Greening-ESPG and PG

Up to now, order has been considered as arising, basically, through a sequence of successions. This is indeed a very com-mon form of order and perhaps the one that is most familiar. In this chapter, however, another kind of order, called the generative order, is introduced. This order is primarily concerned not with the outward side of development, and evolution in a sequence of successions, but with a deeper and more inward order out of which the manifest form of things can emerge creatively. Indeed this order is fundamentally relevant both in nature and in con-sciousness. In the following chapters its relevance to society will also be discussed.

The generative order will be explored with the help of a number of examples drawn from mathematics, physics, and the fields of art and literature. This will lead, in turn, to the implicate order, which is a particular kind of generative order that has been most fully worked out in physics. However, the implicate order will be found to have a broader significance, not only in physics

but also in biology, consciousness, and the overall order of society and each human being.

Although specific proposals for how the generative order may be used will be discussed, it is not the main purpose, in intro-ducing this new notion of order, simply to pursue its application in detail. Rather, it is to use these ideas in order to go more deeply into the meaning of creativity. In succeeding chapters, these notions will provide a base from which to move yet further in the general direction of creativity.

FRACTAL ORDER

In the previous chapter, order, as discussed in terms of similar differences and different similarities, was considered largely as a means of understanding curves, structures, and processes that are already present in nature or in the mind. However, it is equally possible to use such a notion of order, based on similar-ities and differences, to generate shapes, figures, forms, and pro-cesses. For example, starting from a single segment it is possible to generate a line by means of a process of repetition, in which each element is similar (equal to) the next. A polygon can be produced through a similarity of angle and length. In a related fashion all second-degree curves can be generated from an initial difference which is repeated in a way that is similar to itself.

Higher-degree curves require the repetition of more differences, but they can all be constructed in the same fashion.

This idea could be pursued in ever greater refinement. How-ever, for the purposes of this section, a more developed form of order will be used: the mathematical theory of fractals, which was recently invented by B. B. Mandelbrot,¹ which is closely related to the theory of chaos, as discussed in the previous chap-ter. Fractals involve an order of similar differences which include changes of scale as well as other possible changes. A simple example is to start with a base figure, the triangle:

and then consider a generator, which is really a small triangle that can be applied to each side of the basic figure.

In this way a six-pointed star is produced:

In the following step, the generator is reduced in scale and applied again to each line segment, giving rise to the figure:

and then to

Clearly this process can continue indefinitely and results in a figure with extremely interesting properties. The reader may turn to Mandelbrot’s book for details but, for the moment, accept that the circumference of this figure has grown to

be infinite and has no slope.² These are particularly curious properties to have been generated in such a straightforward fashion.

By choosing different base figures and generators, but each time applying the generator on a smaller and smaller scale, Mandelbrot is able to produce a great variety of shapes and figures that have very interesting mathematical properties. Some of these have the appearance of islands, mountains, clouds, dust, trees, river deltas, and the noise generated in an elec-tronic circuit. All are filled with infinitesimal detail and are evocative of the types of complexity found in natural forms. In addition, they reflect the way in which the details of a form appear to be similar over a wide range of scales of size: Often when we “zoom in” on some object in nature it continues to exhibit similarities of form at greater and greater magnifica-tion. Other fractals show ever new detail at smaller and smaller scales.

Mandelbrot points out that the geometry of fractals lies much closer to the forms of nature than do the circles, triangles, and rectangles of Greek geometry. It could be said that traditional geometry, out of which much of mathematics and the tools of physics have evolved, is, in fact, a highly artificial way of describ-ing the world. Somethdescrib-ing closer to the fractal order, on the other hand, should be an appropriate starting point for discussing

nature in a more general way, and for providing better formal descriptions of the processes of physics and biology.

The complex figure generated from the triangle is a little like a very irregular island which, of course, possesses a coast line that is ultimately infinite in length, when analyzed on an indefinitely fine scale. Other fractals begin as simple lines which expand in highly subtle ways until they appear to cover the entire page. An interesting question is therefore generated by these fractal figures: What is their dimension? Are they lines, of one dimension, or planes, of two dimensions? The answer is that a fractal is of fractional dimension, lying somewhere between a line and a plane. (Other fractals may have a dimension that lies between that of a point [zero] and a line [one].) Indeed Mandelbrot argues that the fractal dimension of an object is a significant characteristic and, for example, a river delta or a country’s coastline can be characterized by its particular fractal dimension.

But how can a geometrical figure, drawn on a piece of paper, have a fractional dimension? Consider a plane, this page for example. If a dot A is made on this plane, then any neighboring point B, C, D, or E, no matter where it is printed on the page, will also be in the plane.

This is not, however, true of a simple line XY. Although a point A, for example, is on the line, and the neighboring points B and C are on the line, it is always possible to find neighboring points D, E, and F that are not on this line. Hence one property of a line, which has one dimension, is that points in its immediate neighborhood can be found that do not lie on it.

Now consider a fractal line with its unlimited complexity. As the fractal generator is successively applied, more and more points that previously lay outside this line will be included.

Clearly, in some sense, it has more than one dimension. In the limit, in which the fractal line fills the plane so that no point remains in the plane that does not also lie on the fractal line, it will have become two-dimensional. So, in general, the dimen-sions of a fractal line lie somewhere between one and two.³ And in three dimensions, general fractals can be constructed whose fractional dimension lies between zero and three.

While the fractal figures illustrated so far appear quite complex, they could hardly be called disordered, for they are composed of a quite simple order involving a single similar difference that is repeated at constantly decreasing scale. Moreover, figures of even greater complexity can be created using more than one generator and applying the alternative generators according to some fixed rule. One such rule of application, selected by Mandelbrot, is to use random numbers generated in a computer. In this way, through the introduction of random successive differences, he is able to generate the curves for Brownian motion as well as totally irregular coastlines.

It should be possible to generalize Mandelbrot’s ideas still further by introducing additional categories of differences other

than simple scaling, for example, differences in direction, shape, and so on, to arrive at yet more subtle fractal figures. Indeed, the principles involved in producing fractals may be much closer to those employed by nature than those associated with the figures and structures of traditional geometry. However, since so much attention has been given in the past to sequential order, it may be some time before a large number of concrete applications of Mandelbrot’s ideas are discovered. Rather, the overall notion of generative order should be regarded as a very fruitful area for investigation, which may reflect not only on science but on many aspects of life.

GENERATIVE ORDER

Mandelbrot’s fractals are only one example of a generative order (in the fractal case, a generation which proceeds by repeated applications of a similar shape but on a decreasing scale). Many other generative orders could be constructed in mathematics.

However, the whole idea of generative order is not restricted simply to mathematics but is of potential relevance to all areas of experience.

Generative order can, for example, be seen in the work of a painter. Indeed, in a certain restricted sense the generation of form using Mandelbrot’s fractals can be compared with the vari-ous stages of painting. At least until this century an artist did not generally begin to work with detail but, in the case of a portrait for example, attempted to capture the overall form and gesture of the sitter with an initial sketch on the canvas. Such a painter may have even employed the trick of squinting at the sitter in order to cut down detail and emphasize tone and shadow. Gradually this initial sketch was built up and made more detailed, solidity being indicated by modeling, as the first layer of paint was added. As the painting progressed, detail was created in a progressive way, each time by building on the whole. Just as the complex forms of

nature appear to be generated through successive additions of smaller and smaller detail, so at one level, a painting could be thought of as growing in a similar fashion.

But of course the generative order of a work of art is far more complex than the preceding description might suggest. For many orders of growth are involved which, in a great painting, are united within a single more comprehensive generative order.

The painter may begin with a general idea, a feeling that con-tains, in a tacit or enfolded way, the whole essence of the final work. The next stage may be to observe the general scene and make sketches that rely upon the sense of visual perception. But in addition to the outward perception, there is also an inner perception in operation which is inseparable from the painter’s whole life, training, knowledge, and response to the history of painting. The outward and inward perceptions are, in turn, inseparable from an emotional and intellectual relationship to the theme and even to its literary and social values. Yet this vision is by no means rigid and fixed, for as the painter begins to work on the canvas, a new interaction takes place. He or she is con-stantly faced with both physical limitations and new potentials, in the very muscular activity of painting and in fresh perceptions of the growing painting beneath the brush.

In all this activity, what is crucial is that in some sense the artist is always working from the generative source of the idea and allowing the work to unfold into ever more definite forms.

In this regard his or her thought is similar to that which is proper to science. It proceeds from an origin in free play which then unfolds into ever more crystallized forms. In science as in art it is necessary that what is done with more definite forms should continue at each stage to be open to the kind of free play that is essential to creativity. This holds even if, as with certain artists, such as Matisse, the ultimate form may be a simplification and generalization of what the artist started with, rather than an articulation of greater detail. Matisse’s initial creative perception

was the constant guide to his activity. This can be seen in the large number of sketches and studies that he made for each of hisfinal paintings, prints, and drawings. His generative idea was clearly the motivation for a subtle and meaningful simplification of lines and forms.⁴

While the essence of the generative order of a painting ultim-ately escapes definition, it is clear that this order is very different from that of a machine, in which the whole is built out of the parts (i.e., in which the whole emerges through accumulation of detail). By contrast, one of the most important activities during the creation of a work of art is its unfolding, within a particular medium from the original perception. Something similar can be seen in music. Each composition is played in sequential, tem-poral order, yet its generation can never take place completely within such a sequential way. For that matter the unfolding of the meaning of the music in the mind of a perceptive listener is never totally sequential. This is especially clear in the work of Mozart, who is said to have seen a whole composition in a flash and then to have unfolded it by playing it or rapidly writing it down. Beethoven, by contrast, does not appear to have conceived his works directly as a whole in precisely this fashion, for his notebooks contain themes and sketches worked over long periods of time. Nevertheless, the basic activity in Beethoven’s creative work is clearly still a constant unfoldment from a gen-eral notion of order.

Bach, for his part, appears to have comprehended fairly dir-ectly and as a whole the potential contained within a theme a few bars long, as the following story, told by his son Wilhelm Friedermann Bach, indicates:

After he had gone on for some time, he asked the King to give him a subject for a Fugue, in order to execute it immedi-ately without any preparation. The King admired the learned manner in which his subject was thus executed extempore;

and, probably to see how far such art could be carried, expressed a wish to hear a Fugue in six Obligato Parts. But as it is not every subject that is fit for such full harmony, Bach chose one himself, and immediately executed it to the astonishment of all present in the same magnificent and learned manner as he had done that of the King.⁵

But on returning to Leipzig, Bach was to accept the King’s challenge and compose a six-part fugue, nine canons, and a trio sonata on the Royal Theme which he submitted, along with his original fugue, as a Musical Offering. Clearly, in some implicit way the potential of Bach’s magnificent composition was per-ceived by him as enfolded within the King’s theme.

There is evidence that in speech the whole meaning is simi-larly generated quite quickly, along with the language needed to express it, which comes out as a sequence of words. What is said at any given moment, for example, has never been said in exactly the same way before. In this sense the generative order of language is creative and bears a relationship to artistic and musical creation.

A major feature of a generative order is that through it a pro-cess of creation may begin from some broad encompassing over-all perception. There is a clue from our language, for the word generate has the same root as general and genus. This supports the earlier claim that, in the arts, creative generation is basically from some general perception, which is then unfolded into particular forms. These may move toward greater and greater detail or, as is the case with Matisse, toward an expression of the general.

FOURIER ANALYSIS

In moving between two extremes, such as art and mathematics, the aim has been to suggest the universal and pervasive character of generative order. For the moment, however, the mathematical

side will be stressed, by considering Fourier analysis. For by means of Fourier analysis, a particular arbitrary form can be built out of sets of periodic waves, each of which is of a global order.

Consider such a single wave:

This wave is defined by an order which is similar to itself from period to period. It represents, for example, a wave on a string stretched out in space, or a wave evolving in time. Clearly its order is global in that it repeats itself in a similar way indefinitely.

Now add to the first wave a second of double the frequency:

Adding the two together produces:

The diagrams show how more and more waves can be added together to create shapes of any form whatsoever. While each simple wave represents a global order, when they are put together they add up to produce a complex local order as well.

It is possible to create a well-defined figure in this way based on a generative order which relates the waves of successive fre-quencies together. This indeed is just how a Fourier series is constructed, for any complex figure can be generated, given a series of coefficients which determine the ways in which the global waves are to be related together. As an example of a Fourier summation, consider a music synthesizer in which a series of oscillators each produce a wave of given frequency, a pure tone. The characteristic sound of any instrument, with all its complex local order, can then be generated by turning appropriate dials on the machine and thereby adding different pure tones together. (In fact a synthesizer also adds character-istics for the attack and decay of each note.)

GOETHE’S URPFLANZE

The simple example of a Fourier series demonstrates how a local order may basically follow from a global order, a reversal of the normal point of view in which global order is regarded as the outcome of local order. But as pointed out earlier, genera-tive orders, especially of a global nature, have so far not been used extensively in science. It is interesting to note, however, that Wolfgang Goethe seriously investigated such a notion two centuries ago. In considering the relationships between the many varieties of plants there are in the world, and the variations that exist within a particular family and genus, Goethe was led to the notion of the Urpflanze. Literally this means an original

The simple example of a Fourier series demonstrates how a local order may basically follow from a global order, a reversal of the normal point of view in which global order is regarded as the outcome of local order. But as pointed out earlier, genera-tive orders, especially of a global nature, have so far not been used extensively in science. It is interesting to note, however, that Wolfgang Goethe seriously investigated such a notion two centuries ago. In considering the relationships between the many varieties of plants there are in the world, and the variations that exist within a particular family and genus, Goethe was led to the notion of the Urpflanze. Literally this means an original