the point that I am making is that such realisation is equally consistent with a more comprehensive mathematical interpretation of number, where quantitative and qualitative aspects are equally recognised (through the marriage of reason with the contemplative vision).

So once again, a mathematician may well recognise the important role of intuition with respect to important new discoveries. And this inherently requires to a degree - sometimes marked - the temporary negation of customary rational understanding. This then allows deeper holistic insight to incubate in the unconscious which is essential in enabling an important new breakthrough. But unfortunately such a mathematician will then formally interpret this new finding in a merely reduced rational manner (with the 1st dimension as positive solely recognised).

As I live in Dublin I can identify with the inscription on Brougham Bridge in honour of William Rowan Hamilton.

"Here as he walked by on the 16th of October, 1843, Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication

i^2 = j^2 = k^2 = ijk = - 1"

So this inscription indicates well how the "discovery" essentially relates to a sudden illumination (releasing holistic intuition into consciousness). Notice how this does not happen in the normal sequential manner of successive rational linkages spread out in linear time! Rather it represents the present moment thrust as it were into linear time (where the relationship of all aspects of the problem to each other are understood simultaneously). Indeed so fearful was Hamilton at losing such inspiration that he felt compelled to carve the equation immediately into the stone at the bridge (though alas no record of this now remains).

However in formal terms, Mathematics has nothing to say about the role of intuition in understanding, or its important dynamic interaction with rational type understanding.

So, in the most accurate sense, conventional mathematical interpretation thereby offers but a reduced and ultimately quite distorted account of the nature of mathematical truth.

In other words, in the qualitative mathematical manner that I now use these terms, Conventional Mathematics is entirely defined within a merely (positive) 1-dimensional framework of interpretation, where qualitative is reduced to quantitative meaning. However proper incorporation of quantitative with qualitative requires recognition that all other numbers (as dimensions) have an important potential role to play in mathematical interpretation.

So once the negative 1st dimension - which remains merely implicit in conventional mathematical interpretation - is then explicitly recognised, the very nature of Mathematics changes from an absolute fixed to a relative dynamic approach, which necessarily entails the interaction of both quantitative and qualitative aspects.

Now, we have already looked at the nature of 2 as a dimensional number, which immediately arises through explicit dynamic recognition of the negative aspect of linear understanding. So 2-dimensional interpretation contains both positive and negative aspects, in dynamic relationship with each other (as the complementarity of real opposites).

There is a remarkable evidence of this provided - when appropriately interpreted - by the Riemann Functional Equation. So s, representing a dimensional number (i.e. power) of the Function on the RHS, can be given a complementary expression on the LHS, now expressed with respect to the dimensional number 1 - s.

This suggest therefore that there are intimate connections as between 2 as dimensional number and - 1 (on opposite sides of the equation). What this means in effect is that we must keep switching as between quantitative and qualitative (and qualitative and quantitative) type understanding with respect to interpreting both sides of the equation.

Therefore when we explicitly recognise the holistic intuitive significance of the result for the Function, with - 1 as dimension on the LHS, this immediately leads to a corresponding recognition of the rational nature of the result for 2 (as dimension) on the RHS. In other words whereas the numerical result (π^2)/6 makes sense from a rational linear perspective on the RHS, this is not so with respect to the corresponding result (- 1/12) on the LHS! And the reason for this is that the LHS result does not conform directly to a linear quantitative, but rather a circular qualitative interpretation (of a holistic kind).

The deeper implication of this is that proper interpretation of the nature of the Riemann Zeta Function cannot be carried out from within the conventional mathematical perspective. As the real secret of the primes relates to this fundamental relationship as between its quantitative and qualitative aspects, clearly this is completely missed from a mere 1-dimensional perspective (where qualitative meaning is inevitably reduced in quantitative terms).

So just as the Riemann Zeta Function is uniquely undefined in quantitative terms where s = 1, equally it remains uniquely undefined in qualitative terms (in terms of overall interpretation) likewise where s = 1.

We can now suggest a further important connection with the Riemann Zeta Function.

We have already defined 2 (as dimensional number) as the rational interpretation of the complementarity of opposite real poles.

However the very nature of reason is to separate poles. So we are attempting therefore to express with the number 2 (as positive dimension) what properly relates to the true nature of interdependence in an indirect rational manner (which tends to give it a somewhat independent identity).

Therefore to move to the true intuitive meaning of what is implied by 2 (as dimension) we must negate such rational interpretation.

Then when we successfully negate any lingering independent element we are left with the intuitive recognition of true interdependence (which is nothing in phenomenal terms).

Now this is deeply illuminating as the value of the Riemann Zeta Function (the first trivial zero) for which s is - 2, = 0.

This strongly suggests that this numerical value corresponds directly to the holistic qualitative - rather than specific quantitative - meaning of 0. So once again, whereas we can interpret values on the RHS of the Functional Equation (> 1) in quantitative terms, corresponding values on the left are - relatively - of a qualitative nature.

In short whereas the positive sign with respect to any dimensional number, represents its rational interpretation, the corresponding negative sign represents its direct intuitive recognition (through negation of independent rational elements).

This explains therefore in qualitative terms, why the Riemann Zeta Function = 0 for the trivial zeros (i.e. negative even integer values of s). In all cases, these represent the complementarity of opposites where pure interdependence arises. And such interdependence is directly grasped through intuitive recognition (which implies negation of indirect rational understanding provided through the positive even number dimensions). And this recognition = 0 in phenomenal terms.

My own route to this understanding was based on a deep resonance with the work of St. John of the Cross, who deals very well with the negative dimensions (from a mystical contemplative perspective).

So the "dark nights" or purgations are directly concerned with the experience of negative dimensions (in qualitative mathematical terms).

The active purgations relate to the odd numbered dimensions (especially 1). The passive purgations relate to the even numbered dimensions. And the direct goal of such passive purgation in St. John's terms is "nada" i.e. nothing (= 0 in qualitative terms).

Finally he talks of "nights of sense" and "nights of spirit". The former would relate in scientific terms to empirical perceptions whereas the latter would relate to more deep rooted theories and concepts. And we will later demonstrate a further startling holistic mathematical result that arises through the dynamic interaction of perceptions (as parts) and concepts (as wholes) respectively!

Thus once again we can see in the process of discovery of the greatest scientists and mathematicians (e.g. recently with Andrew Wiles) long periods spent in the intellectual wilderness. These implicitly in a mathematical context, constituted active nights of sense and spirit i.e. where attachment to former customary perceptions and concepts required considerable erosion before essential new insights could successfully develop.

Just to complete this section we return to the fact that what is true in psychological terms has - by definition - a complementary meaning from a physical perspective.

Now just as interdependence in psychological terms leads to the generation of spiritual energy (in the form of holistic intuition) equally interdependence in physical terms leads to the generation of physical energy. However the mysterious feature of such energy as with light, is that it has no phenomenal existence in itself, but rather only indirectly through interaction with other phenomenal processes.

So if you look at the world through contemplative eyes, you will realise that because mass represents just another form of energy, that phenomena essentially do not exist! Rather what we term "physical reality" relates to arbitrary appearances of a merely relative nature that have no ultimate substance.

However the point that I am making is that such realisation is equally consistent with a more comprehensive mathematical interpretation of number, where quantitative and qualitative aspects are equally recognised (through the marriage of reason with the contemplative vision).

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