Fractals 1 (Dynamic Interpretation of Fractal Dimensions)


Mayank started a very interesting thread on the deeper significance of fractals.

I have no doubt that Jung would have been fascinated by the work of Mandelbrot. Conventional Mathematics will never be able to solve the issues deeper that fractal geometry raises. Indeed - as I suggest in this post - the current interpretation of "fractal" dimensions represents - even in mathematical terms - a basic confusion. Such dimensions are more mysterious than the mathematicians care to admit.


In 1968, the French mathematician, Benoit Mandelbrot asked an intriguing question.

"How long is the coastline of Britain?"


The surprising answer is that there is in fact no definite answer. One can indeed approximate an answer by defining one’s scale of measurement. However as these measurements are increasingly refined the answer eventually becomes so large that it is theoretically infinite.


Thus though the (two- dimensional) area is clearly finite, the curve tracing its perimeter is (theoretically) infinite.


Mandelbrot referred to these naturally rough terrains (such as coastlines) as fractals.

Numerous examples can be found in nature displaying fractal features such as leaves, trees, mountains, clouds, galaxy clusters etc.



In so many ways fractals raise the intriguing - and highly important - question of the relationship of finite and infinite notions.


I will illustrate this by just three well-defined mathematical fractals.


A classic example "one-dimensional" is the Cantor Set. Here we start with a straight-line interval. We then delete the middle third of the line. This leaves two smaller intervals. Again we proceed to delete the middle thirds of these new intervals and proceed indefinitely in this same manner.


Ultimately, the total length removed will be equal to the length of the original interval (so that it no longer remains). Yet we will be left with an infinite number of points!.


The best known "two-dimensional" fractal of this type is the Koch snowflake (named after Van Koch).


Here we start by drawing an equilateral triangle (i.e. where the three sides are of equal length.

Again we take the middle third of each side to construct three new equilateral triangles (extending outwards).


Now if say the perimeter of the original triangle was 3 units to start with, then is easy to demonstrate that the perimeter of the new figure will be 4 units.

However we can continue this procedure indefinitely (i.e. constructing new equilateral triangles on the middle third of each exposed line of our figure).


So again we generate a significant paradox that though the area of the "snowflake" tends to a finite limiting value, its actual perimeter will be of infinite length.


Other "two dimensional" examples are the Sierspinski Carpet and Sierspinski Gasket. In the first case we start with a square and cut out the middle area (1/9) of total area.

We then divide the are remaining into smaller squares (8 in total) and apply the same procedure, continuing on indefinitely.

(In the latter case we cut out equilateral triangles rather than squares)


The Sierspinki Carpet and Gasket can be extended to "three-dimensional" cubes (i.e. the Menger Sponge).


Here we start to cut away "middle" sections. Clearly as we create more and more "holes" in the cube its surface area will continually increase and will eventually tend to infinity. However the volume of the sponge will have a finite value.


Mathematical fractals, in fact seriously challenge conventional notions of dimensions.

In conventional terms we define dimensions in terms of unambiguous directional movements.


Thus we trace out a one-dimensional line by moving in a clearly defined direction (length).


We trace out a two dimensional figure (such as a rectangle) by moving unambiguously in two directions (length and height). We trace out a three dimensional cube by moving unambiguously in three directions (length, breath and height).

This is precisely how we define our conventional physical notions of dimensions. Space can be either one-dimensional (as a line), two dimensional (as a plane) or three dimensional (as a solid). The fourth dimension of time is treated as somewhat separate (though it can be treated – as in Relativity Theory as an "imaginary" space dimension).


When we proceed with this integral understanding of dimensions, all measurements appear to be finite.

However our simple mathematical fractals directly challenge this very notion, for here in each case we have a mixture of finite and infinite notions of measurement.


Again the line of zero finite length has contains an infinite set of points.


The snowflake of with a well-defined finite area has a perimeter of infinite length (the Koch Curve).


The sponge with a definite finite volume has a surface area that is infinite.

In fact, with these fractals, conventional notions of directional movement break down leading to an entirely different concept of dimension.

The key feature of the three fractals above is self-similarity (i.e. the parts are similar to the whole but on a reduced scale).

Now if we take a D-dimensional figure and divide it into N entirely similar parts, the similarity ratio, R between the entire figure and a single part is given by R = N raised to the Dth root (i.e. the reciprocal of D).


Thus if we take a straight line and split it into N equal pieces then each piece will clearly be 1/N of the length of the whole. So the similarity ratio R will be N.

This is exactly the value we get from our formula by taking D= 1. (This is what one would expect as the line is one-dimensional).

With the Koch snowflake there is a definite pattern whereby a perimeter length that starts with 3 units is increased to 4 .

So R is equal to 3 and N = 4.

The value of D which gives the dimension of the Koch Curve is obtained by the formula 3 = 4 (raised to the power of 1/D).

This can be simply solved by taking logs of both sides

So that log 3 = (1/D).(log 4).

D in fact has the surprising value of 1.2619 (correct to 4 decimal places).

Now I must take issue here with the conventional mathematical interpretation of this result.

It is generally used to prove that the dimension is fractional.


This strictly speaking is incorrect and betrays the reduced rational bias of mathematics.

Properly speaking a fraction is a rational number.


Now the number we have generated from the formula is not a rational but rather an irrational number (i.e. a finite number with an infinite random expansion of its decimal sequence). It only has a reduced rational interpretation when one approximates the value (correct to a defined number of decimal places).


From a qualitative point of view this is a highly significant point. The very word "fractal" to describe the dimension is something of a misnomer.

The dimension of the Koch Curve is not fractional but rather irrational. We thus have to explain the true meaning of irrational (and not just fractional) dimensions.



Fortunately Holistic Mathematics can throw considerable light on the qualitative significance of "fractals".

Remember a basic axiom of Holistic Mathematics is that every mathematical notion with a quantitative interpretation (in conventional mathematics) has a corresponding qualitative interpretation (in Holistic Mathematics).

Thus Holistic Mathematics can provide a deeper qualitative interpretation as to what is meant both by a fractional dimension and a true irrational dimension (as in this case of the Koch snowflake).

Let us go back now to the conventional notion of dimensions. Remember they are dependent on clearly defined one-directional interpretations of movement leading to unambiguous finite interpretations.

The psychological corollary of this is the use of a rational one-directional logic geared to merely (reduced) finite interpretations of reality.


It is very important to understand properly the key dynamics (or rather lack of dynamics) of linear rational understanding.


In experiential terms we have the continual interaction of perceptions and concepts. Our notions of objects are directly rooted in perceptions while our notions of dimensions are directly rooted in concepts.


Linear understanding – by its very nature – requires that we freeze our interpretation of this dynamic interaction. Without this, definite knowledge of reality would not be possible.


So with linear interpretations concepts are fixed and perceptions are viewed against this static dimensional background.

For example if I take an example of a number e.g. "7", this represents a particular number perception which is viewed against the general background concept of number. Thus we can have one general number concept (serving as a qualitative mental background for multiple number perceptions).


Indeed the conventional notion that time is one-dimensional (i.e. one-directional) is directly rooted in this interpretation of concept. Time thereby serves as the general qualitative background in which we locate "three-dimensional" space objects. Indeed the very ability to form definite perceptions regarding scientific objects regards freezing the dynamic interactions (involving perceptions and concepts) to a marked degree.

Though it is moving ahead of our story the true dynamic interpretation of dimensions is provided by a qualitative interpretation of the complex number system. Here space and time are truly symmetrical and bi-directional (with positive and negative polarities). "Real" space is simply "imaginary" time, and "real" time is "imaginary" space. The well-known Mandelbrot Set (which serves as the basis for the generation of beautiful computer graphics) not surprisingly is based on the (quantitative) counterpart of this dynamic number system.

So quite simply in experience by fixing the qualitative concept as general and universal, we fix time as one-dimensional (and space as three-dimensional).


This enables us to deal with the merely quantitative aspects of holons in a reduced manner.


In conventional quantitative interpretations "lower" level parts are included in "higher" level wholes. However the reverse does not apply. "Higher" level wholes are not included in "lower" level parts.


Thus for example if I cut a cake into a number of slices, in terms of conventional interpretation, each slice (as "lower" part) is included in the "higher" whole (i.e. the cake). However the "higher" whole (i.e. cake) is not included in the "lower" part (i.e. each slice).

Thus by freezing the interaction as between the qualitative and quantitative (concepts and perceptions), we are able to carry out this one-directional quantitative interpretation. This in turn confirms our notions that time is one-dimensional (i.e. moving forward in a positive manner) and that spatial perceptions are three dimensional (with positive directions).


This limited interpretation of true psychological dynamics is the very basis for our notions of rational science. Here quantitative objects as parts are divisible to the nth degree. They are literally viewed qualitatively as rational numbers (i.e. as fractions of larger objects). However the qualitative dimensions are viewed as whole and indivisible.

Science is based on the understanding of nature. It is very interesting how this natural aspect is preserved in terms of our understanding of dimensions. We literally view them in terms of the most common natural numbers.


Our scientific rational understanding only extends therefore to (quantitative) objects.

Surprisingly it does not extend to our understanding of (qualitative) dimensions.


However from a correct dynamic perspective this scientific interpretation is invalid. As physical dimensions are holons, clearly they must have a part (as well as whole) interpretation. However this part interpretation is not reflected in scientific understanding.


The true experiential relationship of concepts and perceptions is bi-directional (not one-directional). It is intuitive as well as rational; it deals with infinite as well as finite reality.


Let me illustrate this point with relation to the interpretation of a number.

Now in rational terms a number perception (e.g. "7) is viewed as invariant with respect to its number concept. In other words the dynamic interaction of the number concept with its corresponding perceptions does not alter our quantitative interpretation of numbers. In other words we have here a reduced one-directional interpretation of number.


Now in dynamic terms number perceptions dynamically interact with number concepts (and number concepts with perceptions).


Thus the number "1" as perception has a unique relationship with the number concept (to which "1" is related). However equally the number concept has a unique relationship with the number perception "1". The relationship is thus bi-directional.


Likewise this two-way relationship applies to the numbers 2, 3 4 etc. and by extension to any number.

In conventional (quantitative) terms we have a one-directional interpretation of the relationship as between whole and part.


The parts are included in the whole; however the whole is not included in the parts.

Both wholes and parts are confused with merely quantitative interpretations of reality.

Thus going back to the cake example, if I divide a cake into 5 slices, each part slice (i.e. 1) is included in the whole (i.e. 5).

However the bi-directional interpretation is far more subtle.


From one perspective the parts are transcended and included in a "higher" level whole. This involves the movement from the quantitative appreciation of perceptions to the qualitative appreciation of corresponding concepts. Thus the individual (quantitative) numbers are transcended and included in the qualitative number concept.


However from the opposite reverse perspective the whole is made immanent and included in each "lower" level part. This involves moving from the qualitative appreciation of the general number concept back to the quantitative perception of a particular number. In this manner the qualitative number concept is made immanent and included in each quantitative number perception.


Thus in this bi-directional interpretation we can see clearly that the parts are included in a "higher" whole (transcendence). Equally however the whole is included in the "lower" parts (immanence). This is the qualitative counterpart to mathematical self-replication psychological terms. Now an alternative way of expressing the same relationship is that objects are included in dimensions (and dimensions in objects).

Thus both objects and dimensions have both part and whole aspects.


Thus in this dynamic sense there is a matching qualitative concept for each quantitative perception.


Thus we never step into the same river twice! When we vary the perception the corresponding concept likewise varies; equally when we vary the concept its (corresponding) perception also varies.


Now this bi-directional interpretation is directly rooted in intuitive rather than rational understanding. Not surprisingly therefore it is entirely missed in rational one-directional interpretations.


The implication of the bi-directional interpretation is that when we now attempt reduced rational translations two equally valid interpretations are possible.


Conventional science concentrates on the merely quantitative aspect of experience (the division of perceptions into multiple parts).

However there is an equally valid qualitative interpretation (the division of concepts into multiple parts).


Thus properly understood it is not only objects that are fragmented in rational experience. The dimensions themselves are equally fragmented. However this inevitably missed in a quantitative one-directional interpretation.

Thus we can give an exciting new holistic mathematical interpretation to the conventional structures of the rational linear level. Properly understood the (horizontal) concrete operational structures involve quantitative fragmentation of perceptions (i.e. rational objects) against a fixed conceptual background (dimensions). However the (vertical) formal operational structures - properly interpreted - represent qualitative fragmentation of concepts (i.e. rational dimensions) against a fixed perceptual background) objects).

The (diagonal) vision-logic stage then involves the two-way interaction of rational objects with rational dimensions.


However invariably in science formal recognition of the role of intuition is largely missing. Thus the qualitative aspect is reduced to the quantitative and intuitive dynamics are themselves greatly impeded.

Thus vision-logic which should lead to "higher" spiritual stages often fails in this role due to lack of sufficient intuition.

Now in holistic mathematics concepts (dimensions) and perceptions (objects) are horizontal and vertical with respect to each other.

Again, the two-way interaction of rational objects with rational dimensions (i.e. the vision-logic) stage is the natural gateway to the "higher" spiritual stages. In holistic mathematical terms these are precisely defined as irrational combining both intuitive and rational interpretations of experience, (which are paradoxical in terms of each other).

The very means of moving to these "higher" more intuitive stages is by beginning to appreciate the two-way complementarity at a rational level.

Now there is an amazing parallel in quantitative mathematical terms. Any (horizontal) rational number raised to a (vertical) rational power or dimension is irrational (except for trivial cases that reduce to integral solutions).


The complementary qualitative interpretation provides a precise description of the true vision-logic stage (in holistic mathematical terms). Recognising the dynamic complementarity of (rational) perceptual and conceptual understanding is the very means of moving on directly to "higher" spiritual (i.e. irrational) understanding.

However considerable confusion exists which is exemplified by Ken Wilber as to the precise significance of the vision-logic stage.

In a previous post "Content and Context", I demonstrated the nature of this confusion. Ken portrays psychological development in transcendent holarchical terms. However his approach to research is very much an example of the reverse immanent approach (where understanding of the "higher" whole is induced through knowledge of the "lower" parts). However because Ken in effect uses a one-directional approach, he is unable to reconcile this apparent inconsistency.

So let me end by tracing the obvious parallels as between the quantitative and qualitative interpretation of "fractals".


Self-similar fractals (i.e. wholes replicated in parts and parts replicated in wholes) lead to paradoxical situations where finite and infinite notions are inseparable.

Conventional notions of integral dimensions break down and are replaced by irrational dimensions.


Rational understanding is geared to finite understanding and intuitive understanding to infinite notions.

Self-similar spiritual understanding (where wholes are included in parts and parts included in wholes) leads to the development of irrational structures. These involve the explicit interaction of both rational and intuitive understanding (which are paradoxical in terms of each other).


Quite literally therefore conventional notions of dimensions break down to be replaced by irrational dimensions.

Quite simply the deeper meaning of mathematical "fractals" cannot be grasped in terms of conventional rational understanding. It can only be understood in terms of "higher" spiritual understanding (where we can give a directly complementary qualitative interpretation).



Fractal Dimensions



My head hurts. Or should I say, the concepts jumping around in my

head seem unrelated and do not fit anywhere. What precisely is

"intuition"? I mean, really? Is it the relationship established and

  communicated from the level (dimension) above the one we are on?

The whole defines the parts, the parts make up the whole. The whole

can not exist without the parts. The whole is infinite and

qualitative, the parts are finite and definitive. The parts can be

measured, the whole can not be. So, vision-logic can see "intuition"

as concept, but can not "understand" it because it represents a

"whole". Vision-Logic takes you up to a point and then deposits you

to the wings of intuition? or is it to the wings of H.M.?

The fractal roads seem to replicate, but not lead anywhere..... Is

this the point? The pattern replicates until a point where it can

consciously alter itself or its environment. At which point, a new

pattern replicates itself ad infinitum....

Perhaps this is as good as it gets...until a new pattern emerges!


  • Juriel


    Re: Fractal Dimensions




    When translated in (indirect) rational terms, intuition always

    represents the coincidence of polar opposites. For example objective

    and subjective in dualistic terms are separate. However through

    (spiritual) intuition we are able to understand both poles as

    ultimately identical.

    Thus the generation of intuition in experience depends on

    maintaining the bi-directional nature of relationships where one

    switches readily between opposite poles.

    Whole and part are another example of polar opposites (this time in

    a vertical sense).

    Properly speaking intuition should not be associated with either

    whole or part (in isolation), but rather with the dynamic

    interaction of both.

    Now in terms of the rational linear level conop (i.e. concrete

    operational thinking), involves the specialised development of parts

    (i.e. perceptions).

    Formop (formal operational thinking) involves the specialised

    development of wholes (i.e. concepts).

    It is only at the vision-logic stage that the full possibilities for

    two-way interaction as between rational concepts and perceptions

    (wholes and parts) can take place. Thus the vision-logic stage is

    the most intuitive of the rational stages paving the way for a

    quasi-integration of experience.

    My key point however is that, due to mistranslation of its true

    nature, the generation of intuition often remains merely implicit

    and does not develop to its fullest potential. So a person then

    tends to plateau at this level of experience (i.e. vision-logic)

    without moving on to the "higher" spiritual levels.

    The fractal patterns I have been talking about in the post (apart

    from the coast line) are purely mathematical and fully


    This would not be the case with the more realistic fractal patterns

    of nature. Again there is self replication, but it is always subtly

    varying. This pattern would be more realistic in terms of actual


    Finally I would not attribute consciousness to physical fractal

    patterns but rather their (complementary) psychological






    Could you clarify in layman's terms?



    > (Mayank started a very interesting thread on the deeper

    significance of fractals. > I have no doubt that Jung would have

    been fascinated by the work of Mandelbrot (with its deep mandalic


    Hello Peter,

    I am new to the forum having found this through a link

    of another author. I was following the thread that Mayank started on

    merging fractal and consciousness as a study, and was greatly

    intrigued. Those that responded besides Judy Kay seemed to knock

    down his thesis and the concept seemed to die on the vine. I taught

    High School mathematics for a brief perod before entering the

    busines acumen, but I have a fond interest in mathematics

    escpecially how it might appy to the spiritual or consciousness. I

    have read a little bit on Chaos Theory and virtual intelligence,

    whcih Judy Kay mentioned. You started out by stating 2 interesting

    propositions, but like Paul , even with my B.S. in mathematics I

    could not follow easily. Could you please try the best you could to

    quaulify the 2 statements you made at the beginning in say layman's

    terminology? I love the forum even when the aruments get a little

    out of hand.

    Bill Cael


    Re: Could you clarify in laymans's terms


    Hello Bill,

    Welcome to the Forum.

    I am not sure which two statements you are referring to in your


    I would of course be glad to try and clarify if you let me know

    which ones you specifically have in mind.





    Re:Re:Could you clarify in laymens terms

    Hello Peter

    I think your ability to visualize the concepts is amazing to say the

    least. But you prompted your first post on this with 2 statements.

    If you would re-read my post I re-printed them from your post at the

    top of the page. Thanks: Bill Cael



    Reply to Bill



    My apologies for the oversight! I hope this reply is of some help.

    Fractals as you know represent a very contemporary development.

    They are fascinating for a number of reasons.

    1. They are widely found in nature. So in many ways fractal

    structures are potentially more relevant than our customary

    idealised scientific notions.

    2. Because of their relevance to many fields they tend to beak down

    the limited specialised boundaries of science facilitating a more

    integrated approach.

    3. They require a different type of understanding than that

    typically associated with scientific understanding.

    The conventional approach is still based on the rational paradigm

    (which is limited in many ways). I would say that fractals require a

    deeper holistic appreciation involving both reason and intuition.

    4. Fractals by their very nature are inherently dynamic. Through

    continual feed back remarkably complex structures can be generated

    from very simple initial states. Also these structures are extremely

    sensitive to changes in initial conditions.

    5. They lead to an entirely different appreciation of the notion of

    dimensions. Conventional notions of 3 dimensions (of space) and 1

    (of time) are so deeply ingrained that we find it difficult to

    imagine alternative notions. However fractal dimensions are entirely

    different in nature and raise very serious questions regarding the

    limitations of customary understanding. Personally I find this the

    most intriguing aspect of fractals.

    6. At another level fractals invite us to reconsider the meaning of

    art, the role of computers in science, and the relationship as

    between science and art. Some of the computer-generated graphics

    that are based on very simple mathematical iterative procedures are

    undeniably very beautiful.

    Fractals lead to a contemporary art form or should I say art-science

    form that is unique.

    Now Mayank realises that natural structures have close

    correspondents in terms of consciousness. Thus the fractal

    structures of nature must thereby in some way be replicated in our

    mental processes. And in several interesting and suggestive posts he

    has explored this important connection. Clearly they have struck a

    certain chord on the Forum. (When I last checked there were 59

    messages in his fractal thread).

    My own view is basically quite similar. The very basis of Holistic

    Mathematics is that every quantitative mathematical relationship has

    a dynamic qualitative counterpart. Thus the fundamental structures

    of reality (including consciousness) are therefore mathematical in

    this holistic sense.

    Therefore fractals offer a very interesting challenge. I am trying

    to demonstrate how one converts - as it were - from conventional

    mathematical notions (number, dimensions, finite, infinite etc.) to

    their dynamic psychological equivalents.

    It is very important not to confuse this holistic appreciation of

    mathematical notions with conventional interpretations. Otherwise

    reductionism abounds. However it is still possible to demonstrate

    that all holistic interpretations are complementary with standard

    analytical interpretations (while maintaining the qualitative


    So if I am doing my job properly, one will be able to take any

    (quantitative) mathematical notion (used in relation to physical

    fractals), and see that I have come up with a complementary (though

    qualitatively different) mathematical interpretation in relation to


    Of course appreciation of these holistic constructs requires a more

    subtle form of understanding that involves the interaction of both

    reason and intuition.

    Yes, I have no doubt that Jung would have been fascinated - if still

    alive - with the work of Mandelbrot.

    Apart from the fact that Jung had a particularly wide range of

    interests I have always maintained that implicitly he adopted a

    holistic mathematical approach to his work. He saw mathematical

    symbols - esp. number as the key archetypes for order and he

    understood this - not just in the conventional quantitative manner -

    but rather in a fundamental qualitative sense.

    As you know Jung was fascinated with those simple geometrical

    patterns i.e. mandalas which he saw as deeply symbolic of the

    psychological quest for integration. Now I think it is reasonable to

    assume therefore that he would have been extremely interested in the

    strange and beautiful graphics generated from within the Mandelbrot


    Clearly these would have represented a new type of mandala for Jung

    with deep archetypal significance. I think it would have focussed

    his mind on the nature of psychological recursive processes (and the

    conditions necessary for generating meaningful patterns of

    integration in experience).

    Another aspect suggested by such "mandalas" is the paradox of

    simplicity and complexity i.e. how the most complex organisation can

    be based on very simple initial principles. I have suggested that at

    its most fundamental the qualitative binary simple can

    satisfactorily explain all transformation processes in nature. I

    think Mandelbrot Sets would have set Jung thinking in like manner.

    Fractals help us to break through old patterns of thought and see

    reality in exciting new ways. Because they inherently involve

    paradox (and the corresponding need for intuitive understanding),

    they can be particularly helpful in breaking down the boundaries

    dividing the scientific and spiritual worldviews.




    Why Brilliance Sometimes Can be Futile

    Dear Peter,


    OK, I could follow about a third of what you said, not because it

    wasn't clear, you are clear, but you have too many referrents going

    on at the same time for a non-mathematician to hold onto.

    Your explanation about what seems to be merely rational

    relationships of measurement, and how the irrational is contained

    with, or exists simultaneaously, or something like that, was very,

    very interesting. But there is just too much there for me to handle.

    I can't hold on to that many concepts at the same time without being

    natural with all the terms, I can't include so much without more

    experience. The "choices" about what demensionality we are focussing

    on at a given moment was also very enlightening.

    Let me give an example, and see if this works for you, and others.

    There is a chord in music called the Neopolitan 6th, based on the

    flatted second tonal degree of the Pythagorean system of acoustical

    divisions, and the resultant tonal center in whatever key, that is

    what is considered fundatmental for that key, which manifests itself

    in the form of a major chord, outside, seemingly, of whatever key

    relationship has been established. It is the previously established

    tonal center which "defines" the function of the chord.

    This chord seems to sound irrational for the established key because

    only one of its members is found naturally in the key itself, that

    being the 4th degree of the scale, no matter what key center we have


    This chord is often found in it's first inversion, hence the term

    "6th" because of the formation of the chord with it's base note, not

    its root, in the bottom note of the chord. There is an interval of a

    6th between the bottom and top note of the major chord, having

    nothing to do with the 6th degree of the established tonality. This

    6th is actually the flatted supertonic of the established key, and

    can be considered, acoustically, to lead, eventually, to the

    establishment of an unrelated key center.

    The acoustic phenomenon generated by the chord is for the purpose of

    either cadence, where it is considered a color chord, and can be

    subsitiuted for the supertonic or the subdominant, or for the

    purposes of modulation to a different key, this being determined not

    only by acoustical laws which have been "set-up" by previous and

    following chords, but is also relevant to the particular era in

    which the chord is being used, ie., Baroque, Classical, Romantic,

    Impressionistic, or Modern. The "era" of the context of the chord is

    somewhat deterministic for it's perceived function. Jazz might be a

    totally different context for the chord, and it's function there

    would not pertain to the above discussion.

    Shall I go on? Maybe this is clear to you, I wouldn't doubt that,

    but we have psychologists and computer science, and art majors on

    the forum, all smart, but I wouldn't expect them to get very much

    out of the above,EVEN IF IT IS CLEAR AND LOGICAL! This is a

    fascinating chord to a musician. Whoever thought of this chord, and

    I guess it came from Italy somewhere because it is called

    "Neopolitan", was a great innovator in music, and opened up new

    understandings about the possibilities of harmonies.

    I wasn't trying to be obtuse, or sarcastic, or anything of the sort.

    I already respect your work very much.

    OK. You are tired of the "lecture" from Paul M. on the subject. You

    might even be mad about it, I'm not sure. And, in a way, I don't

    blame you. Getting good ideas across to people can be the most

    frustrating job in the book. Talk to Wilber or Beethoven, or

    Leonardo, or whomever about that one.

    GEEZ! Ok, this did start to sound a little sarcastic, but, like the

    Duck, I meant well.

    I consider myself to have slightly above average intelligence,

    otherwise I wouldn't be reading Wilber, or the people on the Forum.

    A brilliant intellect I'm not--just kind of good. Speak just a

    little more to the people who are "kind of good" in that department

    and you will get a lot more attention. (Ok, I'm telling you what to

    do, and I shouldn't, so let it go if it isn't useful.)

    Best regards,

    Paul M.


    A Fifth Of Beethoven



    I appreciate your honest reply.

    I take your point with your description of the Neapolitan 6th. In

    one way I found it very interesting. However as I have little formal

    knowledge of music, clearly my appreciation here - compared to your

    considerable knowledge and experience - would be quite limited.

    However Paul, there still remains an inescapable dilemma. My

    particular talent - if one can call it that - is the ability to

    structure reality in mathematical patterns (with a holistic


    This involves the combination of two aspects (which usually are seen

    in opposition to each other).

    On the one hand it requires a very rational way of looking at

    reality and familiarity with standard mathematical notions. However

    equally it requires a very intuitive approach and familiarity with

    the esoteric mystical worldview.

    Then when these two elements are appropriately combined we get

    Holistic Mathematics.

    From one perspective it fulfils the Pythagorean ideal of

    transformation of mathematics through contemplative insight.

    Equally, it provides a precise indirect (and reduced) rational means

    of structuring the various stages of spiritual experience.

    In some important respects Holistic Mathematics is the reflection of

    a particular personality type. I have stated before that it seems to

    marry the key characteristics of the 4 with the 5 (on the well known

    Enneagram scale). I would not be surprised therefore to find that it

    strikes a chord more easily with people on the Forum (who in some

    way share similar personality styles).

    We all have different talents. I know - like so many others - that I

    have good abilities in some fields, moderate ability in others and

    little or no ability in certain important areas.

    Believe me Paul, apart from this Forum I do not mention Holistic

    Mathematics to anyone else (as I would not consider it appropriate).

    So I spend the bulk of my time dealing with the mundane issues of

    life (for which I have no special competence).

    However this is the Ken Wilber Forum and I happen to believe that

    Holistic Mathematics is very relevant to the issues that Ken deals

    with (and sometimes does not deal with) in his writings. I would

    hope that as time goes by some would gradually get a sense of what I

    am on about.

    Basically I am trying to outline the methodology of what I would

    consider an integrated scientific approach. I consider Ken's

    approach - though brilliant in its own right - to be a

    multi-differentiated (rather than truly integrated).

    These are significant claims and I believe that I can support them.

    I appreciate however that many may experience difficulties with the

    initial posts. I therefore welcome follow-up discussion as a means

    of clarifying issues raised.