## Saturday, September 22, 2012

### Number and Transformation

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

I have been long fascinated by the fact that the two binary digits (1 and 0)
when used in a quantitative manner can potentially encode all
information processes.

I am therefore of the opinion that the same two digits when used in an
appropriate qualitative manner can likewise potentially encode all transformation processes.

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

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

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

At a deeper level this circular aspect relates to the manner in which the
fundamental polarities - which necessarily underlie all phenomenal
relationships - are configured.

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

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

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

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

So the nature of 2-dimensional understanding bears a close relationship
therefore with the two roots of 1, i.e. + 1 and - 1. However whereas with
standard quantitative appreciation, these two values are separated, in
holistic qualitative terms they are interdependent. Thus 2-dimensional
understanding can be therefore expressed as the complementarity of (real)
opposites in the dynamic interaction of poles which are positive and
negative with respect to each other.

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

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

So in short, each number as dimension, relates to a unique manner of
configuring the dynamic interaction of the two fundamental sets of
polarities. So rather like with a compass with four starting coordinates, we
can obtain ever more detailed notions of direction, likewise starting with
the two fundamental polarity sets we can give ever more refined expression
to the dynamic interaction between opposite coordinates through moving to
higher dimensional numbers! So once again each number in this qualitative
sense represents a unique manner of configuring the dynamic interaction as
between the essential polarities that necessarily underlie phenomenal experience.

This key issue is avoided completely in conventional scientific (and
mathematical) terms through sole concentration on the special limiting case
where understanding in formal terms is 1-dimensional.

Now, I believe that this qualitative holistic notion of dimension intimately
applies to the true nature of space and time. So if we were to map
space-time reality, we could validly say that it is truly multi-dimensional
where the ceaseless interweaving of these qualitative numbers are involved. Going even further, the distinctive qualitative features that phenomena possess, thereby represent multi-dimensional configurations with respect to space and time that are ultimately rooted in the qualitative notion of number.

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

In fact, from this perspective, we can say that such phenomena (which
possess no ultimate substance) fundamentally represent but the dynamic
configurations of number (with respect to both their quantitative and
qualitative aspects).

From a geometrical perspective the quantitative shape of all phenomena can
be understood in terms of the interplay of both linear and circular
properties in varying dimensions.

The corresponding qualitative "shape" of these phenomena in their uniquely
distinctive features can likewise be understood in terms of both specific
and holistic features again with respect to the combined interplay of
multiple dimensional numbers (which again represent a distinctive manner in which the fundamental polarities are dynamically configured).

In a direct sense I would see the quantitative aspect of understanding as
relating to form, with the qualitative relating to the mysterious
transformation of this form.

So if we are to isolate what is common to all patterns of transformation, it
is the intersection of this holistic qualitative aspect with established
quantitative notions of form.

However when one accepts that the very nature of the standard paradigm of
science and mathematics is to attempt to reduce this interaction in a merely quantitative manner, then one can perhaps appreciate why it is inimical to transformation.

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

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

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

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

My key point again is this!

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

## Friday, September 21, 2012

### Connections to Taoism

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

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

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

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

For example on P. 152 we have this statement

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

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

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

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

"Tao produced Unity; Unity produced Duality: Duality produced Trinity; and Trinity produced all existing objects.
These myriad objects leave darkness behind them and embrace the light, being harmonised by contact with the Vital Force."

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

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

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

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

## Saturday, September 8, 2012

### What is Number?

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

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

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

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

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

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

We can easily illustrate this with respect to 1.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thus once again, 1 - 1 = 0.

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

So we have 1 + 1 = 2.

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

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

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

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

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

## Friday, September 7, 2012

### Binary Wonder

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Wednesday, September 5, 2012

### Prime Movers

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

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

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

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

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

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

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

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

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

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

It is the same with respect to the prime numbers.

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

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

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

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

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

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

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

## Tuesday, September 4, 2012

### Reality as Number

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

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

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

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

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

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

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

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

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

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

It is the same with respect to the prime numbers.

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

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

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

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

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

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

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

## Monday, September 3, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Saturday, September 1, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

Real transcendental dimensions are of the most refined type whereby one understands all phenomenal relations - not in terms of (separate) linear or (circular) holistic notions - but rather as the relationship between both.
The positive refer to the subtle rational appreciation of this relationship; the negative then relate to direct intuitive realisation.

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

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

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

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

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

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

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

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

Now conventionally the Euler Identity is expressed by the equation,

e^(πi) + 1 = 0;

Therefore e^(πi) = - 1.

Then by squaring both sides

e^(2πi) = 1.

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

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

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

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

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

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

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

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

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

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

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

"Form is nor other than Emptiness

Emptiness is not other than Form."

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

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