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).

## Thursday, August 30, 2012

## 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).

## Wednesday, August 22, 2012

### Multidimensional Nature of Time and Space (10)

As we have seen, Conventional Science is based on rational understanding of a linear logical kind which directly conforms with the qualitative interpretation of 1 (as a number dimension). And of course it is the very nature of such interpretation that qualitative notions are thereby reduced (for any relevant context) in a merely quantitative manner!

Also, as we have seen, directly associated with this approach is the interpretation of time also as 1-dimensional (where it moves in a single positive direction).

However, once we recognise that associated with all numbers is a corresponding qualitative - as well as recognised - quantitative interpretation, then potentially we can have an unlimited number of mathematical interpretations (all of which assume a certain limited validity within their appropriate relative context). This likewise entails that time (and space) itself - when appropriately understood - possesses a potentially unlimited number of possible directions (associated with varying configurations of real and imaginary aspects).

Later, we shall explore the enormous significance of this finding, in providing a fundamental explanation for the distinctive qualitative attributes that all phenomena inherently possess!

However, in this entry I wish to explore the significance of the first of the negative dimensions i.e. where time and space move in a single backward direction (both physically and psychologically) and how this dimension, though formally unrecognised, is intimately involved in all scientific understanding, especially of a creative kind.

Once again the basis of 1-dimensional interpretation (as the paradigm of what we conventionally call "science") is that polar reference frames are clearly separated with respect to formal interpretation of reality.

So typically the scientist, as for example in the case of Einstein, attempts to understand the physical world as objective (and thereby separate from subjective interaction). Even when this assumption is no longer strictly tenable as with the findings of Quantum Mechanics, the conventional paradigm fundamentally remains unchallenged. So physicists, while admitting that the findings of Quantum Mechanics appear deeply paradoxical, do not thereby accept that this exposes a fundamental problem with the existing paradigm. So, for all practical purposes, they just carry on, regardless of its non-intuitive findings.

The other key separation takes place with respect to the quantitative and qualitative poles, relating in turn to the whole and part aspects of reality. So science is understand in terms of precise quantitative type measurement (which thereby excludes interaction with its related qualitative aspects).

However momentary reflection on the matter will indicate that actual experience requires that these poles - which are formally separated in conventional scientific terms - must necessarily interact with each other. So in a richer scientific appreciation of reality, we must seek to understand the dynamic nature of such interaction in experience.

In this way, we can perhaps begin to appreciate that the conventional paradigm - whereby interaction is completely ignored in formal terms - represents but an extreme limiting case. Once again this corresponds directly with 1-dimensional interpretation. However in exploring the full range of possible dynamic interactions that can take place between poles, all the other natural numbers (and number types) as dimensions ultimately become involved.

So how does the 1st negative dimension arise?

A more refined interpretation of scientific reality entails the interaction of both external and internal aspects. So for example scientists' understanding of objective phenomena as external, cannot be ultimately divorced from their corresponding mental perceptions - which - relatively are of an internal nature. So a ceaseless dialogue therefore takes place in experience as between both objects and perceptions and at an even deeper level as between object classes and more generalised internal concepts.

The crucial point is that the actual switch from external to internal in experience always requires, to some degree, the temporary negation of what has already been posited phenomenally in an external manner. Likewise, in reverse, the corresponding switch from internal to external requires the temporary negation of what has been posited in an internal mental manner.

Put another way actual experience entails the ceaseless interaction as between both conscious and unconscious with the primary role of the unconscious in this regard to facilitate ready switching as between both poles.

Now when this takes place to a marked extent, a substantial amount of intuitive energy becomes available in experience (due to the successful fusion of both positive and negative polarities).

However when the role of the unconscious is not properly recognised, though a certain degree of switching must still implicitly be involved, little intuitive energy will be generated. Thus, rigid understanding of a conscious nature will result. Here, interpretation of objective phenomena will tend to confirm, in a somewhat absolute manner, corresponding perceptions and concepts (of these objects).

And indeed this is one great unrecognised limitation of the conventional paradigm, in that by formally ignoring altogether the role of the unconscious in scientific experience, it directly fosters such rigidity!

Therefore, though informally it may well be recognised that high levels of intuition are indeed required for truly creative scientific research, from a formal perspective its important role is completely screened out of interpretation. Not surprisingly, this leads ultimately to a somewhat distorted perspective on scientific truth.

Thus we can now perhaps better appreciate the nature of negative linear understanding (corresponding to the qualitative interpretation of - 1).

So first one posits objective phenomena externally in a rational linear fashion. This corresponds + 1 as qualitative dimension. However to posit corresponding mental constructs in a - relatively - internal manner, one must negate to a degree these objective phenomena. And then to posit phenomena once more in an external fashion, one must likewise negate the internal constructs (perceptions and concepts).

Thus, in dynamic terms, a continual process of positing and negating occurs, which enables the generation of holistic intuition. Thus in healthy scientific understanding, rational understanding is continually fuelled by holistic intuition of an unconscious kind. And such intuition can only be properly explained in a dynamic context (where opposite polarities continually interact).

So, we now see that in a true experiential context, the negative (as well as positive) 1st dimension must necessarily be involved.

This likewise entails that insofar as the negation of phenomena is concerned that time (and space) move in a - relatively - backward direction.

We can perhaps understand this better through looking psychologically at the process involved in great scientific breakthroughs.

For example, following his initial key insight regarding the equivalence of gravity and acceleration, Einstein laboured for many years in considerable darkness. Truly original work requires the development of radical new insights (of a holistic intuitive nature). However before these can shine through into consciousness, a long painful process may be required, whereby one is gradually weaned off customary rational understanding.

So quite literally, a considerable negation with respect to conventional understanding of reality takes place. and while this process is underway, it does genuinely feel as if one is, somehow, psychologically moving backwards in space and time.

Normally, from the positive linear perspective, when one works at a project, one expects a gradual accumulation of knowledge to take place. So as time and space move forward in a positive direction, one's knowledge thereby increases in a similar manner.

However where truly original insight is required to enable a decisive new breakthrough, the opposite can occur, whereby one's customary knowledge is gradually eroded with the forward movement of time (and space). One seems therefore to be going back to an earlier stage in one's development when one's knowledge was considerably less and this can thereby be associated with the experience of time (and space) moving - relatively - in a backward direction. This seemingly negative progress typically therefore leads to a feeling of disillusionment, tempting one to abandon the problem altogether.

However, it is through this process of negation (of what was formerly rationally posited in experience) that holistic intuition of a deep kind gradually incubates in the unconscious. Then when sufficiently developed in relation to the problem considered, it can then burst forth into consciousness in a new Eureka moment of wonderful discovery.

However such a key moment of enlightenment tends to be much more intuitive than rational (though later may indeed be used to support a new rational framework of understanding).

And once again in formal terms, though vital to the new discovery, such intuition is then screened entirely out of formal interpretation, which is presented misleadingly in a - solely - rational manner.

However an important observation needs to be made here. Though the process of scientific discovery in many ways is broadly similar to the process by which spiritual enlightenment is attained, one key distinction needs to be made.

The case of Einstein is in fact highly instructive in this regard. Though his General Theory of Relativity represents perhaps the single greatest breakthrough ever in physical understanding, it did not however lessen Einstein's commitment to the conventional scientific paradigm. So he basically maintained the dualistic view that objective reality could be successfully understood as independent of the enquiring mind.

However where authentic spiritual enlightenment of an advanced level is involved, the negation that takes place (with respect to former posited understanding) is so profound that one's very belief in the dualistic nature of reality becomes seriously undermined.

In other words, when applied to scientific and mathematical understanding, this entails that one can no longer accept its dualistic assumptions, for such assumptions misrepresent the nature of reality in a fundamental manner.

Thus, in normal scientific (and mathematical) discovery, both negative (allowing for the deepening of intuition) and positive linear understanding (of a rational kind) are involved. However here the (nondual) intuition involved is insufficient to undermine commitment to the overall (dualistic) paradigm of interpretation employed.

However with authentic spiritual development (leading to contemplative enlightenment) the negative aspect can be of such a profound nature that it seriously undermines commitment to this universal paradigm.

In other words, the clear implication is that one must now (explicitly) go beyond mere 1-dimensional understanding with respect to scientific (and mathematical) interpretation.

As always the psychological and physical aspects of reality are complementary in dynamic terms . This therefore employs that the negative 1-dimensional nature of time (and space) that we have just illustrated in a psychological context, equally applies to all phenomenal interactions in nature. Of course when we apply conventional 1-dimensional interpretation (of a merely positive kind) to such interactions, time (and space) will indeed appear to move in a solely forward direction!

However when we accept that physical interactions are governed by the same polarities (such as external/internal and whole/part) then the negative 1st dimension necessarily arises in the dynamic switching as between these opposite poles. Then in the extreme case, where the interaction is so dynamic that independent particles can scarcely exist, positive and negative aspects will fuse immediately in pure energy (as the physical counterpart to pure intuition).

We can see this most clearly in relation to the very nature of physical light where each photon - by definition - corresponds to its own anti-photon. So in terms of a beam of light, the positive movement in time (and space) of a photon is cancelled out entirely by the corresponding negative movement as its own anti-photon, so that light "travels" in the continual present moment! In fact within its own reference frame, both particles and waves of light have no phenomenal meaning and only - literally - appear through interaction with phenomena travelling at less than light speed!

However though the conventional paradigm preserves to a considerable degree the myth that movement in time (and space) takes place solely in a positive direction, in truth in dynamic relative terms, both positive and negative movement is necessarily involved with respect to all processes (physical and psychological).

Also, as we have seen, directly associated with this approach is the interpretation of time also as 1-dimensional (where it moves in a single positive direction).

However, once we recognise that associated with all numbers is a corresponding qualitative - as well as recognised - quantitative interpretation, then potentially we can have an unlimited number of mathematical interpretations (all of which assume a certain limited validity within their appropriate relative context). This likewise entails that time (and space) itself - when appropriately understood - possesses a potentially unlimited number of possible directions (associated with varying configurations of real and imaginary aspects).

Later, we shall explore the enormous significance of this finding, in providing a fundamental explanation for the distinctive qualitative attributes that all phenomena inherently possess!

However, in this entry I wish to explore the significance of the first of the negative dimensions i.e. where time and space move in a single backward direction (both physically and psychologically) and how this dimension, though formally unrecognised, is intimately involved in all scientific understanding, especially of a creative kind.

Once again the basis of 1-dimensional interpretation (as the paradigm of what we conventionally call "science") is that polar reference frames are clearly separated with respect to formal interpretation of reality.

So typically the scientist, as for example in the case of Einstein, attempts to understand the physical world as objective (and thereby separate from subjective interaction). Even when this assumption is no longer strictly tenable as with the findings of Quantum Mechanics, the conventional paradigm fundamentally remains unchallenged. So physicists, while admitting that the findings of Quantum Mechanics appear deeply paradoxical, do not thereby accept that this exposes a fundamental problem with the existing paradigm. So, for all practical purposes, they just carry on, regardless of its non-intuitive findings.

The other key separation takes place with respect to the quantitative and qualitative poles, relating in turn to the whole and part aspects of reality. So science is understand in terms of precise quantitative type measurement (which thereby excludes interaction with its related qualitative aspects).

However momentary reflection on the matter will indicate that actual experience requires that these poles - which are formally separated in conventional scientific terms - must necessarily interact with each other. So in a richer scientific appreciation of reality, we must seek to understand the dynamic nature of such interaction in experience.

In this way, we can perhaps begin to appreciate that the conventional paradigm - whereby interaction is completely ignored in formal terms - represents but an extreme limiting case. Once again this corresponds directly with 1-dimensional interpretation. However in exploring the full range of possible dynamic interactions that can take place between poles, all the other natural numbers (and number types) as dimensions ultimately become involved.

So how does the 1st negative dimension arise?

A more refined interpretation of scientific reality entails the interaction of both external and internal aspects. So for example scientists' understanding of objective phenomena as external, cannot be ultimately divorced from their corresponding mental perceptions - which - relatively are of an internal nature. So a ceaseless dialogue therefore takes place in experience as between both objects and perceptions and at an even deeper level as between object classes and more generalised internal concepts.

The crucial point is that the actual switch from external to internal in experience always requires, to some degree, the temporary negation of what has already been posited phenomenally in an external manner. Likewise, in reverse, the corresponding switch from internal to external requires the temporary negation of what has been posited in an internal mental manner.

Put another way actual experience entails the ceaseless interaction as between both conscious and unconscious with the primary role of the unconscious in this regard to facilitate ready switching as between both poles.

Now when this takes place to a marked extent, a substantial amount of intuitive energy becomes available in experience (due to the successful fusion of both positive and negative polarities).

However when the role of the unconscious is not properly recognised, though a certain degree of switching must still implicitly be involved, little intuitive energy will be generated. Thus, rigid understanding of a conscious nature will result. Here, interpretation of objective phenomena will tend to confirm, in a somewhat absolute manner, corresponding perceptions and concepts (of these objects).

And indeed this is one great unrecognised limitation of the conventional paradigm, in that by formally ignoring altogether the role of the unconscious in scientific experience, it directly fosters such rigidity!

Therefore, though informally it may well be recognised that high levels of intuition are indeed required for truly creative scientific research, from a formal perspective its important role is completely screened out of interpretation. Not surprisingly, this leads ultimately to a somewhat distorted perspective on scientific truth.

Thus we can now perhaps better appreciate the nature of negative linear understanding (corresponding to the qualitative interpretation of - 1).

So first one posits objective phenomena externally in a rational linear fashion. This corresponds + 1 as qualitative dimension. However to posit corresponding mental constructs in a - relatively - internal manner, one must negate to a degree these objective phenomena. And then to posit phenomena once more in an external fashion, one must likewise negate the internal constructs (perceptions and concepts).

Thus, in dynamic terms, a continual process of positing and negating occurs, which enables the generation of holistic intuition. Thus in healthy scientific understanding, rational understanding is continually fuelled by holistic intuition of an unconscious kind. And such intuition can only be properly explained in a dynamic context (where opposite polarities continually interact).

So, we now see that in a true experiential context, the negative (as well as positive) 1st dimension must necessarily be involved.

This likewise entails that insofar as the negation of phenomena is concerned that time (and space) move in a - relatively - backward direction.

We can perhaps understand this better through looking psychologically at the process involved in great scientific breakthroughs.

For example, following his initial key insight regarding the equivalence of gravity and acceleration, Einstein laboured for many years in considerable darkness. Truly original work requires the development of radical new insights (of a holistic intuitive nature). However before these can shine through into consciousness, a long painful process may be required, whereby one is gradually weaned off customary rational understanding.

So quite literally, a considerable negation with respect to conventional understanding of reality takes place. and while this process is underway, it does genuinely feel as if one is, somehow, psychologically moving backwards in space and time.

Normally, from the positive linear perspective, when one works at a project, one expects a gradual accumulation of knowledge to take place. So as time and space move forward in a positive direction, one's knowledge thereby increases in a similar manner.

However where truly original insight is required to enable a decisive new breakthrough, the opposite can occur, whereby one's customary knowledge is gradually eroded with the forward movement of time (and space). One seems therefore to be going back to an earlier stage in one's development when one's knowledge was considerably less and this can thereby be associated with the experience of time (and space) moving - relatively - in a backward direction. This seemingly negative progress typically therefore leads to a feeling of disillusionment, tempting one to abandon the problem altogether.

However, it is through this process of negation (of what was formerly rationally posited in experience) that holistic intuition of a deep kind gradually incubates in the unconscious. Then when sufficiently developed in relation to the problem considered, it can then burst forth into consciousness in a new Eureka moment of wonderful discovery.

However such a key moment of enlightenment tends to be much more intuitive than rational (though later may indeed be used to support a new rational framework of understanding).

And once again in formal terms, though vital to the new discovery, such intuition is then screened entirely out of formal interpretation, which is presented misleadingly in a - solely - rational manner.

However an important observation needs to be made here. Though the process of scientific discovery in many ways is broadly similar to the process by which spiritual enlightenment is attained, one key distinction needs to be made.

The case of Einstein is in fact highly instructive in this regard. Though his General Theory of Relativity represents perhaps the single greatest breakthrough ever in physical understanding, it did not however lessen Einstein's commitment to the conventional scientific paradigm. So he basically maintained the dualistic view that objective reality could be successfully understood as independent of the enquiring mind.

However where authentic spiritual enlightenment of an advanced level is involved, the negation that takes place (with respect to former posited understanding) is so profound that one's very belief in the dualistic nature of reality becomes seriously undermined.

In other words, when applied to scientific and mathematical understanding, this entails that one can no longer accept its dualistic assumptions, for such assumptions misrepresent the nature of reality in a fundamental manner.

Thus, in normal scientific (and mathematical) discovery, both negative (allowing for the deepening of intuition) and positive linear understanding (of a rational kind) are involved. However here the (nondual) intuition involved is insufficient to undermine commitment to the overall (dualistic) paradigm of interpretation employed.

However with authentic spiritual development (leading to contemplative enlightenment) the negative aspect can be of such a profound nature that it seriously undermines commitment to this universal paradigm.

In other words, the clear implication is that one must now (explicitly) go beyond mere 1-dimensional understanding with respect to scientific (and mathematical) interpretation.

As always the psychological and physical aspects of reality are complementary in dynamic terms . This therefore employs that the negative 1-dimensional nature of time (and space) that we have just illustrated in a psychological context, equally applies to all phenomenal interactions in nature. Of course when we apply conventional 1-dimensional interpretation (of a merely positive kind) to such interactions, time (and space) will indeed appear to move in a solely forward direction!

However when we accept that physical interactions are governed by the same polarities (such as external/internal and whole/part) then the negative 1st dimension necessarily arises in the dynamic switching as between these opposite poles. Then in the extreme case, where the interaction is so dynamic that independent particles can scarcely exist, positive and negative aspects will fuse immediately in pure energy (as the physical counterpart to pure intuition).

We can see this most clearly in relation to the very nature of physical light where each photon - by definition - corresponds to its own anti-photon. So in terms of a beam of light, the positive movement in time (and space) of a photon is cancelled out entirely by the corresponding negative movement as its own anti-photon, so that light "travels" in the continual present moment! In fact within its own reference frame, both particles and waves of light have no phenomenal meaning and only - literally - appear through interaction with phenomena travelling at less than light speed!

However though the conventional paradigm preserves to a considerable degree the myth that movement in time (and space) takes place solely in a positive direction, in truth in dynamic relative terms, both positive and negative movement is necessarily involved with respect to all processes (physical and psychological).

## Tuesday, August 21, 2012

### Multidimensional Nature of Time and Space (9)

So far we have investigated how a unique experience of time (and space) is associated with all the positive integers (representing dimensional numbers).

And as such an experience directly complements the nature of physical reality, this implies likewise that a corresponding structure of time (and space) likewise exists in the physical world with respect to all positive integers.

And once again the very reason why this is not readily apparent is that the conventional scientific paradigm is based solely on the linear use of reason (with 1 as the default dimensional number). This then explains the customary interpretation of time as 1-dimensional (where it is believed to move in a single forward direction).

Likewise we have seen that a fundamental distinction divides, as it were, interpretation of time (and space) with respect to the even and corresponding interpretation with respect to the odd dimensional numbers. Basically - when understood appropriately in a dynamic relative manner - integration in nature (physical and psychological) always relates directly to even numbered, whereas differentiation is associated with the odd numbered dimensions.

So in this context Conventional Science - based on 1 as dimension - is properly suited merely as a means of differentiated interpretation of reality (and then only at the simplest unrefined level). Such science is therefore directly of an analytical rather than holistic variety.

However if we are to adopt a proper integral notion of science, it must be based on an even integer as dimensional number. Thus the simplest version of a holistic scientific approach, that is truly integral in nature, is 2-dimensional, allowing for direct complementarity with respect to the fundamental polar opposites underlying phenomenal reality (such as internal/external, quantitative/qualitative, form/emptiness etc.)

Of course the most comprehensive approach to science must necessarily combine differentiated and integrated aspects in a dynamic relative manner (allowing therefore for both odd and even numbered dimensional interpretations).

Using my own terminology. I refer to the differentiated (analytical) aspect of science as a Type 1 approach. However once again Conventional Science in fact represents but the simplest version of the Type 1 approach (where interpretation is of a basic 1-dimensional nature).

The corresponding integral (holistic) aspect of science represents the Type 2 approach. Once again the simplest integral approach is 2-dimensional in nature (using circular type reason based on the direct complementarity of "real" opposites)!

Most of my own work in recent decades has been geared to exploration of the precise implications of the 2, 4 and 8-dimensional integral approaches for science respectively.

Finally the most comprehensive is the Type 3 approach (which I formerly referred to as "radial"). This attempts to combine both the differentiated (Type 1) and integral (Type 2) aspects of science in a coherent dynamic fashion. So this would entail use of both the odd and even number integers (as dimensions). And as we shall now begin to see, it involves much more besides!

Again to briefly recap, my basic starting point is that all numbers with a recognised existing quantitative interpretation can equally be given a coherent qualitative meaning.

So far we have investigated the nature of such qualitative meaning with respect to the positive integers (both positive and negative). Therefore a distinctive dimensional meaning is associated with every positive integer, which intimately applies to the nature of time (and space) in both physical and psychological terms.

However we can have negative as well positive integers (in quantitative terms). Likewise we can have rational (fractional) values that are not integers. Then we can also have irrational number quantities (both algebraic and transcendental) as well as imaginary and complex values.

Thus, in principle, each of these number notions (as quantities) can thereby be given a corresponding qualitative meaning (as dimensions) which again intimately apply to the nature of time (and space) in both physical and psychological terms.

Therefore we start with this ongoing investigation. by first attempting to clarify the notion of negative dimensions and the important manner in which they dynamically apply to all processes in nature!

And as such an experience directly complements the nature of physical reality, this implies likewise that a corresponding structure of time (and space) likewise exists in the physical world with respect to all positive integers.

And once again the very reason why this is not readily apparent is that the conventional scientific paradigm is based solely on the linear use of reason (with 1 as the default dimensional number). This then explains the customary interpretation of time as 1-dimensional (where it is believed to move in a single forward direction).

Likewise we have seen that a fundamental distinction divides, as it were, interpretation of time (and space) with respect to the even and corresponding interpretation with respect to the odd dimensional numbers. Basically - when understood appropriately in a dynamic relative manner - integration in nature (physical and psychological) always relates directly to even numbered, whereas differentiation is associated with the odd numbered dimensions.

So in this context Conventional Science - based on 1 as dimension - is properly suited merely as a means of differentiated interpretation of reality (and then only at the simplest unrefined level). Such science is therefore directly of an analytical rather than holistic variety.

However if we are to adopt a proper integral notion of science, it must be based on an even integer as dimensional number. Thus the simplest version of a holistic scientific approach, that is truly integral in nature, is 2-dimensional, allowing for direct complementarity with respect to the fundamental polar opposites underlying phenomenal reality (such as internal/external, quantitative/qualitative, form/emptiness etc.)

Of course the most comprehensive approach to science must necessarily combine differentiated and integrated aspects in a dynamic relative manner (allowing therefore for both odd and even numbered dimensional interpretations).

Using my own terminology. I refer to the differentiated (analytical) aspect of science as a Type 1 approach. However once again Conventional Science in fact represents but the simplest version of the Type 1 approach (where interpretation is of a basic 1-dimensional nature).

The corresponding integral (holistic) aspect of science represents the Type 2 approach. Once again the simplest integral approach is 2-dimensional in nature (using circular type reason based on the direct complementarity of "real" opposites)!

Most of my own work in recent decades has been geared to exploration of the precise implications of the 2, 4 and 8-dimensional integral approaches for science respectively.

Finally the most comprehensive is the Type 3 approach (which I formerly referred to as "radial"). This attempts to combine both the differentiated (Type 1) and integral (Type 2) aspects of science in a coherent dynamic fashion. So this would entail use of both the odd and even number integers (as dimensions). And as we shall now begin to see, it involves much more besides!

Again to briefly recap, my basic starting point is that all numbers with a recognised existing quantitative interpretation can equally be given a coherent qualitative meaning.

So far we have investigated the nature of such qualitative meaning with respect to the positive integers (both positive and negative). Therefore a distinctive dimensional meaning is associated with every positive integer, which intimately applies to the nature of time (and space) in both physical and psychological terms.

However we can have negative as well positive integers (in quantitative terms). Likewise we can have rational (fractional) values that are not integers. Then we can also have irrational number quantities (both algebraic and transcendental) as well as imaginary and complex values.

Thus, in principle, each of these number notions (as quantities) can thereby be given a corresponding qualitative meaning (as dimensions) which again intimately apply to the nature of time (and space) in both physical and psychological terms.

Therefore we start with this ongoing investigation. by first attempting to clarify the notion of negative dimensions and the important manner in which they dynamically apply to all processes in nature!

## Thursday, August 9, 2012

### Multidimensional Nature of Time and Space (8)

In dealing with the nature of 2, 4 and 8 dimensions of time (and space) respectively, we saw how that in each case they are characterised by the complementarity of opposite poles.

So 2-dimensional reality is characterised by the complementarity of the "real" poles (i.e. external and internal).

4-dimensional is then characterised by the additional complementarity of the "imaginary" poles (i.e. whole and part).

8-dimensional is finally chracterised by the additional complementarity of special complex poles where both real and imaginary parts are of equal magnitude. This can then be understood as relating to the ultimate interaction as between form and emptiness (i.e. where the dynamic interaction of phnomena are so refined that they do not even appear to arise). So in this sense the dynamic nature of form becomes inseparable from emptiness.

Now it must be understood that further distinct structures with respect to the nature of time (and space) are associated with all other even numbered integers.

However they all possess one important common feature in that they are characterised - however intricately - by the complemenentarity of opposite poles.

This can be simply appreciated with reference to the fact that the qualitative dimension entailed (for each even integer) is inversely related to its corresponding number of roots of 1 (in quantitative terms).

And as all even numbered roots can be arranged in a complementary manner (with half of the roots expressed as the negative of the other half) this entails in turn that the dynamic dimensional structure for all even numbers applying to time (and space) is based directly on the complementarity of opposites.

However this principle clearly does not apply to the odd integer dimensions.

For example in the important case of 1 i.e. the linear mode, which again characterises conventional scientific understanding, time is not understand in dynamic complementary terms (where its ultimate nature is paradoxical).

And in an important more refined manner, this linear view likewise characterises all the odd integer dimensions.

For example, to examine the dimensional structure associated with 3, we need to look at the corresponding three roots of 1 i.e. 1, - .5 + .866i, and - .5 + .866i (expressed correct to 3 decimal places).

As we can see the first root here is 1, which in a sense stands out on its own (as independent of the other roots). The remaining roots - necessarily of an even number - always appear as pairings of complex conjugates (with the imaginary part arranged in a complementary manner).

So how do we explain the nature of such odd numbered dimensions? How do we, for example, attempt to describe the nature of time as experienced in 3 dimensions?

Now this is an issue that I have given an enormous amount of thought to over the last 30 years or so. However suffice it to say that the odd numbered dimensions (apart from 1) are much more difficult to clearly explain than their even counterparts.

I would describe it this way.

The standard dualistic rational approach - where phenomena are treated as independent - is of a linear (1-dimensional) nature.

The standard nondual contemplative approach - where phenomena are treated in merely relative - and ultimately illusory - fashion is based on the complementarity of opposites. So associated with the ascending scale of even integers are ever more refined contemplative (nondual) experiences of reality.

However the odd integers (≥ 1) are associated with a hybrid of both approaches (and ultimately are incompatible with each other).

In this respect I can draw on my own experience of 3 dimensions. Indeed I remember suggesting a proposal for a thesis on a dynamic methodology for Economics (that I only realised many years later, was, in holistic mathematical terms, of a 3-dimensional nature).

On the one hand I was here trying to preserve the validity of the standard conventional model (i.e. 1-dimensional). On the other hand I was trying to reconcile this with the 2-dimensional approach based on the complementarity of opposites. So I was attempting to balance the traditional linear with a new dynamic 2-dimensional approach (based on a circular logic). However from an experiential perspective this led to inevitable conflict. Commitment to the linear aspect fostered - what in spiritual terms is referred to as - dualistic attachment; meanwhile the 2-dimensional aspect experientially required the very erosion of such attachment. So in the end I abandoned that approach (though later was able to return to it from a more refined perspective).

So this led me to the view that it in experience odd dimensional structures are necessarily asymmetrical in nature and ultimately inconsistent with the nondual perspective.

So in terms of psycho spiritual growth, one starts with dualistic 1-dimensional understanding, which is of a strongly differentiated nature. Then 2-dimensional understanding provides the "lowest" level with respect to integrated experience (of a nondual contemplative kind). However because both dual and nondual are dynamically related, to reach higher levels of integration represented by the even dimensions, one must equally embrace higher levels of differentiation represented by the odd dimensions (which inevitably are of an inconsistent temporary nature).

So in terms of space and time the very nature of the odd numbered dimensions entails that space and time can no longer be experienced in a complementary manner. This means that new forms of phenomenal rigidity i.e. new forms of matter, necessarily arise with the odd numbered dimensions.

If we apply this understanding to the dynamic nature of "lower" physical reality, this implies that associated with each higher odd dimension are new matter particles, whereas with the appropriate even dimensions these quickly dissolve in the creation of energy.

And as all dimensions in a sense co-exist simultaneously, there is an unending trail of matter as it were waiting to be discovered. In other words the higher the dimensions we can access (which in a sense is the task of particle accelerators) the more new forms of matter will arise.

Equally from a psycho spiritual perspective, the higher the dimensions one can access through advanced contemplation, the more refined one's experience can become so that one can actually now "see" such matter arising from a deep unconscious level of experience.

From a physical perspective, the interaction of odd with even integer dimensions relates to the continual transformation of matter into energy (and gravity). And there is no limit to this process with ever more dynamic short-lived transformations associated with the higher dimensions.

From a corresponding psychological perspective, the interaction of odd with even integer dimensions relates to the continual transformation of dualistic matter phenomena into (contemplative) spiritual energy. And again there is no strict limit to this process with ever more dynamic short-lived transformations associated with the higher number dimensions.

So 2-dimensional reality is characterised by the complementarity of the "real" poles (i.e. external and internal).

4-dimensional is then characterised by the additional complementarity of the "imaginary" poles (i.e. whole and part).

8-dimensional is finally chracterised by the additional complementarity of special complex poles where both real and imaginary parts are of equal magnitude. This can then be understood as relating to the ultimate interaction as between form and emptiness (i.e. where the dynamic interaction of phnomena are so refined that they do not even appear to arise). So in this sense the dynamic nature of form becomes inseparable from emptiness.

Now it must be understood that further distinct structures with respect to the nature of time (and space) are associated with all other even numbered integers.

However they all possess one important common feature in that they are characterised - however intricately - by the complemenentarity of opposite poles.

This can be simply appreciated with reference to the fact that the qualitative dimension entailed (for each even integer) is inversely related to its corresponding number of roots of 1 (in quantitative terms).

And as all even numbered roots can be arranged in a complementary manner (with half of the roots expressed as the negative of the other half) this entails in turn that the dynamic dimensional structure for all even numbers applying to time (and space) is based directly on the complementarity of opposites.

However this principle clearly does not apply to the odd integer dimensions.

For example in the important case of 1 i.e. the linear mode, which again characterises conventional scientific understanding, time is not understand in dynamic complementary terms (where its ultimate nature is paradoxical).

And in an important more refined manner, this linear view likewise characterises all the odd integer dimensions.

For example, to examine the dimensional structure associated with 3, we need to look at the corresponding three roots of 1 i.e. 1, - .5 + .866i, and - .5 + .866i (expressed correct to 3 decimal places).

As we can see the first root here is 1, which in a sense stands out on its own (as independent of the other roots). The remaining roots - necessarily of an even number - always appear as pairings of complex conjugates (with the imaginary part arranged in a complementary manner).

So how do we explain the nature of such odd numbered dimensions? How do we, for example, attempt to describe the nature of time as experienced in 3 dimensions?

Now this is an issue that I have given an enormous amount of thought to over the last 30 years or so. However suffice it to say that the odd numbered dimensions (apart from 1) are much more difficult to clearly explain than their even counterparts.

I would describe it this way.

The standard dualistic rational approach - where phenomena are treated as independent - is of a linear (1-dimensional) nature.

The standard nondual contemplative approach - where phenomena are treated in merely relative - and ultimately illusory - fashion is based on the complementarity of opposites. So associated with the ascending scale of even integers are ever more refined contemplative (nondual) experiences of reality.

However the odd integers (≥ 1) are associated with a hybrid of both approaches (and ultimately are incompatible with each other).

In this respect I can draw on my own experience of 3 dimensions. Indeed I remember suggesting a proposal for a thesis on a dynamic methodology for Economics (that I only realised many years later, was, in holistic mathematical terms, of a 3-dimensional nature).

On the one hand I was here trying to preserve the validity of the standard conventional model (i.e. 1-dimensional). On the other hand I was trying to reconcile this with the 2-dimensional approach based on the complementarity of opposites. So I was attempting to balance the traditional linear with a new dynamic 2-dimensional approach (based on a circular logic). However from an experiential perspective this led to inevitable conflict. Commitment to the linear aspect fostered - what in spiritual terms is referred to as - dualistic attachment; meanwhile the 2-dimensional aspect experientially required the very erosion of such attachment. So in the end I abandoned that approach (though later was able to return to it from a more refined perspective).

So this led me to the view that it in experience odd dimensional structures are necessarily asymmetrical in nature and ultimately inconsistent with the nondual perspective.

So in terms of psycho spiritual growth, one starts with dualistic 1-dimensional understanding, which is of a strongly differentiated nature. Then 2-dimensional understanding provides the "lowest" level with respect to integrated experience (of a nondual contemplative kind). However because both dual and nondual are dynamically related, to reach higher levels of integration represented by the even dimensions, one must equally embrace higher levels of differentiation represented by the odd dimensions (which inevitably are of an inconsistent temporary nature).

So in terms of space and time the very nature of the odd numbered dimensions entails that space and time can no longer be experienced in a complementary manner. This means that new forms of phenomenal rigidity i.e. new forms of matter, necessarily arise with the odd numbered dimensions.

If we apply this understanding to the dynamic nature of "lower" physical reality, this implies that associated with each higher odd dimension are new matter particles, whereas with the appropriate even dimensions these quickly dissolve in the creation of energy.

And as all dimensions in a sense co-exist simultaneously, there is an unending trail of matter as it were waiting to be discovered. In other words the higher the dimensions we can access (which in a sense is the task of particle accelerators) the more new forms of matter will arise.

Equally from a psycho spiritual perspective, the higher the dimensions one can access through advanced contemplation, the more refined one's experience can become so that one can actually now "see" such matter arising from a deep unconscious level of experience.

From a physical perspective, the interaction of odd with even integer dimensions relates to the continual transformation of matter into energy (and gravity). And there is no limit to this process with ever more dynamic short-lived transformations associated with the higher dimensions.

From a corresponding psychological perspective, the interaction of odd with even integer dimensions relates to the continual transformation of dualistic matter phenomena into (contemplative) spiritual energy. And again there is no strict limit to this process with ever more dynamic short-lived transformations associated with the higher number dimensions.

## Wednesday, August 8, 2012

### Multidimensional Nature of Time and Space (7)

We have already looked at the 4-dimensional nature of time, relating ultimately to the dynamic complementary interaction with respect to (i) internal and external and (ii) whole and part aspects. This notion of time strictly applies to all phenomenal processes (both physical and psychological).

In my own writings I have concentrated on the 1, 2 and 4-dimensional interpretation of relationships culminating with the even more refined 8-dimensional structure. Now before going on to look at the crucial distinction as between the even and odd integer dimensions with respect to the interpretation of time, we will now look briefly at the nature of this 8-dimensional interpretation.

Once again the 8 dimensions of time (and space) as qualitatively interpreted are inversely related to the 8 roots of 1 (in quantitative terms). We have already encountered the first 4 of these roots 1, - 1, i and - i when looking at 4-dimensional interpretation. However four additional roots arise i.e. 1/k(i + i), 1/k (1 - i), 1/k(- 1 + i) and 1/k(- 1 - i) respectively (where k = square root of 2).

So the question then arises as to what these additional roots signify in complementary qualitative terms!

As we have seen, 1 as a real number relates to specific (conscious) interpretation with respect to the actual, whereas i, as an imaginary number, relates to holistic (unconscious) interpretation with respect to the corresponding potential nature of phenomena. Whereas the real corresponds directly with (linear) rational understanding, the imaginary indirectly corresponds to (linear) reason (i.e. as a means of translating circular type logic in an indirect linear manner).

Now the equality of real and imaginary parts - which applies with respect to all four roots - likewise signifies a perfect balance with respect to both the real and imaginary aspects of interpretation. From a psychological perspective, this implies that both conscious and unconscious aspects of understanding are now so refined that neither aspect can assume a phenomenal identity (which would signify a degree of separate independence).

Fascinatingly, if we attempt to represent these 4 complex roots in geometrical terms they will appear as null lines with magnitude = 0.

Now from a psychological perspective, such an experience of time would be identified with pure contemplative awareness i.e. spiritual light. Equally in complementary fashion, physical light is likewise characterised by this very notion of time.

Since Einstein, the absolute nature of light has been appreciated. So though the speed of light serves as reference frame for all phenomena (travelling at lesser speeds) within its own frame of reference, time does not pass for a beam of light. So physical light in this context simply exists in the present moment just as in complementary fashion spiritual light (through pure contemplative awareness) likewise exists in the present moment.

However as we have seen though light (in both physical and spiritual terms) can be represented in null terms (= 0), likewise it can be represented in both cases as comprising both real and imaginary aspects (of equal magnitude). This entails with respect to physical light, that it equally comprises both particle and wave aspects.

Now by definition when these are not separated light remains in its null form. However both the particle and wave forms can appear as real though interaction with more rigid phenomena. So when the particle aspect appears as real, the wave aspect (as its physical shadow) remains imaginary; likewise when the wave aspect now appears as real, the particle aspect (as its physical shadow) likewise remains imaginary.

A complementary recognised interpretation likewise applies with respect to spiritual (contemplative) light. When the particle aspect (as immanence) operates, the spiritual light is revealed within particles; however the shadow wave aspect (as transcendence) where the spiritual light is understood as beyond all phenomena remains hidden. Then in reverse, when the wave aspect (as transcendence) is revealed the shadow particle aspect (as immanence) remains hidden.

This directly implies that the purest contemplative experience (as an emptiness with respect to all phenomena) necessarily entails the equal balance of both immanent (particle) and transcendent (wave) aspects.

However there are 4 complex roots suggesting therefore 4 distinctive interpretations. It has now long been my belief that what these roots have their qualitative equivalent in the holistic mathematical nature of the fundamental four forces. So we have illustrated one of these (i.e. electromagnetic energy of which natural light is a component). However in principle the other 3 would then apply to the three remaining forces. Then in complementary spiritual terms, we have four forces. Indeed I have relayed in other places how mystical personalities generally conform to one of these possible four forces. So a force in this sense relates to psychological - rather than physical - motion, in what could be accurately referred to as motivation (pertaining to the will).

Indeed proper understanding of the nature of 2, 4 and 8 dimensions leads to what I refer to as a holistic (or integral) TOE in that it has the capacity to deal with all the fundamental interactions in nature.

So the 2-dimensional approach relates to the dynamic interaction of phenomena (external and internal) in complementary physical and psychological terms.

The 4-dimensional approach relates to the additional dynamic appreciation of the combined interaction of phenomena in the context of space and time (whole and part).

Finally the 8-dimensional approach attempts to finally unify phenomena and dimensions (physically and psychologically). This leads to an appreciation of such unification being contained within - what we recognise - as forces.

So in the recognition of the ultimate nature of such forces (physical and spiritual) phenomena of a relative nature no longer arise. Thus in the strictest sense, phenomenal reality does not exist, as phenomena, in any absolute sense are ineffable. Rather - what we call - reality is necessarily characterised by phenomenal appearances (of a merely relative nature).

I have long recognised the 1, 2, 4 and 8 dimensional perspectives as being especially important with respect to an integrated scientific approach. However all other possible dimensions have their own significance.

So we will next look briefly at odd integer dimensions.

In my own writings I have concentrated on the 1, 2 and 4-dimensional interpretation of relationships culminating with the even more refined 8-dimensional structure. Now before going on to look at the crucial distinction as between the even and odd integer dimensions with respect to the interpretation of time, we will now look briefly at the nature of this 8-dimensional interpretation.

Once again the 8 dimensions of time (and space) as qualitatively interpreted are inversely related to the 8 roots of 1 (in quantitative terms). We have already encountered the first 4 of these roots 1, - 1, i and - i when looking at 4-dimensional interpretation. However four additional roots arise i.e. 1/k(i + i), 1/k (1 - i), 1/k(- 1 + i) and 1/k(- 1 - i) respectively (where k = square root of 2).

So the question then arises as to what these additional roots signify in complementary qualitative terms!

As we have seen, 1 as a real number relates to specific (conscious) interpretation with respect to the actual, whereas i, as an imaginary number, relates to holistic (unconscious) interpretation with respect to the corresponding potential nature of phenomena. Whereas the real corresponds directly with (linear) rational understanding, the imaginary indirectly corresponds to (linear) reason (i.e. as a means of translating circular type logic in an indirect linear manner).

Now the equality of real and imaginary parts - which applies with respect to all four roots - likewise signifies a perfect balance with respect to both the real and imaginary aspects of interpretation. From a psychological perspective, this implies that both conscious and unconscious aspects of understanding are now so refined that neither aspect can assume a phenomenal identity (which would signify a degree of separate independence).

Fascinatingly, if we attempt to represent these 4 complex roots in geometrical terms they will appear as null lines with magnitude = 0.

Now from a psychological perspective, such an experience of time would be identified with pure contemplative awareness i.e. spiritual light. Equally in complementary fashion, physical light is likewise characterised by this very notion of time.

Since Einstein, the absolute nature of light has been appreciated. So though the speed of light serves as reference frame for all phenomena (travelling at lesser speeds) within its own frame of reference, time does not pass for a beam of light. So physical light in this context simply exists in the present moment just as in complementary fashion spiritual light (through pure contemplative awareness) likewise exists in the present moment.

However as we have seen though light (in both physical and spiritual terms) can be represented in null terms (= 0), likewise it can be represented in both cases as comprising both real and imaginary aspects (of equal magnitude). This entails with respect to physical light, that it equally comprises both particle and wave aspects.

Now by definition when these are not separated light remains in its null form. However both the particle and wave forms can appear as real though interaction with more rigid phenomena. So when the particle aspect appears as real, the wave aspect (as its physical shadow) remains imaginary; likewise when the wave aspect now appears as real, the particle aspect (as its physical shadow) likewise remains imaginary.

A complementary recognised interpretation likewise applies with respect to spiritual (contemplative) light. When the particle aspect (as immanence) operates, the spiritual light is revealed within particles; however the shadow wave aspect (as transcendence) where the spiritual light is understood as beyond all phenomena remains hidden. Then in reverse, when the wave aspect (as transcendence) is revealed the shadow particle aspect (as immanence) remains hidden.

This directly implies that the purest contemplative experience (as an emptiness with respect to all phenomena) necessarily entails the equal balance of both immanent (particle) and transcendent (wave) aspects.

However there are 4 complex roots suggesting therefore 4 distinctive interpretations. It has now long been my belief that what these roots have their qualitative equivalent in the holistic mathematical nature of the fundamental four forces. So we have illustrated one of these (i.e. electromagnetic energy of which natural light is a component). However in principle the other 3 would then apply to the three remaining forces. Then in complementary spiritual terms, we have four forces. Indeed I have relayed in other places how mystical personalities generally conform to one of these possible four forces. So a force in this sense relates to psychological - rather than physical - motion, in what could be accurately referred to as motivation (pertaining to the will).

Indeed proper understanding of the nature of 2, 4 and 8 dimensions leads to what I refer to as a holistic (or integral) TOE in that it has the capacity to deal with all the fundamental interactions in nature.

So the 2-dimensional approach relates to the dynamic interaction of phenomena (external and internal) in complementary physical and psychological terms.

The 4-dimensional approach relates to the additional dynamic appreciation of the combined interaction of phenomena in the context of space and time (whole and part).

Finally the 8-dimensional approach attempts to finally unify phenomena and dimensions (physically and psychologically). This leads to an appreciation of such unification being contained within - what we recognise - as forces.

So in the recognition of the ultimate nature of such forces (physical and spiritual) phenomena of a relative nature no longer arise. Thus in the strictest sense, phenomenal reality does not exist, as phenomena, in any absolute sense are ineffable. Rather - what we call - reality is necessarily characterised by phenomenal appearances (of a merely relative nature).

I have long recognised the 1, 2, 4 and 8 dimensional perspectives as being especially important with respect to an integrated scientific approach. However all other possible dimensions have their own significance.

So we will next look briefly at odd integer dimensions.

## Tuesday, August 7, 2012

### Multidimensional Nature of Time and Space (6)

Perhaps before moving on further it might be instructive to elaborate a little more on what the imaginary nature of time entails.

As we have seen in psychological terms the notion of imaginary time arises due to the unconscious aspect of human experience. This then accords in complementary fashion with the holistic aspect of physical phenomena.

Now this holistic aspect remains very difficult for conventional scientists to accept as it implies that - what can only be understood as - holistic intelligence strictly applies with respect to all physical processes.

Last night while watching a TV programme on slime mold, a superb example was given of this holistic intelligence. So even though the slime mold represents a simple organic system comprising a single cell it can create complicated networks that are as efficient (if not more efficient) than those designed by professional engineers. So I would describe this as an innate holistic capacity that has emerged through evolution. However, much as we might want to avoid the issue, this inevitably relates to some inherent psychic capacity within matter. So whereas we may accept that in still remains with respect to the slime mold at an extreme innate level, yet we cannot deny its existence (at a deep unconscious level).

Of course I would go much further than this as I have already suggested with respect to prime numbers and the Riemann Hypothesis. It is now becoming apparent - at least in some quarters - that underlying the conventional "real" prime numbers (as discrete integers) is an intricate harmonic system of number wave patterns that are imaginary in nature.

So in a very direct sense this alternative imaginary number system (that is intimately connected with the "real" primes) represents the holistic basis of the number system.

A deeper understanding of this combined number system (of discrete number particles and continuous waves) implies that it is already inherent in the most basic of phenomenal physical processes. Thus a profound mystery, that we are only now able to start grappling with after so much evolution in human consciousness, can be seen to have already been inherent in physical processes from the very beginning of their phenomenal existence.

Again this really points to the fact that the fundamental scientific paradigm that we have been using to interpret physical reality (as if in some way independent of its psychological aspect) is ultimately totally unfounded.

And one of the key implications of combining holistic mind with specific phenomena of matter is the realisation that all phenomenal processes - both physical and psychological - necessarily unfold in both real and imaginary time (and of course also in complementary real and imaginary space).

Like many other I have been looking at the London Olympics over the past week. When the joyous athletes who have just won the gold medal are questioned about this defining moment in their careers, so often they express it as the realisation of a dream dating from early childhood.

So the "dream" relates essentially to the holistic unconscious aspect of experience that starts as mere potential for fulfilment. Thus it is clear that these athletes see the eventual prospect of winning the gold medal as providing them with a deeper holistic meaning in providing a true sense of fulfilment. And then in the wonderful moment of actual achievement the potential dream is finally realised.

Normally we think of dreams (as in sleep) as of an imaginary nature i.e. fantasy. And in very true sense such dreams (as expressive of the holistic unconscious) do indeed pertain to imaginary time (and space).

However the unconscious likewise play a big role in our waking lives (as we have just illustrated with Olympic athletes). Likewise the unconscious is especially important with respect to all creative inspiration. Strictly, even the most mundane of conscious tasks would not be possible if not oiled as it were with some measure of holistic (unconscious) insight.

Therefore all events, that psychologically unfold, contain both specific (conscious) and holistic (unconscious) aspects. And this implies that all such events unfold with respect to both real and imaginary time (and space). And once again because physical and psychological are dynamically complementary, this equally implies that in the physical world, all phenomenal events unfold in both real and imaginary time (and space).

Once again the real aspect relates to time (as independent) where opposite polarities are relatively separated. The imaginary aspect relates to time (as ultimately interdependent) where opposite polarities are relatively united. Thus an athlete may spend many long years of training in real time. However in the moment of fulfilling the dream this all melts into the present moment. So imaginary time - though linear in nature - is always expressive of a present moment that continually exists. Thus the more deeply aware one is of this underlying present moment the more imaginary is one's actual experience of time i.e. as a secondary manifestation in linear terms of what inherently is present and unchanging.

Not surprisingly therefore the purest experience of the imaginary nature of time (and space) unfolds at the "higher" more refined levels of contemplative experience.

However such contemplation in turn represents but a refinement in intuitive capacity (which is an integral part of all experience).

And once again - because physical and psychological are complementary - we can accurately say that the earliest i.e. most primitive physical interactions belong more to a world of imaginary - rather than real - time (and space). So virtual particles - for example - largely inhabit imaginary time (and space).

It is only when dynamic interaction sufficiently slows down to enable independent particles to form that the world of real time (and space) properly unfolds. And this world is necessarily underlined by a corresponding world of imaginary time (and space).

As we have seen in psychological terms the notion of imaginary time arises due to the unconscious aspect of human experience. This then accords in complementary fashion with the holistic aspect of physical phenomena.

Now this holistic aspect remains very difficult for conventional scientists to accept as it implies that - what can only be understood as - holistic intelligence strictly applies with respect to all physical processes.

Last night while watching a TV programme on slime mold, a superb example was given of this holistic intelligence. So even though the slime mold represents a simple organic system comprising a single cell it can create complicated networks that are as efficient (if not more efficient) than those designed by professional engineers. So I would describe this as an innate holistic capacity that has emerged through evolution. However, much as we might want to avoid the issue, this inevitably relates to some inherent psychic capacity within matter. So whereas we may accept that in still remains with respect to the slime mold at an extreme innate level, yet we cannot deny its existence (at a deep unconscious level).

Of course I would go much further than this as I have already suggested with respect to prime numbers and the Riemann Hypothesis. It is now becoming apparent - at least in some quarters - that underlying the conventional "real" prime numbers (as discrete integers) is an intricate harmonic system of number wave patterns that are imaginary in nature.

So in a very direct sense this alternative imaginary number system (that is intimately connected with the "real" primes) represents the holistic basis of the number system.

A deeper understanding of this combined number system (of discrete number particles and continuous waves) implies that it is already inherent in the most basic of phenomenal physical processes. Thus a profound mystery, that we are only now able to start grappling with after so much evolution in human consciousness, can be seen to have already been inherent in physical processes from the very beginning of their phenomenal existence.

Again this really points to the fact that the fundamental scientific paradigm that we have been using to interpret physical reality (as if in some way independent of its psychological aspect) is ultimately totally unfounded.

And one of the key implications of combining holistic mind with specific phenomena of matter is the realisation that all phenomenal processes - both physical and psychological - necessarily unfold in both real and imaginary time (and of course also in complementary real and imaginary space).

Like many other I have been looking at the London Olympics over the past week. When the joyous athletes who have just won the gold medal are questioned about this defining moment in their careers, so often they express it as the realisation of a dream dating from early childhood.

So the "dream" relates essentially to the holistic unconscious aspect of experience that starts as mere potential for fulfilment. Thus it is clear that these athletes see the eventual prospect of winning the gold medal as providing them with a deeper holistic meaning in providing a true sense of fulfilment. And then in the wonderful moment of actual achievement the potential dream is finally realised.

Normally we think of dreams (as in sleep) as of an imaginary nature i.e. fantasy. And in very true sense such dreams (as expressive of the holistic unconscious) do indeed pertain to imaginary time (and space).

However the unconscious likewise play a big role in our waking lives (as we have just illustrated with Olympic athletes). Likewise the unconscious is especially important with respect to all creative inspiration. Strictly, even the most mundane of conscious tasks would not be possible if not oiled as it were with some measure of holistic (unconscious) insight.

Therefore all events, that psychologically unfold, contain both specific (conscious) and holistic (unconscious) aspects. And this implies that all such events unfold with respect to both real and imaginary time (and space). And once again because physical and psychological are dynamically complementary, this equally implies that in the physical world, all phenomenal events unfold in both real and imaginary time (and space).

Once again the real aspect relates to time (as independent) where opposite polarities are relatively separated. The imaginary aspect relates to time (as ultimately interdependent) where opposite polarities are relatively united. Thus an athlete may spend many long years of training in real time. However in the moment of fulfilling the dream this all melts into the present moment. So imaginary time - though linear in nature - is always expressive of a present moment that continually exists. Thus the more deeply aware one is of this underlying present moment the more imaginary is one's actual experience of time i.e. as a secondary manifestation in linear terms of what inherently is present and unchanging.

Not surprisingly therefore the purest experience of the imaginary nature of time (and space) unfolds at the "higher" more refined levels of contemplative experience.

However such contemplation in turn represents but a refinement in intuitive capacity (which is an integral part of all experience).

And once again - because physical and psychological are complementary - we can accurately say that the earliest i.e. most primitive physical interactions belong more to a world of imaginary - rather than real - time (and space). So virtual particles - for example - largely inhabit imaginary time (and space).

It is only when dynamic interaction sufficiently slows down to enable independent particles to form that the world of real time (and space) properly unfolds. And this world is necessarily underlined by a corresponding world of imaginary time (and space).

## Monday, August 6, 2012

### Multidimensional Nature of Time and Space (5)

We have already looked at both the linear (1-dimensional) and circular (2-dimensional) perspectives on time. Once again from the 1-dimensional perspective, time is conceived in somewhat absolute fashion as having one positive direction. Then from the corresponding 2-dimensional perspective, time is conceived in relative terms as having two (complementary) directions that are positive and negative with respect to each other.

So the the nature of time conforms (in physical and psychological terms) directly with the holistic mathematical notion of dimensions, whereby each integer (representing the number of qualitative dimensions) bears an inverse complementary relationship with its corresponding number of roots with respect to unity, in quantitative terms. Thus the 2-dimensional qualitative structure (of both time and space) is inversely related to the 2 roots of 1 i.e. + 1 and - 1 (in quantitative terms).

However this mathematical notion of dimensions can then be generalised for any number giving a coherent structure for the 3, 4, 5, ... n dimensions of time (and space).

Indeed ultimately we can give meaning to the notion of such dimensions as fractional, negative, irrational (algebraic and transcendental) imaginary etc.

Thus associated with any number (in quantitative terms) is a corresponding (qualitative) dimensional meaning that intimately applies to the dynamic relative nature of both time (and space) in physical and psychological terms!

To illustrate these ideas a little further we will now look at the especially important case with respect to the true dynamic structure of time (and space) from a 4-dimensional perspective.

So 4 here (representing qualitative dimensions) is inversely complementary with the 4 roots of 1 i.e. 1 - 1, i and - i (in quantitative terms).

Now the first pair here replicate the qualitative structure associated with 2 (as representing dimensions). So once again time (and space) here have two complementary real directions that are positive and negative with respect to each other.

However the second pair i and - i are now also associated with two imaginary directions of time (and space).

Once again it may be instructive to concentrate a little on what these imaginary directions precisely entail!

In conventional scientific terms the relationship between whole and part is misrepresented in merely a conscious real fashion.

Once again as an illustration, if we take the concept of number this represents a whole notion, that potentially applies in infinite terms to all perceptions within its class. However actual numbers (as parts) are necessarily of a finite nature. So strictly, in the interaction of concept (whole) with perceptions (parts), we have the interaction of (potential) infinite with (actual) finite notions.

However in conventional scientific interpretation, the potential aspect of the whole is then reduced in a merely finite manner, with the whole concept now mistakenly assumed as applying to all actual finite part cases within its class.

So in the most general sense - as is the case with mathematical proof - Conventional Science assumes a direct correspondence in real terms as between (whole) concepts and (part) perceptions in a conscious manner. In this sense the assumed real nature of phenomena corresponds directly with the merely conscious nature of interpretation involved.

However when we understand the relationship as between concepts and perceptions in a more refined dynamic manner, we must necessarily allow for holistic (unconscious) as well as specific (conscious) aspects of experience. So strictly the experience of a concept, as potentially applying in an infinite manner, is of an intuitive (unconscious) nature, whereas the reduced interpretation of such a concept, as applying directly to all finite perceptions, is rational (conscious).

Likewise from the opposite perspective, the interpretation of an actual part phenomenon as finite and specific is rational, whereas its understanding, as in somehow reflecting the infinite whole, is once again of an intuitive (unconscious) nature.

Thus the true physical relationship of whole and part, properly entails both holistic (infinite) and specific (finite) aspects, entailing a corresponding psychological interaction that is both of an intuitive (unconscious) and rational (conscious) nature.

Now the holistic mathematical significance of "imaginary" is that it represents an indirect finite manner of representing, in physical terms, what is inherently of a potential infinite nature. In complementary psychological terms, it represents the manner of conveying what is properly of a holistic intuitive nature in an indirect rational fashion.

Thus in conventional terms, phenomena are treated in real terms as whole-parts, where every each phenomenon is interpreted in reduced quantitative terms as part of a larger whole.

However if we are to preserve the crucial qualitative distinction as between whole and part we must understand the relationship in complementary imaginary terms. Thus the part in this context - while maintaining its unique nature - reflects the infinite whole (that is qualitatively distinct). Likewise the (holistic) whole in a sense reflects all its specific parts in collective terms (again in a qualitatively distinct manner).

So in Jungian terms, each specific object is now understood in some sense as a unique immanent archetype of a universal whole (that is infinite); in reverse terms the whole, while collectively embodying all its parts, serves as a transcendent archetype, that retains a qualitative distinction.

And this two-way relationship can be indirectly expressed in rational terms as the complementarity of opposites of an imaginary nature.

Thus when we allow for the quantitative/qualitative distinction with respect to all phenomena in nature, then time necessarily possesses both real and imaginary directions that are positive and negative with respect to each other.

So from one perspective, our rational understanding of phenomena (as reduced whole/parts) takes place in real time with - relatively - positive (external) and negative (internal) directions.

However equally from a refined holistic perspective, indirectly translated in a rational manner, our intuitive understanding of phenomena (as symbols or archetypes of a more universal meaning) takes place in imaginary time with again - relatively - positive (whole) and negative (part) aspects.

Thus from the dynamic 4-dimensional (circular) perspective, time has both real and imaginary aspects with both both positive and negative directions.

And as space and time are complementary, this likewise entails that space has real and imaginary aspects with both positive and negative directions.

Now, this might seem to imply that we now have 8 dimensions in total. However in dynamic terms, the imaginary aspect of space always coincides with the real aspect of time in experience; likewise the imaginary aspect of time coincides with the real aspect of space.

Once again in experiential terms this strongly accords with Jungian notions, whereby the function that is at any time conscious in experience has a corresponding complementary shadow that remains hidden and unconscious! The clear implication of this therefore is that when the unconscious aspect of experience is not properly recognised our actual experience of phenomenal interactions in space and time becomes increasingly rigid!

So when interpreted in a more refined dynamically interactive manner, all scientific and mathematical understanding entails both real (conscious) and imaginary (unconscious) aspects in psychological terms; then in corresponding physical fashion this entails that all phenomena likewise entail both real (specific) and imaginary (holistic) aspects.

This necessarily entails therefore that all phenomena (in both physical and psychological terms) interact qualitatively in 4-dimensional time (and space) with - relatively - real and imaginary aspects (in both positive and negative directions).

So the the nature of time conforms (in physical and psychological terms) directly with the holistic mathematical notion of dimensions, whereby each integer (representing the number of qualitative dimensions) bears an inverse complementary relationship with its corresponding number of roots with respect to unity, in quantitative terms. Thus the 2-dimensional qualitative structure (of both time and space) is inversely related to the 2 roots of 1 i.e. + 1 and - 1 (in quantitative terms).

However this mathematical notion of dimensions can then be generalised for any number giving a coherent structure for the 3, 4, 5, ... n dimensions of time (and space).

Indeed ultimately we can give meaning to the notion of such dimensions as fractional, negative, irrational (algebraic and transcendental) imaginary etc.

Thus associated with any number (in quantitative terms) is a corresponding (qualitative) dimensional meaning that intimately applies to the dynamic relative nature of both time (and space) in physical and psychological terms!

To illustrate these ideas a little further we will now look at the especially important case with respect to the true dynamic structure of time (and space) from a 4-dimensional perspective.

So 4 here (representing qualitative dimensions) is inversely complementary with the 4 roots of 1 i.e. 1 - 1, i and - i (in quantitative terms).

Now the first pair here replicate the qualitative structure associated with 2 (as representing dimensions). So once again time (and space) here have two complementary real directions that are positive and negative with respect to each other.

However the second pair i and - i are now also associated with two imaginary directions of time (and space).

Once again it may be instructive to concentrate a little on what these imaginary directions precisely entail!

In conventional scientific terms the relationship between whole and part is misrepresented in merely a conscious real fashion.

Once again as an illustration, if we take the concept of number this represents a whole notion, that potentially applies in infinite terms to all perceptions within its class. However actual numbers (as parts) are necessarily of a finite nature. So strictly, in the interaction of concept (whole) with perceptions (parts), we have the interaction of (potential) infinite with (actual) finite notions.

However in conventional scientific interpretation, the potential aspect of the whole is then reduced in a merely finite manner, with the whole concept now mistakenly assumed as applying to all actual finite part cases within its class.

So in the most general sense - as is the case with mathematical proof - Conventional Science assumes a direct correspondence in real terms as between (whole) concepts and (part) perceptions in a conscious manner. In this sense the assumed real nature of phenomena corresponds directly with the merely conscious nature of interpretation involved.

However when we understand the relationship as between concepts and perceptions in a more refined dynamic manner, we must necessarily allow for holistic (unconscious) as well as specific (conscious) aspects of experience. So strictly the experience of a concept, as potentially applying in an infinite manner, is of an intuitive (unconscious) nature, whereas the reduced interpretation of such a concept, as applying directly to all finite perceptions, is rational (conscious).

Likewise from the opposite perspective, the interpretation of an actual part phenomenon as finite and specific is rational, whereas its understanding, as in somehow reflecting the infinite whole, is once again of an intuitive (unconscious) nature.

Thus the true physical relationship of whole and part, properly entails both holistic (infinite) and specific (finite) aspects, entailing a corresponding psychological interaction that is both of an intuitive (unconscious) and rational (conscious) nature.

Now the holistic mathematical significance of "imaginary" is that it represents an indirect finite manner of representing, in physical terms, what is inherently of a potential infinite nature. In complementary psychological terms, it represents the manner of conveying what is properly of a holistic intuitive nature in an indirect rational fashion.

Thus in conventional terms, phenomena are treated in real terms as whole-parts, where every each phenomenon is interpreted in reduced quantitative terms as part of a larger whole.

However if we are to preserve the crucial qualitative distinction as between whole and part we must understand the relationship in complementary imaginary terms. Thus the part in this context - while maintaining its unique nature - reflects the infinite whole (that is qualitatively distinct). Likewise the (holistic) whole in a sense reflects all its specific parts in collective terms (again in a qualitatively distinct manner).

So in Jungian terms, each specific object is now understood in some sense as a unique immanent archetype of a universal whole (that is infinite); in reverse terms the whole, while collectively embodying all its parts, serves as a transcendent archetype, that retains a qualitative distinction.

And this two-way relationship can be indirectly expressed in rational terms as the complementarity of opposites of an imaginary nature.

Thus when we allow for the quantitative/qualitative distinction with respect to all phenomena in nature, then time necessarily possesses both real and imaginary directions that are positive and negative with respect to each other.

So from one perspective, our rational understanding of phenomena (as reduced whole/parts) takes place in real time with - relatively - positive (external) and negative (internal) directions.

However equally from a refined holistic perspective, indirectly translated in a rational manner, our intuitive understanding of phenomena (as symbols or archetypes of a more universal meaning) takes place in imaginary time with again - relatively - positive (whole) and negative (part) aspects.

Thus from the dynamic 4-dimensional (circular) perspective, time has both real and imaginary aspects with both both positive and negative directions.

And as space and time are complementary, this likewise entails that space has real and imaginary aspects with both positive and negative directions.

Now, this might seem to imply that we now have 8 dimensions in total. However in dynamic terms, the imaginary aspect of space always coincides with the real aspect of time in experience; likewise the imaginary aspect of time coincides with the real aspect of space.

Once again in experiential terms this strongly accords with Jungian notions, whereby the function that is at any time conscious in experience has a corresponding complementary shadow that remains hidden and unconscious! The clear implication of this therefore is that when the unconscious aspect of experience is not properly recognised our actual experience of phenomenal interactions in space and time becomes increasingly rigid!

So when interpreted in a more refined dynamically interactive manner, all scientific and mathematical understanding entails both real (conscious) and imaginary (unconscious) aspects in psychological terms; then in corresponding physical fashion this entails that all phenomena likewise entail both real (specific) and imaginary (holistic) aspects.

This necessarily entails therefore that all phenomena (in both physical and psychological terms) interact qualitatively in 4-dimensional time (and space) with - relatively - real and imaginary aspects (in both positive and negative directions).

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