On first sight the Mandelbrot Set does not appear unduly impressive. It looks something like a lack beetle or alternatively of blob of ink splattered on paper.
However appearances can be deceptive. When one examines it more closely an intricate web of detail around its edges displays itself which becomes ever more wondrous with magnification.
Indeed there is an excellent demonstration of this to be found in the well known CD-Rom Encylopaedia "Encarta". As I imagine that many Forum participants have ready access to Encarta, I invite them now to find "fractals" and then click on "Exploring Fractals Interactivity". Then go to the "Magnify a Fractal" box and click on the white square as instructed. As the magnification increases one cannot but be impressed by the immense amount of detail contained within this fractal image (which is theoretically infinite). After about 7 Ievels of magnification one can see the image of the same "black beetle" once again reappearing. By level 9 it is clearly visible.
This demonstrates one of the remarkable features of fractal images i.e. self-similarity. Ultimately this feature challenges the very way we look at reality.
Conventional notions of reality are based on - whatever the level of investigation - starting with building blocks (i.e. the parts) from which we attempt to derive the whole picture. So if in the construction of the human body we start with atoms (as the basic parts) we can trace a progressive picture of increasing holistic organisation from atoms to molecules to cells to organs etc. Whereas at any level we would maintain that the "lower" parts are contained in the "higher" whole we would not however maintain that the "higher" whole is contained in the "lower" parts. Thus in conventional scientific terms molecules are necessarily made up of atoms; however atoms are not made up of molecules.
This is the holarchical view of development and may appear as common-sense. Once again according to this view the "lower" parts are contained in the "higher" whole. However the "higher" whole is not contained in the "lower" parts.
Fractal images clearly challenge this asymmetrical viewpoint. Clearly with a fractal image the "lower" parts are contained in the "higher" whole. Indeed through magnifying the whole image we can better access the detail of these parts. However remarkably the "higher" whole is equally contained in the "lower" parts. The Encarta exercise ably demonstrates this feature. So fractal images are very important not just for their strange and fascinating beauty but also philosophically as they challenge the very basis of conventional scientific notions.
To appreciate this it is necessary to know a little more about the nature of the Mandelbrot Set.
The Mandelbrot Set represents the special use of a recursive mathematical procedure.
Quite simply recursive procedures involve a continual well-defined feedback (or iteration) process. We start with an existing value and subject it to a mathematical transformation (by which it acquires a new value).
We then use this newly generated result as our starting value and continue the same procedure once more. Once again we generate a new result which we use as our starting value. We can of course carry on in this fashion ad infinitum.
For example let us take the simple equation y = 2x (with x = 1).
So y (from the equation) = 2.
Now this result of y becomes the new value of x (i.e. x = 2).
Now y (from the equation) = 4.
This result now becomes the new value of x (i.e. x = 4).
So we can carry on this recursive feedback procedure indefinitely (whereby we keep replacing the value of x with that of y).
Now though this is a simple example of a linear feedback mechanism (i.e. the value of the independent variable x, on which y depends is one-dimensional).
Though perfectly valid, in mathematical terms such linear feedback mechanisms are not very useful. They do not lead to interesting transformations.
This position is replicated in qualitative psychological terms.
Linear feedback mechanisms (i.e. where understanding is confined to the rational linear level do not lend themselves to interesting personality transformations).
Indeed both in qualitative as well as quantitative terms non-linear feed-back mechanisms have the capacity to generate more creative and dynamic transformations.
Now the simplest example of a (quantitative) non-linear transformation would be given by the equation y = x (raised to the power of 2). If the starting value for x = 2, then y = 4.
This then becomes the new starting value for x so y = 16.
Once again this becomes the new starting value for x and y = 256.
We can carry on indefinitely in this manner. If we plot this graph (with x as the horizontal and y the vertical axes respectively), then we will obtain a curved (non-linear) figure termed a parabola that is symmetrical about the x axis.
What is involved here is a continual horizontal to vertical transformation (in quantitative terms).
y is defined as one-dimensional (i.e. implicitly raised to the power of 1). All such real numbers (in the number system) literally lie on a horizontal line.
However x is defined in two-dimensional terms (i.e. is raised to the non-trivial power of 2). It therefore has a distinct vertical meaning (in terms of this non-unitary dimensional number).
However conventional mathematics necessarily reduces the vertical aspect (which is inherently qualitative) in horizontal (quantitative) terms. Thus 2 (raised to the power of 2) has a reduced quantitative interpretation as 4 (implicitly raised to the power of 1).
Thus in this non-linear feedback loop, we have a continual process by which a starting number that is horizontally defined in quantitative terms, is subjected to a (qualitative) vertical transformation which is then given a (reduced) quantitative transformation. However even here we can appreciate that underlying the quantitative feedback loop is a (hidden) qualitative dimension.
The Mandelbrot Set is based on a special set of numbers (i.e. complex numbers). This in fact involves a more sophisticated quantitative interpretation of both the horizontal and vertical aspects of number.
Here we draw a horizontal line (in a two-dimensional plane) which we call the real axis.
We then draw a vertical line which intersects the horizontal line at the point 0 which we call the imaginary axis. Thus any number in this plane (which is called the complex plane) can be defined by two numbers (co-ordinates) one which is real and the other imaginary. An imaginary number is a real (i.e. ordinary) number multiplied by the square root of -1 (written as i).
A complex number is thus one which has both a real part (relating to the position of a number in relation to the horizontal axis) and an imaginary part (relating to the position in relation to the vertical axis).
Though it is not properly recognised in conventional mathematics, imaginary numbers are really an indirect way of expressing the qualitative aspect of number (in quantitative terms). Though these indirect expressions are still treated as quantities they are now at least distinct from real (horizontal) quantities.
By using complex (rather than the real) numbers the Mandelbrot iteration procedure is able to generate, from a simple numerical starting point, highly complex patterns which are filled with an undeniable qualitative beauty.
Now the equation which generates the Mandelbrot Set is y = z (raised to the power of 2) + c where c is a complex number.
To make it simple we start with z = 0. So y = c (which has both real and imaginary number components). We then square this number and add it to c which gives us a new complex number.
We then starting with this new number we once again square it and add it to c. The resulting result again becomes the new start number. So we can continue on using this iteration procedure ad infinitum.
Not all starting values for c will generate the Mandelbrot Set.
Indeed values which define the Mandelbrot Set is very tightly constrained.
(These range of these values are as follows. The real part of c can vary from -2.25 to .75; the imaginary part can vary from -1.5 to 1.5).
So in each case the range of the variable is 3.
So every point in the complex plane lies either inside the Mandelbrot Set or outside. It all depends on the seed values for the real and imaginary parts of c.
Other starting values for c (which exemplify the more general Julia Sets) can also generate fascinating fractal patterns. Some revolve around a single point (i.e. have a single attractor). Others revolve around several points (have a number of point attractors). Yet others trail off to infinity (have an infinite attractor). These images usually degenerate very quickly with an uninteresting pattern. However if the starting values are close to the region of the Mandelbrot Set values, then fascinating dust like structures can emerge.
The Mandelbrot Set is the most complicated mathematical object known. However what is amazing is that it can be generated (in principle) very easily. It is based on a very simple equation (with the initial values precisely defined in complex terms). It is then subject to an iteration procedure where new values are constantly generated and fed back into the equation.
Of course before the development of high-powered computers it was not possible in practice to generate these fractal patterns. However that has quickly changed. (They are therefore very much a product of our computer age).
Before I move on to develop the qualitative significance of these developments I will once again list the main features of the Mandelbrot Set.
The task now is to move from the (conventional) quantitative to a holistic qualitative appreciation of the Mandelbrot Set.
1. As we see in the first point above, it is based on a simple feedback procedure. The task now is to apply this notion to psychological experience.
Psychological reality is indeed based on constant recursive procedures. One of the most important manifestations of this is the manner by which perceptions and concepts continually interact in experience.
Every phenomenon has both perceptual and conceptual aspects. Indeed this applies intimately to such mathematical entities as numbers. If I am to understand a particular number perception (e.g. 4), I must be able to see this number as relating to the general concept of number.
Thus though we may here directly focus on its particular (concrete) aspect, without implicit recognition of its general (formal) aspect, this number would have no meaning.
Likewise when we concentrate on the (general) formal notion of number as in algebra, it implies the background knowledge of specific (concrete) numbers. Without this related (concrete) particular aspect, (formal) general understanding has no meaning.
Thus parts are inextricably related to wholes and wholes to parts, in the very manner that experience of phenomena takes place.
At each level of the Spectrum (concrete) particular knowledge at first unfolds followed by (formal) general development.
However the most dynamic experience is in terms of the interaction of both aspects.
Vision-logic for example exemplifies this at the rational linear level (L0).
Here a dynamic iteration procedure continually takes place. We start with concrete phenomena. These are then transformed in conceptual terms leading to new starting phenomena in experience. And the process continues in this manner s leading to an ever-changing interpretation of reality.
So a psychological feedback mechanism is inherent in the very manner we experience phenomena.
2. Though phenomena are conventionally interpreted in (reduced) linear terms, strictly speaking they are non-linear.
Let us now examine closely the process by which perception turns to conception in experience.
Let us take a simple physical phenomenon (e.g. a box). Now the concrete recognition of this phenomenon takes place in quantitative terms. This represents the horizontal aspect of the phenomenon.
However the general appreciation of the concept "box" involves a qualitative transformation in understanding. Strictly speaking the concept has a holistic identity (as applying potentially to all boxes). Thus the concept of "box" represents the quality of "box" and does not have a direct quantitative meaning. However in conventional interpretation, this meaning is quickly reduced in quantitative terms. So we now understand the concept of "box" as applying to all (quantitative) boxes.
Thus concepts literally provide the dimensional (qualitative) aspect of experience. Relative to perceptions (which are horizontally defined in quantitative terms), concepts have a vertical (qualitative) interpretation.
Thus we now see that the psychological feedback process involves a continual movement as between (horizontal) perceptions and (vertical) concepts. The starting values are one-dimensional. In other words the perceptions are understood in linear rational terms. However the transformed values are two-dimensional. In other words the decisive qualitative shift from concrete (particular) to formal (holistic) understanding implicitly involves intuition (which combines polar opposites in experience and is two-dimensional).
Thus properly interpreted the understanding process is non-linear. However in conventional terms it is interpreted in (reduced) linear (i.e. one-dimensional) terms.
3. The Mandelbrot Set uses complex rather than real numbers.
In like manner - when appropriately interpreted - actual experience of reality takes place in a complex rather than real manner.
In scientific terms what is "real" is defined in solely quantitative terms. One of the great weaknesses therefore of conventional science is that it has no means (within its own paradigm) of translating the equally important qualitative aspect of objects. It must necessarily reduce the qualitative in "real" terms.
Now as we have seen the qualitative aspect relates directly to intuition (rather than reason).
Now this cannot be directly translated in rational terms. However it can be precisely translated in indirect rational manner.
As I have stated many times before intuition is generated through the negation (undoing) of rational phenomena. This intuition (formed in the spiritual unconscious) is two-dimensional (with positive and negative polarities). It can only be expressed in indirect conscious fashion through projection. Thus the two-dimensional experience (related to the negative direction of phenomena) is expressed in one-dimensional conscious terms.
Now this is an exact complement of the mathematical notion of an imaginary number.
Thus the qualitative aspect of phenomena has an appropriate translation as "imaginary" in holistic mathematical terms. Here symbols do not represent quantities (in "real" terms). Rather they represent qualities (i.e. as holistic archetypes) which are "imaginary".
Thus properly understood all perceptions and concepts have both quantitative "real" and qualitative "imaginary" interpretations. The "real" aspect of a perception is its (particular) quantitative interpretation. Its "imaginary" aspect is its (holistic) qualitative aspect (as related to its corresponding concept).
The "real" aspect of a concept is its actual quantitaive interpretation (as related to concrete quantities). Its "imaginary" aspect is its potential relationship to all perceptions.
Thus "real" and "imaginary" in dynamic terms are strictly relative terms.
Thus we now see the psychological iteration process involving both perceptions and concepts should be translated in "complex" terms (with "real" quantitative and "imaginary" qualitative aspects).
When this qualitative "imaginary" aspect is properly recognised extremely rich scientific patterns can be generated (where appreciation involves the interaction of both quantitative and qualitative aspects).
4. Again this rich experience of reality is based on a very simple procedure. (Quantitative) parts are transformed into (qualitative) wholes and then reduced once again as (quantitative) parts.
In experience these terms are relative. We can equally start with (qualitative) parts that are transformed into (quantitative) wholes. This is simply to state that both wholes and parts at all levels of experience can be given both a quantitative and qualitative interpretation. In holistic mathematical terms they have both "real" and "imaginary" interpretations.
This experience is infinitely rich in detail. This simply entails that the very interaction of qualitative and quantitative aspects is itself the means though which spiritual awareness is developed.
The great potential richness of science is largely lost through adopting a solely "real" interpretation of phenomena.
5. Fractal images as stated lead to (qualitative) artistic interpretation. Likewise scientific experience that is based on "complex" experience leads to artistic interpretation.. In other words from this perspective the rich tapestry of scientific phenomena continually serves as a source of great wonder and joy. The cognitive aspect becomes progressively intertwined with affective and ultimately both with spiritual understanding.
6. Fractal images exhibit self-similarity. This is perhaps the most characteristic feature of fractals.
Properly interpreted all phenomenal experience exhibits self-similarity. However this is not at all apparent from the conventional perspective.
Self-similarity involves two interrelated features. The (lower) parts are included in the (higher) whole. Equally however the (higher) parts are included in the "lower" whole.
This needs careful explanation.
The scientific "real" interpretation simply reduces the qualitative aspect of phenomena in quantitative terms. "Lower" parts are then included in the "higher" (quantitative) whole. Thus a slice of a cake (part) is included in the cake (whole).
However the cake (whole) is clearly not included in the slice (part) in this quantitative definition.
Thus the conventional (quantitative) interpretation is one-directional and asymmetrical. A true interpretation requires giving equal attention to both its quantitative and qualitative aspects. In holistic mathematical terms - as we have seen - this means treating it in complex terms (with both "real" and "imaginary" aspects).
So if we start with a part that is quantitative, the transformation to a whole state takes place in qualitative terms. In this sense the quantitative part is transcended in a new qualitative whole. In holistic mathematical terms the "real" part is transcended (and included) in an "imaginary" whole.
However a reverse process must now take place whereby a transformation to a part state takes place in quantitative terms. In this sense the qualitative whole is made immanent in a new quantitative part. In holistic mathematical terms the "imaginary" whole is now made immanent (and included) in a new "real" part.
Let me illustrate with respect to a number, again if I start with a specific number, say 4, this is initially understood in quantitative terms. The transformation to a whole state takes place when one recognises that "4" possesses the general property of "number"(which holistically applies to all numbers). This whole state - strictly speaking - is qualitative (rather than quantitative). So in holistic mathematical terms, the "real" part "4" is transcended and thereby included in an "imaginary" whole (i.e. the holistic property of number).
A reverse number process also takes place involving a transformation to a part state (in quantitative terms). This happens when one recognises that the general property of number is related to each particular number. This holistic property of number is thereby made immanent in the number "4". In holistic mathematical terms, the "imaginary" whole
(i.e. the qualitative property of number) is made immanent and thereby included in a "real" part (i.e. the specific number "4).
This bi-directional relationship as between whole and part (and part and whole) can only be maintained by properly distinguishing quantitative and qualitative aspects of experience. This requires translating phenomena in "complex" rather than "real" terms.
Any "real" translation of phenomena inevitably reduces the qualitative to the quantitative.
Thus the "real" parts are transcended and included in a new whole. However this whole is interpreted in reduced "real" terms. So we are left with the common sense view that the "real" parts are included in the "real" whole.
As the opposite perspective that the "real" whole is included in the "real" parts would then be paradoxical this is immediately discarded in the conventional paradigm.
This mistranslation of the true relationship between whole and parts (and parts and whole), I would consider the biggest problem in conventional science. Putting it more formally it represents gross mistranslation of complementary opposites (in vertical terms).
Even Ken Wilber consistently mistranslates this relationship. For Ken development is synonymous with transcendence (i.e. the holarchical view of evolution where "lower" level parts are transcended and included in "higher" level wholes. However development is equally synonymous with immanence (i.e. the partarchical view where "higher" level wholes are made immanent and included in "lower" level parts). However this complementary view of development is largely missing from Ken's writings.
The message is clear. Just as in conventional terms mathematicians have realised that the comprehensive number system is complex (with real and imaginary members), likewise in holistic mathematics the comprehensive number system is also "complex".
This enables proper translation of the interrelationship of quantitative and qualitative (parts and wholes).
Thus the very manner in which parts and wholes interact at all levels of reality is complex (in a holistic mathematical sense). Likewise the complementary manner in which perceptions and concepts interact in experience is complex.
Thus reality is literally incredibly complex. However the paradox is that the dynamic recognition of this complexity leads directly to simplicity (in the continual fusion of these vertical opposites). Complexity is simply a polarised phenomenal expression of simplicity. Thus a truly simple approach to reality (i.e. purely spiritual) is capable of generating and sustaining an incredible amount of complexity.
7. The starting values for the Mandelbrot Set are tightly constrained.
This equally applies to experience. As so often stated the spiritual path is extremely narrow and requires a high level of vigilance and great sensitivity to any form of imbalance.
The Mandelbrot Set has a single attractor. Likewise the authentic spiritual life has a single attractor (i.e. the pure point or focus of concentrated awareness).
Some fractals outside the Mandelbrot Set have several attractors. Fractal patterns keep oscillating between these points.
Again it is very similar with spiritual development. So often followers suffer from divided attention and are never willing to give singular commitment. They literally have several competing attractors.
Finally we have the extreme cases of an infinite attractor. This would signify a life that has lost all sense of balance and is heading headlong in a destructive direction.
8. The generation of fractal images depends on the power of modern computers.
Ultimately this involves the encoding of information in binary terms. As I have stated there is a qualitative version of the binary system (based on an alternative logical system) which is the root means of encoding all transformation processes.
The application of the qualitative binary system to transformation processes could in itself serve as a important means of achieving this transformation.
Thus all the important features of the Mandelbrot Set have psychological qualitative equivalents.
However there are important differences.
Mathematical feedback mechanisms are deterministic. Thus if two people start with the same initial values for the Mandelbrot Set, they will generate the same fractal image.
However psychological feedback mechanisms are more open. Thus continual correction can take place literally modifying one's initial starting values. In some cases considerable correction can take place (as with dramatic conversion experiences).
Though the mathematical procedure for generating fractals is non-linear (in quantitative terms), it is based on a paradigm which is in fact linear. Thus its approach is linear in qualitative terms.
The holistic mathematical procedure however is non-linear (in qualitative terms) and better suited for assessing the philosophical implications of fractals.
Finally it is important to point out that the fractal patterns in nature are very different from mathematical fractals and only exhibit self-similarity to a limited degree. Mountains, clouds, leaves etc. do indeed exhibit fractal features. Thus at a lower level of scaling the structure of a leaf may indeed closely resemble its larger scale counterpart. However degree of scaling possible is not infinitely divisible. Indeed in reacting to traditional geometrical notions, I feel that Mandelbrot has gone overboard on the fractal features of nature. Indeed much of nature is only fractal to a very limited extent. What is really needed therefore is a paradigm that combines traditional linear with new non-linear notions.
It is very similar with psychological experience. The fractal paradigm - qualitatively interpreted - can explain the non-linear aspects. Ultimately this self-similar viewpoint reaches its zenith with pure spiritual contemplation (which is both transcendent and immanent simultaneously). All the parts are understood as being in the whole; likewise the entire whole is understood as being in each of the parts. This is possible because the true basis of both is spiritual. However linear and non-linear understanding have an equally valid role in experience. Both viewpoints ultimately must be integrated.