The Pennsylvania State University The Graduate School Department of Philosophy

PICTURES OF THOUGHT: THE REPRESENTATIONAL FUNCTION OF VISUAL MODELS

A Thesis in Philosophy by Karim Joost Benammar

Copyright 1993 Karim Joost Benammar

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

December 1993 Table of Contents

Chapter 4. Fractal images as visual models ...... 83

4.1. Self-similarity and the projection of mathematical monsters ...... 85 4.2. Fractal geometry and fractal dimension . . . . 92 4.3. Fractal images as models ...... 99 4.4. The limits of modelling ...... 105

Chapter 5. Strange attractors ...... 110

5.1. Strange attractors in the study of dynamical systems ...... 110 5.2. Modelling and approximation ...... 115 5.3. Mathematics and the world ...... 120 5.4. Generating interpretations ...... 122

Chapter 6. Generative models and representation . . . . 125

6.1. Necessity and analogy ...... 126 6.2. Projection and representation ...... 133 6.3. Generative models ...... 137

Chapter 7. Visual models and knowledge of the world . . 144

7.1. Visual models and language ...... 145 7.2. Visualization and the visible ...... 151 7.3. The construction of reality ...... 154

Bibliography ...... 158 List of figures

4.1. The Koch curve ...... 87 4.2. The Koch Snowflake ...... 87 4.3. Construction of the Cantor set ...... 87 4.4. The Sierpinski Gasket (1) ...... 89 4.5. The Sierpinski Gasket (2) ...... 89 4.6. The Sierpinski pyramid ...... 89 4.7. The fractal dimension of the coast of Norway . . . . 94 4.8. The Mandelbrot set and corresponding Julia sets . . 96 4.9. The Mandelbrot set ...... 97 4.10. Fractal planetrise ...... 102

5.1. A Poincaré section ...... 113 5.2. A strange attractor ...... 113 5.3. The Lorenz butterfly with equations ...... 116 5.4. A more detailed Lorenz butterfly ...... 117

6.1. the necessary function of visual models . . . . . 132 6.2. Analogies between relations ...... 139 6.3. Generative representation between theory and empirical system ...... 140 6.4. Properties of theory and analogical model . . . . 140 6.5. Properties of the relation between generative model and theory ...... 141 6.6. Properties of the relation between theory and model ...... 141 6.7. Characteristics of generative models ...... 142

7.1. Correspondences between theory and model . . . . . 150 7.2. Projection and representation ...... 150 83

4. Fractal images as visual models

Fractals are self-similar images; parts of the image are exactly the same as the whole. They are produced by a variety of different mathematical techniques. Self-similar transformations of images, geometrical projection of numerical sets, or plotting the results of iterative equations on a complex plane are three methods which produce fractal images. Fractals form a new, "fractal" geometry, since the dimension of a fractal image need not be an integer, but can instead be a fraction. This dimension has proved to be the most important mathematical property of fractals thus far. Benoit Mandelbrot is widely credited for his dogged perseverance in trying to interest the scientific community in fractals. He coined the word "fractal," provided a definition of fractal dimension, and studied the impact of fractals as models for a wide variety of natural and artificial phenomena (Mandelbrot 1982). He resurrected the obscure equations and images that had already been studied at the turn of the century by mathematicians who lacked the technology to further visualize their creations. Fractal images are unusual and perplexing; infinitely complicated images spring from simple equations, 84 mathematical constructs turn out to model empirical systems, and complex phenomena can be visualized, in a restricted sense of the term. What kind of visual model is a fractal? What empirical system does it represent? What is the representational relation between the theory of fractal images and images and objects in the world? Fractals images are strangely beautiful images that are produced by mathematical equations or from measurements of empirical processes; this makes them visual models par excellence. Fractals visually model a wide variety of physical and non-physical phenomena, ranging from clouds and lightning strikes to population dynamics. Sometimes, this modelling is connected to an image of an empirical phenomenon (the way a thing "looks"), while at other times it is a visualization of abstract variables, comparable to the graphical imaging of turbulence. These two senses of visual modelling will allow us to sharpen the distinction between projection and resemblance which was raised in the second chapter. In the last chapter we discussed the increased complexity in projection methods and graphical manipulation. The increase in complexity of projection method used for images will come from the radical concept of fractal dimension, and from novel projection methods based on plotting iterated results. These projections are not simply topological transformations. The use of fractal dimension 85 for measuring a wide variety of graphical images modelling phenomena has been a tremendous, though controversial, boost in the effort to link fundamental properties of systems to the topological properties of the visual models that represent them. The furor over fractals has now abated somewhat, and the new perception is that they have been overrated as a revolutionary scientific tool. They are not a panacea for all visualization quandaries, and cannot yet be fruitfully applied in all fields. But this only seems a predictable backlash after their meteoric rise in the scientific and lay consciousness, which has not always led to revolutionary results.

4.1. Self-similarity and the projection of mathematical monsters

Although the widespread fascination with fractal images and their applications is barely more than a decade old, visionary mathematicians at the turn of the century thought about endlessly self-similar geometrical objects to illustrate paradoxical conclusions about infinity and dimension. These "mathematical monsters" (Mandelbrot 1982, 35-6; Voss 1985, 25), provide a good introduction to fractals produced by geometric projection methods and to the 86

Hausdorff dimension DH, the most common and straightforward fractal dimension. These fractal images are constructed by starting with a geometrical shape, the initiator, and endlessly repeating the same geometrical transformation, called the generator. After several generations, the image has become so detailed that it can no longer be reproduced exactly on paper. However, since the image is exactly self-similar, zooming in on any part of the image gives us another representation of the whole: exactly self-similar fractal images stretch on into infinity. Infinite fractals are virtual images; they can never be completely represented. Figure 4.1 details the construction of the Koch curve, named after the Swedish mathematician Helge von Koch who proposed it in 1904 as an example of a continuous curve that is nowhere differentiable (Schroeder 1990, 7). Figure 4.2 shows the Koch "snowflake," another geometrical construction of the same curve. Figure 4.3 explains the construction of the Cantor set obtained by repeatedly erasing the middle third of a line segment. After infinitely many repetitions of this operation, we are left with an infinite number of points which have no total length; this has been aptly named "Cantor dust." Georg Cantor proposed this set to demonstrate the possibility of sets with uncountably many numbers and measure (length) zero (Schroeder 1990, 16). 87

Figure 4.1. The Koch curve.

Figure 4.2. The Koch Snowflake.

Figure 4.3. Construction of the Cantor set. 88

Figures 4.4 and 4.5 show the construction of a two- dimensional Cantor set called the Sierpinski gasket. Figure 4.6 shows a three-dimensional Sierpinski "pyramid". The fractal (Hausdorff) dimension can be calculated by a simple formula1; it is a way of measuring the space- filling capacity of our fractal images. This is an intuitive concept: a very squiggly line will fill more space than a straight line. We can distinguish three different dimensions here: the topological dimension of a line, which is always equal to 1; its Euclidean dimension, which is 1 for a straight line and 2 for a curved line (since it is represented in a plane); and its fractal dimension, which depends on the squiggliness and complexity of the line, and is superior to 1 in all cases other than the straight line (Fan, Neogi, and Yashima 1991, 15). The Koch curve and snowflake in figure 4.1 and 4.2 have

a Hausdorff dimension DH = 1.26..., indicating their moderate space-filling capacity. The Cantor set has fractal

dimension DH = 0.63..., since it is less than a line, but this dimension is much more than 0, which could be suggested

by infinite dust. The Sierpinski gasket has dimension DH =

1.58..., and the pyramid has dimension DH = 2, although it is embedded in three dimensions. The computation of the

1 The Hausdorff dimension DH is given by the formula: DH = lim log N , where N is the number of straight-length r- >0 log (1/r) segments of length r needed to step along the curve from one end to the other (Schroeder 1990, 9-10). 89

Figure 4.4. The Sierpinski Gasket (1). From Schroeder 1990, 17.

Figure 4.5. The Sierpinski Gasket (2). From Schroeder 1990, 29.

Figure 4.6. The Sierpinski pyramid. From Mandelbrot 1982, 143. 90 fractal dimension is a powerful mathematical tool to typify fractal images, and so to differentiate between the space- occupying capacity of different dusts, curves, and solids. These early mathematical monsters function as visual representations of mathematical properties, demonstrating the existence of peculiar sets or curves. But they need not be visual images, and can be conceived as sets of numbers; there are arithmetic transformations which correspond to the geometric operations to generate self-similar images. The Cantor set can be represented by a series of numbers, and numerical transformations correspond to the geometrical operation of cutting out the middle third (Schroeder 1990, 162-64). These mathematical monsters are fractal images produced by geometrical transformations, and they illustrate theories about numbers. They are visual models of properties of a theory; but they are not maps, because they are not projections of a territory or empirical system.

We can focus on the extension of similarity to self- similarity to explain one of the increases in complexity of the representational function. The importance of the mathematical notion of similarity for projection is demonstrated by the mapping algorithms used in turbulence data projections (3.3). Earlier, I argued that this mathematical, formal concept of similarity could not be expanded into a philosophical notion of similarity which 91 would explain resemblance; the looser notion of similarity used in everyday language is based on an intuitive and vague comparative value-judgement. But it is possible to expand the mathematical concept of similarity into the more complex recursive concept of "self-similarity." Self-similarity is an extension both of the recursive nature of iterative equations, and the visual sense of an image within an image. The notion of self-similarity can also be defined as symmetry across scales: fractals exhibit symmetry not just in the usual two- or three-dimensional space (although these early mathematical monsters tend to exhibit these symmetries as well), but symmetry across scale. Self-similarity or symmetry across scales is a good definition of a fractal image: although Mandelbrot, the modern "father" of fractal images, defines any geometrical object with a fractal dimension as a fractal, Lauwerier points out that this definition is restrictive, because some self-similar images (for example the Levy curve, or the Sierpinski pyramid above) have natural dimensions (Lauwerier 1991, 83). The notion of infinite self-similarity producing endlessly self-similar images within images, and the idea that scale is a negligible characteristic of a shape if the image is symmetrical across scale are the legacy of the early mathematical monsters. This legacy provides the basis not only for the production of fractal images through iterative equations, but also establishes self-similarity as 92 a critical feature of many physical phenomena, such as coastlines or clouds. Within a rather large range, it is not possible to tell from a photograph of a coastline or a cloud what the scale is. There is, however, a fundamental difference between images of empirical phenomena and constructed, self-similar fractal images; fractal images are infinitely self-similar, whereas images of phenomena in the world, such as biological systems, coastlines, and clouds, are limited by their inner composition and limits of physical reality such as the size of molecules. Biological systems are typically only self- similar across two orders of magnitude, physical systems across four (Peitgen and Richter 1986, 5), and clouds across a remarkable eight (Burrough 1985, 156-7). Because self- similarity and symmetry across scale are typical features of fractal images but only of images of empirical systems within a specific range, the modelling relation of virtual fractal images is restricted to that range only. The structure and shape of reality, unlike that of the creations of mathematicians' imagination, has quantifiable limits.

4.2. Fractal geometry and fractal dimension

The fractal Hausdorff dimension DH can be calculated for any curve, whether it is self-similar or not. Another 93 way of conceptualizing this dimension is by asking, with Mandelbrot: how long is the coast of England? A little reflection will show that this depends on the length of the measuring rod used: using rough approximation on a map will yield a much smaller overall length than if we actually walked around every cove or cliff with a yardstick (Mandelbrot 1982, 25-29). Figure 4.7 shows Jens Feder's measurement of the coast of Norway, which, as one could expect from the many fjords and inlets, has a very high fractal Hausdorff dimension

DH = 1.52. The insert illustrates the box-counting algorithm, which is a very useful way of measuring the Hausdorff dimension of "practical" fractals which are not constructed geometrical entities. Self-similarity, and fractals in particular, can also be achieved by other means. Manfred Schroeder explains: Iteration ... is one of the richest sources of self- similarity. Given the proper jump start, the repeated application of some self-same operation, be it geometric, arithmetic, or simply symbolic, leads almost invariably to self-similarity. (Schroeder 1990, 49)

And indeed, the most stunning fractal images can be constructed on a computer screen by plotting the results of a simple iterative equation for every point. Simple iterative equations produce the endlessly rich complexity of the Julia sets in the complex plane, named in honor of the French mathematician Gaston Julia, who conceived of them in 94

Figure 4.7. The fractal dimension of the coast of Norway. From Schroeder 1990, 212 and 214. 95

1919 and discovered many of their mathematical properties but could not represent them because of technological limitations. The equally famous Mandelbrot set consists of the points in the complex plane where the Julia sets are connected (Lauwerier 1991, 149). The Mandelbrot set is thus a very complex boundary between connected and disconnected Julia sets. The Mandelbrot set consists of the points in a complex plane for which the solutions to the iterative equation "square, then add the original complex number; repeat ad infinitum" are finite. The level of detail in this picture is achieved by the number of iterations made before deciding about the finiteness of the result (whether the corresponding Julia set is connected). Figure 4.8 shows the relation between the Mandelbrot set in the center and some of the possible Julia sets found at various points along it. Figure 4.9 shows a color rendition of the beautiful and infinite complexity of the Mandelbrot set itself. The many beautiful colored renderings of the Mandelbrot set, with their infinite complexity of cardioid shapes, curling branches, and "seahorses" found at different levels of magnification, have 96

Figure 4.8. The Mandelbrot set with corresponding Julia sets. From Peitgen and Richter, ii. 97

Figure 4.9. The Mandelbrot set. From Schroeder 1990, color insert. 98 fascinated scientists, computer programmers, and artists: they have found a niche in the scientific and lay imagination. The Mandelbrot set is produced through the plotting of an iterative equation of complex numbers projected on a two-dimensional plane. The sense of complexity and intricacy, of recurring shapes found on widely different scales, is in sharp contrast to the elegant, even naive simplicity of the equation. It takes more or less fourteen zooms into the Mandelbrot set to travel from the scale of the universe to that of an atom, but the Mandelbrot set, like other fractals, is infinitely complex, beyond human or even cosmological scale. What does the Mandelbrot set represent? What is it a visual model of? The Mandelbrot set does not represent any physical or virtual system, but represents a boundary, a limit which determines the essential property of corresponding Julia sets: whether they are connected or not. This may seem rather modest--after all, what is the delimitation of such an abstract mathematical boundary compared to a visual model representing an important empirical system or which helps us visualize some aspect of reality which has never been rendered visible before? But there have never been boundaries this complex, not even in the wildest imagination of visionary mathematicians, nor has the technology been available to project these infinitely complex images with such ease. The Mandelbrot set conceived 99 as a boundary reinforces what has become one of the most forceful paradigms of "chaos theory": simple problems do not necessarily have simple solutions (Gleick 1987, 303). Despite the fact that it is not a visual model of a physical system, the Mandelbrot set is a visual model and vindicates the priority given to projection models. If the projection of such an innocuous mathematical equation can yield a picture of such endless complexity and subtlety, then the objection that nothing new or interesting can be found by a mere projection of data or measurements rings somewhat hollow. The Mandelbrot set also shows the limits of conceiving of models as analogies: what possible relation of analogy could be established between the picture of the Mandelbrot set and a physical system, or between the equation and the picture? We are a long way from atoms conceived as billiard balls.

4.3. Fractal images as models

In what sense are fractal images visual models of the external world, if they are constructed by self-similar geometrical transformations or from iterative equations? The Mandelbrot set represents a boundary, but a highly abstract boundary between properties of mathematical sets. Fractal images are images which are clearly and exclusively produced by the application of mathematical theory. The 100 image is prior to any application of it. The image is not constructed to represent a reality that cannot be perceived or is not known; the image is the given. Fractal images are models without our initially having any sense of what we are modelling, whereas maps model an empirical process. When modelling turbulence, the projection and mapping process are complex, but the goal is still to turn empirical data, with these indirect means, into images. The pictures produced by mapping and fractal images are thus projected from opposite ends; maps are projections of empirical data, while fractals are projected from theories. Some visual models with fractal properties, such as the strange attractors described below, combine the two; they are projections from empirical data as well as from a series of mathematical equations. In the case of strange attractors, both mathematics and the empirical system produce an image, without one being formally modelled on the other. In the case of simulations, the numbers are produced by theories and equations, in the same way fractal images are projected. Fractal images can thus be compared to models produced by simulations.

Strangely enough, fractal images produced by formal projections can easily be manipulated to "look like" things in the world: clouds, ferns, coastlines and landmasses, plants, even works of art. Starting from simple simulations 101 of Brownian motion, it is possible to manipulate fractal images by adding color and assigning different features to produce virtual mountains, coastlines, landscapes, and so on. Richard Voss, a collaborator of Mandelbrot, has been particularly successful in using fractal images for the creation of virtual images of the world: figure 4.10 is the famous "fractal planetrise" which graces the cover of Mandelbrot's book. Peitgen and Richter talk about a "phenomenology" of the world analyzed through fractal geometry, rather than forcing the models of Euclidean geometry upon a world that is certainly not Euclidean (Peitgen and Richter 1986, vi). The difference between Euclidean and fractal geometry is indeed striking: fractal geometry can produce remarkable perceptual models of coastlines, mountains, clouds, or fern leaves. There are two important restrictions on fractal- based modelling of natural phenomena. We saw above that fractal images are infinitely self-similar, while biological and physical systems are self-similar only over a certain range of scales. Fractal images are thus virtual models of natural phenomena; no actually existing mountain, cloud, or coastline is exactly modelled by a fractal model. This approximation is also evident in applying classical geometrical shapes to the world: Mount Fuji is not an exact truncated cone. Fractal geometry can produce, with considerable manipulations, "realistic" renditions of 102

Figure 4.10. Fractal Planetrise. From Mandelbrot 1982, cover. 103 natural phenomena. These may be useful for characterizing or describing the phenomena, and are often far superior to descriptions relying on classical shapes. Many of the non-fractal models used in science are virtual as well. As Kenneth Falconer observes: "There are no true fractals in nature--for that matter, there are no inextensible strings or frictionless pulleys either!" (1990, 265).

The visual images that look like coastlines, mountain ranges, scattered islands or ferns are not visual models; they are "forgeries" (Voss 1985), fractals constructed through clever manipulations to "mimic" nature (Peitgen and Richter 1986, vi). This feature of fractal images now shows us why reliance on similarity, resemblance, "looking like," and even analogy is misleading for establishing the function of visual models. In science, what matters is not so much that things "look alike," because we do not understand what many abstract physical systems should look like anyway. Fractal images are not scientifically accurate or interesting visual models of real ferns, real clouds, turbulence, or real coastlines; this is their Hollywood function, their value as a powerful tool for the graphic designer. Fractal images are accurate and interesting visual models for levels of disorder in natural or artificial systems such as the 104 variations in population of gypsy moths, the fluctuation of commodities or stock exchange prices over time, background noise in communication, or various experimental settings. None of these systems, especially at the level of abstraction at which they are represented, looks like anything that can be perceived in the world. What matters is that manipulating the visual model allows us to manipulate the theory; through our interaction with one, we interact and learn about the other. The manipulation of fractal images to produce mountains, clouds and ferns does not increase our understanding of real mountains, real clouds, or real ferns. Mathematical equations do not allow us to represent coastlines that exist anywhere in the world, although specific claims about the fractal dimension of coastlines can be made.

The representational function of fractal images is complex and can operate at more than one level of representation. In some instances, visual models projected from empirical systems really do look like fractal images projected from equations. For example, some visual models produced by the study of the property of materials such as iron, or the aggregation of particles in diffusion-limited aggregation (DLA), are fractal images; their fractal dimension can be determined, and simulation of these 105 processes from iterative equations can be constructed (Schroeder 1990, 196-99). What is the relation between nature and fractal images? How is it that some fractal images look natural? Michael Barnsley has developed a system to produce strikingly realistic images with a only a few fractal variables. His Iterated Function Systems (IFS) appear very promising for drastic compression and storing of data of pictures. This does not imply, however, that nature could "store" very complex information with the same economy of means as iterative equations. Flowers do not sprout point by point on in a three-dimensional solid, but evolve organically. Another development is the production of so-called "turtle algorithms," designer tools which produce strikingly "realistic" and "lifelike" images through iterated and automated drawing functions (Schroeder 1990, 259-61). What is the "strikingly realistic" turtle algorithm-designed image compared to? A photograph, or a line drawing on a plane? What level of order of the object is represented by the level of complexity in drawing the image?

4.4. The limits of modelling

Fractal images extend the representational function of visual projection models beyond maps and graphical imaging of turbulence data in two ways. Fractal images make the 106 visualization of mathematical objects, properties, and theories possible in ways which were literally inconceivable two decades ago. Fractal dimension is being developed as a promising new tool for research. The debate about visualization and the existence of mathematical objects, even in such simple cases as an isosceles triangle, has hinged on the relation of images to equations. The development of fractal geometry and its infinitely rich and complex, self-similar images can thus be understood as an extension of classical geometry. Yet it also signals a departure: the imbalance between the infinite complexity of the Julia and Mandelbrot sets and the deceptively simple equations needed to generate them have given visual models and visualization a new impetus, and established them at the forefront of mathematical research. The creations springing from the imagination of mathematicians may turn out to look like nothing on earth, or alternatively they may mirror shapes of nature. In both cases, fractal images provide fuel for the discussion about the existence of mathematical objects. Fractal geometry can be viewed as an extension of classical geometry (Barnsley 1988, 1). The fractal dimension is the most important part of this extension; it is both a mathematical value which can be calculated by various methods, and a property of the graphical representation. Images in classical geometry are often 107 reversible; the image, given a suitable grid, can be turned back into an equation again. Some of the early mathematical monsters are reversible; but images which are the product of iterative equations, since they are plotted rather than projected or drawn, are not reversible. The projection methods of some fractal images still occur within the topological space defined by classical geometry; although the image has fractal properties, the space in which it is represented is still defined by Cartesian coordinates. Other fractal images, such as the Mandelbrot set or strange attractors, are plotted: the image is drawn by calculating its value for each point of a chosen grid. The image is not drawn continuously, but each point of the grid is calculated to see if it is part of the set of points that make up the image. This is how the famous zooms into the Mandelbrot set are possible: the grid can easily be redefined to consist of a small area of the previous grid. Plotting represents an extension of the projection method, which is the fundamental basis of visual projection models. The kind of image that results from plotting is radically different from a drawn image; in a sense, the whole grid is the image. Plotting an image may involve solving thousands of complicated equations; it has become feasible only with the development of computers and high-level graphics. Nowhere is the interaction of the 108 development of a scientific methodology and of technological progress more apparent than here. Fractal dimensions are a mathematical property of an image which clearly links an algebraic formula of a theory with the geometrical quality of the image. Fractal dimensions allow us to measure a pictorial attribute or pictorial quality of a visual model. The calculation of fractal dimensions thus functions as a bridge between two scientific methods--one mathematic, the other pictorial. There are many different kinds of fractal dimension, including the Hausdorff dimension, the Kolmogorov capacity, the Minkowski dimension, the correlation dimension, the information dimension, and so on.2 These fractal dimensions are measured using different techniques, some of which are more applicable than others, depending on the situation. For simple fractals, all these various dimensions tend to be equal, whereas they usually vary in the case of multifractals (Schroeder 1990, 200-206). The various dimensions are used to establish a mathematical property of an image. Calculating these different dimensions is not essentially different from the aforementioned example of assigning a numerical value to the squiggliness of a coastline.

2Part of the difficulty with all these fractal dimensions is that they are defined differently by different researchers. The fractal dimensions are also given different names, according to the figure in the history of mathematics who is deemed most deserving of the honor. 109

The fractal dimension emerges not only as the defining property of fractal images, but as the mathematical property and tool to measure empirical systems and to determine important qualities about them. The mathematical dimension of fractals, not necessarily their visual aspect, makes them models of phenomena and empirical systems. The computation of fractal dimension has added a whole new methodology to the use of visual models in science; now it is the visual representation of a state of a system that can yield characteristic numerical values. 110

5. Strange attractors

5.1. Strange attractors in dynamical systems

Science has until very recently been the study of predictable, regular, repeatable, and calculable phenomena. Nature, however, never quite behaves as though she were held in a scientific straightjacket, and small disturbances always turn the neatly promulgated laws into ideals, limit- cases of the irregular behavior of experimental systems.3 The solution to this situation has been to restrict the domain of inquiry to regular patterns in nature, and to describe and explain ideal systems not subjected to friction and other disturbances. When confronted with mathematical equations for which solutions can not be calculated, the strategy has been to reduce them to solvable equations, and to limit the inquiry to these specific cases.4 These tactics led to and in turn were fueled by the metaphysical assumption that the universe was mechanistic and predictable.

3Nancy Cartwright argues that all laws of physics, and of science in general, are abstractions which never hold for any actual, experimental setting; but that is exactly what allows them to be laws. The philosophical question is: in what sense do these laws then describe the world? (Cartwright 1983). 4A seemingly simple problem in classical mechanics, the relations of mutual attraction between three bodies, cannot be solved mathematically; this was shown by Poincaré. 111

The study of dynamical systems is the study of phenomena which exhibit irregular behavior and which cannot be reduced or idealized. These phenomena are all around us: the flow of gases and fluids, the weather, the Brownian motion of molecules, the study of turbulence in fluids. There has been a recent and rather sudden shift of emphasis: instead of taking the regular phenomena to be primary and the irregular phenomena to be scientifically uninteresting side-effects, current researchers realize that the study of idealized and simplified systems has serious limitations, and that natural phenomena need to be studied in all their dynamic irregularity and complexity. The study of irregular and unpredictable patterns in dynamical systems received an enormous boost with the gradual interest in what has been somewhat misleadingly coined "chaos theory." The term "chaos theory" is misleading because chaos theory is not a theory and not about chaos (Kellert 1993b). Chaos theory is not a single theory in a single discipline; it is the name given to a new methodology which is being used by, among others, physicists, chemists, meteorologists, seismologists, metallurgists, probability theorists, and psychologists, although its theoretical base is in mathematics (Gleick 1987). Nor does chaos theory investigate or explain completely chaotic phenomena. The systems studied by chaos 112 theory are not predictable, but they are determined (Hunt 1987; Stone 1989).

"Strange attractors" are visual models produced in the study of dynamic systems. An attractor, represented by a point in phase space, is a stable state toward which the system tends. An attractor can consist of a single point or finitely many points. A strange attractor consists of infinitely many points; it is called a strange attractor because it of its remarkable and unexpected shape.5 Strange attractors use the principle of phase space, in which the complete state of knowledge about a dynamical system, mechanical or fluid, collapses to a point. The history of the system time can then be charted by the moving point, tracing its orbit through phase space with the passage of time. This is called a Poincaré map; its construction is explained in figures 5.1 and 5.2. Strange attractors portray patterns of motion, as well as information about the periodicity and change of the system that would be invisible otherwise. The dynamical system represented by the strange attractor can be a dissipative

5The name "strange attractor" was coined by the mathematicians David Ruelle and Floris Takens, because, as Lauwerier remarks, "strange reflects how amazed they were, even though as mathematicians they are accustomed to a lot of strange things, when they saw [its picture] appear on the screen" (Lauwerier 1991, 137-38). 113

Figure 5.1. A Poincaré section. The Poincaré sections shows the intersections of the solutions of function with a plane S. From Hall 1992, 269.

Figure 5.2. A strange attractor. The strange attractor becomes visible on the Poincaré section, a slice through a three-dimensional torus. The successive images show an increasing number of iterations. From Gleick 1987, 143. 114

natural system which hovers on the boundary between a stable and chaotic state, or it can come from a set of equations simulating such a system. The image which is produced renders visible totally new forms of information about the system: the level of its disorder, the possibility of it achieving certain states, and its general "shape." Strange attractors are fractal images; their fractal dimension can be measured, and it is sometimes possible to correlate the shape of the image with specific properties of the system under consideration. Strange attractors have played a pivotal role in the selection and elaboration of theories of turbulence and a wide variety of other dynamical systems; they are generative models.6 Strange attractors can be plotted from simulations of systems made up of a series of mathematical equations, but also from measurements of an empirical system; they function at the boundary of mathematical simulation and experimental results, and may constitute a pictorial link between these two methods. Strange attractors exemplify a shift in the level of abstraction of the representational function of visual models. They do not represent a system that can be

6There are two senses of "generative" here: strange attractors are generated by projection methods; and they are generative models, generating new interpretations and elaborations of theories. 115 perceived in nature, but visually model a higher level of order of that system.

5.2. Modelling and approximation

Edward Lorenz ran a computer simulation of equations describing cellular convection, designed as a minimalist model of the atmosphere and given by a simple set of equations (Lorenz 1963). This simulation produced the first image of what would later be called a strange attractor. Figure 5.3 shows the "Lorenz butterfly," the strange attractor first seen by Lorenz, in three dimensions. Note the three equations from which the image is projected. Figure 5.4 shows a more detailed, two-dimensional version of the Lorenz butterfly. Lorenz's simulation reveals another dramatic effect. Any minute change in the parameters of the equations produces rapidly diverging paths on the attractor, although the overall shape of the attractor remains the same. This means that a future state of the system as a whole is determined by initial conditions which have to be unrealistically precise. This is a central tenet of chaos theory called "sensitive dependence on initial conditions." Since the system that Lorenz chose is taken to model atmospheric conditions, Lorenz concludes his article by stating that long-term weather predictions are impossible, 116

Figure 5.3. The Lorenz butterfly with equations. From Ruelle 1991, 62. 117

Figure 5.4. A more detailed Lorenz butterfly. From Hall 1992, 271. 118

since they would require impossibly precise measurements (Lorenz 1963, 141). Sensitive dependence on initial conditions is also known as the "butterfly effect"; the flap of a butterfly in Brazil may set off a tornado in Texas. The example of the Brazilian butterfly merely points to the degree of precision required of all the parameters of atmospheric conditions to make long-range predictions. The example does not imply a causal link, and should not be "interpreted in a way that suggests that the careless insect actually causes the tornado" (Kellert 1993a, 13n).

The sensitive dependence of some dynamic systems on initial conditions has serious consequences for modelling and simulation. A fundamental assumption of scientific method had always been that small differences in the mathematical formalization of a system would cancel each other out and not qualitatively affect the description of that system (Berlinski 1975, 212; Kellert 1993a, 44). Modelling an empirical system through mathematical, visual or linguistic models always involves approximation; this is inevitable and was considered inconsequential as long as the approximation errors were kept to a minimum. The mathematical simulation of empirical systems on computers relies on the same assumption. 119

However, if dynamical systems show sensitive dependence on initial conditions, then no approximation of these systems can be made because minute initial errors would soon be dramatically magnified, with the result that the approximation would no longer describe the same system. Modelling and simulation become meaningless. Sensitive dependence on initial conditions has forced a reevaluation of metaphysical assumptions about the predictability of empirical systems. The link between predictability and determinism has been severed; although dynamic systems may not be predictable, they are determined. Methodological assumptions about the applicability of models and simulations have also had to be revised. Models always represent some system, but the certainty that a model represents a particular empirical system has disappeared.

The crisis of modelling caused by sensitive dependence on initial conditions reinforces the need for accurate interactive visual models based on projection. The projection of data into visual form provides the researcher with an excellent tool to represent a system or simulation. Sensitive dependence on differences in the variables of the chosen systems will alter the projected pictures. Therefore, contrasting pictures projected from almost identical systems will contribute to defining the kind and extent of sensitive dependence. Although some fundamental 120 metaphysical assumptions about the nature of modelling have been shaken, visual modelling is shown to be more important than ever.

5.3. Mathematics and the world

The image of a strange attractor functions as a bridge between the mathematical simulation of a system and the experimental measurements of dynamical systems: both produce strange attractors. Today, the image is the only link between equations and the empirical system; the empirical data cannot be reduced to equations, nor do the equations completely and accurately describe any naturally occurring system. Strange attractors may be plotted from data of dynamical systems but also, for example, from three- variable mathematical equations. In both cases, the data or equation solutions have to be calculated for each point, necessitating the number-crunching and graphics capabilities of computers. It is not possible, however, to formally correlate these two projection methods: the measurement of data will never yield a mathematical equation which describes the system correctly or completely, and, conversely, a mathematical equation will never of itself yield the description of any dynamical system which can be observed. 121

The picture of the strange attractor does not allow this chasm between nature and mathematics to be bridged. Even though the strange attractor cannot formally reconcile simulation and experiment, it can function as a link between these two realms. The representational relation between the empirical system and the equations is defined through the image. Poincaré maps of the strange attractor provide the researcher with information about the dynamical system. A researcher familiar with the topological properties of Poincaré maps can make some educated guesses about the mathematical equations needed to provide a simulation of the system. Strange attractors provide a little-understood and fascinating link between mathematical equations and empirical systems, contributing to the discussion about the value of simulation experiments. Strange attractors function as a representational relation between theory and empirical system (see section 1.5, figure 1.10).

Whatever the possibility of the discovery of a formal link between the image of a strange attractor and mathematical equations, the constructed image is original. The strange attractor is not a copy of a pre-existing image. The constructed picture is a way of representing measurement (it is a map), radically different from the interrelation of numbers with mathematical symbols, but just as primordial. 122

The information about the dynamical system provided by the image is a unique but potentially confusing way of representing what is known (what can be measured) about this system.

5.4. Generating interpretations

The information obtained in strange attractors is even more abstract than that obtained in the graphic representation of turbulence data variables. It makes no sense to speak of the model representing a visual reality by visual reality, because we do not perceive the development of reduced systems over time. The representational function of the image takes place in a different realm of representation. The image, projected from data, guides our understanding of the system and our research on it. The projection method is not that of a numerical function along axes (this cannot be done because of the kind of equations involved); the image is plotted point by point on a chosen grid by applying the set of equations. The image is not "drawn" continuously, but appears as a ghost out of nowhere, the points at first scattered seemingly at random, but progressively coalescing into a more and more definite shape. The explosion in complexity of the level of representation of the visual model occurs because of the 123 different realm of representation. The term "realm" is used here to indicate the multi-levelled aspect of the information that is rendered visible and intelligible by the visual model. The system is not predictable, but another kind of information is rendered visible by the image of the strange attractor: the level of complexity of the dynamical behavior of the system, and the limit values beyond which the system will not go. The level of disorder and the structure of the chaotic state of the dynamical system produce an original shape, a picture sui generis.

The perceptual aspect of dynamical system, whether a leaking tap, economic data, or weather fluctuations, is not represented by the strange attractor. The image of the strange attractor is a representation of the behavioral pattern of the dynamical system. It is possible to take almost any of the variables of the initial system as the basis of the projection and always get a strange attractor (Holden and Muhammad 1986, 17). The dynamical behavior of the system under consideration is thus neither regulated nor predictable, but it is bounded; the image makes visible the level of disorder of the system. As a necessary, dynamic and interactive visual model, the strange attractor is a prime example of a generative model. The image itself provides information about the system which could not be obtained without it. The strange 124 attractor plays a pivotal role in the elaboration of theories, since it provides information about the state of the dynamical system that cannot be gleaned from the equations, but must be derived from the picture: it is thus a necessary visual model. The strange attractor is also dynamic: it can be projected from any variable, and even constructed by taking a simple variable and projecting it in more than one dimension. It can be projected in two, three or even higher dimensions, and yields an infinite number of derived visual models through Poincaré maps. Finally, the strange attractor is interactive: as a visual model, the strange attractor represents information which allows us to manipulate the system, which in turn changes the picture in unexpected ways. The picture of the strange attractor permits us to fine-tune our understanding of the system. 125

6. Generative models and representation

We have concluded our magical mystery tour of geographic maps, images of vorticity in turbulence, the Sierpinski gasket, the Julia and Mandelbrot sets, the Lorenz butterfly, and other pictures functioning as visual models. How has the study of fractal images, maps, strange attractors, and vorticity imaging altered our understanding of representation and the function of visual models?

The representational relation of visual models, even though it is still in the process of being understood and refined in the scientific programs that spawned it, challenges the restricted understanding of representation in epistemology, theory of language and aesthetics. The extension of projection methods in the sciences and the concomitant explosion in the complexity of the representational function of models provide a rich and powerful concept of representation to explain visual modelling. Generative visual models play key roles in the elaboration and choice of theories. The shifts of understanding explored in the previous chapters produce disparate claims about the representational function of visual models. We have made claims about the fundamental bases for modelling, the properties of 126 projection models, and topological properties. These claims shift our understanding of the representational function of visual models away from an analogy-centered perspective towards a generative perspective. Generative visual models are necessary visual models, and cannot be conceived as having merely a didactic or heuristic function. The fundamental basis for modelling is not a comparison based on analogy, but a projection of data as a visual image. We must shift the burden of explanation from an overused and unproductive concept, analogy (and a series of synonymous concepts such as correspondence, isomorphism, parallelism, and similarity) onto another concept, generative representation.

6.1. Analogy and necessity

The theory that models are essentially analogies is the undisputed favorite in most philosophical and scientific discussions of the role and function of models (Hesse 1966, Dambska 1969, Leatherdale 1974, Redhead 1980, Forge 1983, Giere 1985, Harré 1988a, and others). Part of the reason for this prominence must be that the same hackneyed examples, such as the wave analogy for sound or electric waves, or the billiard balls analogy for the kinetic theory of gasses, crop up again and again (Achinstein 1964, 332- 33). If the domain of inquiry is restricted to necessary 127 visual models, however, the theory of models as analogies becomes unusable. There are three main problems with conceiving of necessary visual models as analogies. The relation between a picture and theory is not an analogy but a resemblance. Analogies by their very nature perform a heuristic function, or substitute temporarily for the complexity of a theoretical description, but they are not necessary for the establishment of the theory. The theory must be known independently of the model for the analogy to be established in the first place. The explanatory power of analogy is dubious because analogy involves an initial, intuitive comparison between two systems.

Some thinkers believe that models are merely simplified systems or diagrams used in teaching or to popularize theories too complicated for a lay audience. They claim that models have only a heuristic or didactic function and give priority to mathematical theory and language in theories. Others argue that pictures are necessary and that theories cannot be formulated without the help of a visual model. I will show that visual models are necessary for the elaboration of theories and for the representation of empirical systems. Necessity is a property of generative visual models. Mary Hesse (1966) examines whether models have a didactic or a heuristic function. She casts the two 128 opposite schools of thought on this issue in a dialogue between a continental "Duhemist" scientist and his British "Campbellian" counterpart. The Duhemist maintains that scientific theories exist only as purely mathematical formalisms, and that models only serve as extraneous, and ultimately superfluous, examples. The more practically and empirically oriented Campbellian holds that models are scientific products and serve heuristic as well as merely pedagogical functions. Hesse argues for a Campbellian position, showing the existence of positive, neutral and negative analogies between model and theories. The positive analogies determine the modelling relations, while the negative analogies show when the model cannot be applied to the theory. The model influences the theory because the imagination of the scientist turns neutral analogies into positive analogies. The initially non-relevant aspects of the model suggest correspondences with the theory being modeled and further critical inquiry about possible extension of the theory. Neutral analogies have predictive power, which accounts for the heuristic function of models (Hesse 1966, 9).

If a positive analogy can be drawn between aspects of the theory and an extraneous model, then that particular aspect of the theory must already be sufficiently understood. The positive analogies make the model a merely didactic, and so a truly "Duhemist" device. The neutral 129 analogies however, those where there is no initial correspondence between model and theory, can perform a heuristic role in generating new aspects of the theory. Neutral analogies between model and theory cannot always be found; in those cases, the heuristic function of the model is limited or nonexistent. Even when there are neutral analogies, it can be difficult to distinguish them from positive or negative analogies. Moreover, the existence of a neutral analogy between model and theory offers no assurance that a heuristic extension of the theory according to the neutral analogy will be useful or significant.

Michael Ruse provides an example of a heuristic extension in his analysis of Darwin's theory of evolution (1973a; 1973b). Ruse operates with a wider definition of models where one theory functions as a model for another theory. Ruse shows how Darwin appropriates a central tenet of Malthus' theory about the natural limits of population growth to bolster his theory of the survival of the fittest. Although Ruse takes modelling to consist in a series of analogies (here between theories), he argues that in this case the notion of a formal analogy cannot explain the modelling process. The extension of the theory being modeled (Darwin's) through the application of elements from the theory as model (Malthus') only shows the fundamental use of material analogy. Ruse argues that this is sufficient to explain the transfer from model to theory. 130

This example shows that formal analogies in the structure of two theories may not have any explicative or heuristic power.

I claim that visual models are necessary for the visual theories in which they appear, and distinguish three kinds of necessity. Roger Krohn argues that visual models make it possible for large amounts of numbers to be represented graphically; the "pictorial imagination" makes sense of the results thus plotted (Krohn 1991). Our perceptual and cognitive powers are limited in their capacity for comprehending page after page of numbers. If the numerical data for turbulence imaging in Chapter 3 were to be printed out, they would fill dozens of volumes. But Krohn's claim that models are central to theories remains modest, since it is restricted to theories where visualization of numbers is required, and does not privilege one picture over another.

A stronger sense of necessity is provided by the concept non-eliminable, proposed by James Griesemer in his study of the role of models in path analysis in statistics. In this case, the same statistical conclusions cannot be reached without applying the picture which illustrates the distribution (Griesemer 1991). The visual model, the constructed picture, cannot be excised from the theory without altering the theory. Griesemer's argument is technically persuasive, but "non-eliminable" is a negative 131 characterization, and does not clarify what the necessity of models consists in. It is possible to extend the previous notions of necessity by claiming that the role of visual models is pivotal for the theories in which they function. The visual model will be instrumental in deciding between different theories or conflicting interpretations of theories. The strange attractor visually provides the coefficients which determine whether the system is regular, periodic, or chaotic, and has thus come to play a pivotal role in the theory of dynamic systems.

On the view that models function as analogies, the visual model is related to the empirical system by analogy with the way the theory itself is related to the empirical system. The sole mediation between model and theory is the empirical system which makes the model an apt representation of the theory in question. This is an indirect link; to argue for an analogical relation is to argue against a direct relation between image and theory. I hold that models are necessary: the relation between the theory and the empirical system has to pass through the visual model. The complete relation between theory and empirical system cannot be accounted for solely by representational relations which do not make use of the 132 visual model. We can now return to figure 1.9. and propose a significant modification:

visual models necessary

mathematical models (equations) theory empirical linguistic description system

??? - other models

representation

Figure 6.1. the necessary function of visual models.

The theory of analogy states that theory and visual model are independent and often incompatible structures. The analogy itself is never necessary to the comprehension of the theory, although it may contribute to the extension of the theory. Izydora Dambska writes that the use of analogies to compare objects and structures in the world is based on isomorphism, and is a deep-seated, almost instinctive, component of the human mind (Dambska 1969, 34). Proposing an analogy is often an intuitive move, which involves a judgement as to the applicability and appropriateness of the claim. But for visual models, it is hard to see what an analogy would consist in; if the analogy is to be based on resemblance, it gives far too high a priority to the way a visual model looks. Thinking in terms of analogies for visual models reduces to endorsing the 133 mimetic theory of representation. The visual models studied here are dynamic models: the projection and manipulation functions of maps, graphic images and some fractals make them into matrices of images. The appearance of an image is therefore largely the result of choices made in the construction; visualization is a process, not an informed comparison of pre-existing and fixed images.

The use of analogy is important in its own right, involving an astute comparison between two different entities often operating in different domains. When we claim that one entity functions by analogy with another we transfer an interpretive schema applicable to one entity onto another. Analogies are crucial to advances in sciences because they suggest comparisons of systems based on their function or form. But analogies do not explain why the initial comparison was made, nor do they describe the relation of modelling involved. Describing modelling relations exclusively as analogies is incomplete and misleading.

6.2. Projection and representation

The projection methods used in the construction of visual models have produced an explosion in the complexity of the representational function of models. The development 134 of projection methods is spurred on by the need for mapping systems beyond the reach of the visible, by the technological development of the computer, and the possibility for topological measurements based on fractal geometry. The explosion in complexity is possible because of the formal nature of the initial projection process, which is rooted in mathematical transformations. The fundamental concept of projection is very simple: use geometry and topology to turn quantitative data (numbers) into graphical data (images). The relation of projection established between visual model and empirical system will remain clear and analyzable as long as it is based on a mathematical transformation of numbers and geometrical projection. New possibilities for graphical imaging and new relations of modelling will result from advances in mathematical projection methods. Interpreting and understanding pictures produced by novel projection methods will also depend on the cultural context.

The representational relation of visual models is fundamentally based on projection, which displaces mimesis as the relevant relation. The projected picture does not resemble another image and is not a copy. Pictures of the world are not images created with the help of tracing paper laid on a self-sufficient and mute reality. Images in the sciences can stand on their own; they do not need to 135 represent anything outside of themselves. A visual model is an image, but the representational relation is not one of mirroring, of reflection. The visual model in the sciences is a representation of states of the empirical system, and interacts with the theoretical constraints under which the available data is analyzed. Visual models follow clear rules of projection: there is a formal link between data and visual image. It is difficult to abandon the metaphysical claim that images are intrinsically copies, reflections of an ontologically and epistemologically prior original object and reality. In the world of art, the beginning of this century saw a massive effort to escape the aesthetic and cultural constraints of representative painting; artists created abstract art which can communicate and express emotions, feelings, or information from its own perspective. Even in art, representation does not have to be mimetic: the image itself can carry meaning, express and represent a vision of reality. The image need not represent anything outside of itself. The image is its own original; it is self-generating.

An isomorphism is a mathematical projection function which maps every single element of one domain onto a single element of another domain. Not all projections are isomorphisms, but isomorphisms and other projections all 136 consist of a formal relation. Similarity, in the sense that it has in daily life and not in the formal sense of a topological transformation, contains an implicit value- judgement in the comparison between the elements of two domains. This accounts for the close connection between similarity and analogy, and the implicit judgements about the picture and what it represents in both cases. On the mimetic theory, images are compared by the way they look, not by the empirical systems they are projections from. Resemblance is a synonym for similarity, restricted to the visual domain.

It may appear as though projection is inherently far too simplistic a relation to account for the complexity inherent in representation, and that it makes for a very poor and limited link between theory and world. There may be some unease about reducing the complex notion of representation, which has proved so hard to define in the past, to a set of techniques for projecting numbers in a topological space and some extraneous manipulations. These reservations, however, are caused by misunderstanding (misrepresenting) the nature and scope of the claim being made about visual projection models. Defining representation as projection does not subvert and replace other theories of representation, but stakes out a group of models for which none of the other theories of representation make much sense. The modest claim is that 137 representation as projection is an additional way to conceive of representation, valid for a limited group of representational relations. A more daring claim is that representation as projection is always an element in any modelling relation, and cannot be ignored or passed over. In both cases, conceiving of representation fundamentally as a projection not only acknowledges the vast amount of projection models used in the scientific endeavor, but also pushes the conceptual understanding of the representational relation of visual models, and of the concept of the visible, further than ever before. Representation remains both complex and elusive if it is thought of as a universally applicable notion; but defined clearly as projection for specific purposes, it explains the basis of the modelling function of the visual models examined here.

6.3. Generative models

The properties of generative visual models are: they are constructions conceived as re-presentations, not copies or mimetic representations of data; their representational function is usually multiple, many-leveled, and complex. Generative visual models are the result of pictorial manipulations of scale, color, dimension, signs and symbols according to specific pictorial conventions. Both the projection method and the manipulations become an essential 138 part of the dynamic aspect of visual models: the picture is an active matrix of possible images produced in a visual domain. Generative visual models are dynamic systems producing families of visual models according to the projection function and various pictorial manipulations. The representational function of generative visual models consists of relations, of multiple interactions between theories and the empirical system. Generative visual models perform a crucial role in the development of theories; they have the potential to generate new interpretations of data which produce altered models and adjustments in theories.

Generative visual models cannot be understood in terms of analogy, similarity, balance, and reduction; they require concepts such as projection, interaction, necessity, and dynamic nature. These models mediate between a theory and an empirical system, and are not distinct entities represented separately. Generative models are necessary, dynamic and interactive. For a visual model to be dynamic, it must be able to be adapted, developed or manipulated to model the empirical system in ways which challenge the theory. This kind of visual model is not a single picture or image, but a technique for rendering a structure of the empirical system visually. A visual model is not just one specific picture, but a possible exemplar from a series of images. There is a family of images, all of which are produced by the same technique or principles which 139 constitute their kinship ties. The visual model is dynamic because it can be developed or manipulated, but the dynamic of change may not always be under the control of the experimenter. While the phenomena delimit, constrain, and produce the visual model, the visual model itself models, exemplifies, or illustrates the theory: the visual model is interactive. If conceived as an analogy, the relation between visual model and theory completes itself in a two-dimensional circle. The interactive visual model, however, is a three- dimensional spiral, where the points of origin are constantly shifting, and the interaction back and forth create a multitude of paths.

The differences between models conceived as analogies and generative models are summed up in the diagrams below (figures 6.2 - 6.7). Analogies hold between two relations; the relation of the theory to the empirical system and that of the model to the empirical system.

EMPIRICAL SYSTEM

THEORY VISUAL MODEL analogies

Figure 6.2. Analogies between relations. 140

In the case of generative representation, the visual model constitutes the relation between theory and empirical system.

THEORY VISUAL MODEL EMPIRICAL SYSTEM generative representation

Figure 6.3. Generative representation between theory and empirical system.

Models as analogies are taken to be didactic and heuristic devices to extend the theory. The properties of the theory are opposed to the properties of the model:

THEORY MODEL complex simplified sketchy diagrammatic essential superfluous eliminable unwieldy manageable impenetrable accessible non-conceptualizable conceptualizable

Figure 6.4. Properties of theory and analogical model.

Generative models are not opposed to theories, since they constitute the relation between model and theory; this relation has the following properties: 141

RELATION BETWEEN GENERATIVE MODEL AND THEORY complex constructed picture necessary: - central - non-eliminable - pivotal manageable accessible conceptualizable

Figure 6.5. Properties of the relation between generative model and theory.

The relation between model and theory on the analogy view can be characterized as follows: THEORY MODEL balance matching reductive replacement substitution ISOMORPHISM SYMMETRY PARALLELISM SIMILARITY RESEMBLANCE

ANALOGY

Figure 6.6. Properties of the relation between theory and model.

The generative model can be characterized by these properties: 142 GENERATIVE MODEL imbalance mismatching projection non-reductive dynamic interactive ASYMMETRY

Figure 6.7. Characteristics of generative models.

Generative models can influence theories and worldviews, provide the basis for new investigative strategies and methods, play the pivotal role of deciding between rival theories, or contribute to the establishment of new theories. Projection shifts the representational function from a passive and balanced relation based on the possibility of symmetry and isomorphism towards the visual model as the interactive and dynamic embodiment of unbalanced and asymmetrical relations.

The three properties of necessity, dynamism and interaction are not a simple test which could be applied to any given visual model to determine if the model is generative. There are gradations of necessity which apply differently to different models. The level of interaction between visual model and theory can be difficult to determine and evaluate. The dynamic nature of the visual model depends largely on the context in which the picture is constructed and the technological means available. In 143 principle, every picture could be manipulated to render it dynamic; but doing so is not always meaningful. We can use the definition of generative models in two reciprocal ways. We can find visual models which influence the development of theories, and define attributes of these models. Or we can define generative models as necessary, dynamic and interactive, and focus on these attributes when we study and evaluate visual models. In both cases, generative visual models illustrate the concept of generative representation. Generative representation is a fruitful concept for the evaluation of models and for the theory of knowledge. We have gone beyond the restrictive concepts of representation as analogy and mimetic representation. 144

7. Visual models and knowledge of the world

Philosophy has been concerned exclusively with the representational property of words and signs in a linguistic system, and has reduced representation to an inquiry into meaning and signification. The study of images and pictures has remained largely parasitical on research into the representational function of language or systems of signs. Yet any reformulation of a concept will remain a linguistic exercise involving our understanding of the linguistic context: a change in language games. The analysis of notions such as analogy, isomorphism, resemblance and correspondence for visualizing and picturing has pointed to severe limitations for basing one of these concepts on another. We have modelled the notion of pictorial representation on linguistic representation. We now turn this around, and come to conclusions about the representational function of language through our analysis of pictorial representation.

Generative visual models are necessary, dynamic and interactive: what makes the representational function and qualities of a generative models different from that of other models? I have shown that generative visual models 145 are not copies of an ontologically prior reality, but are their own originals, or self-generating. How are we to integrate this claim in philosophical discussions about reality and appearance, or constructivism and realism? What consequences does this have for our understanding of an image? What is the epistemological status of a picture of knowledge? What role do pictures play in our understanding of the external world? Our understanding of pictures influences our understanding of the world through the formulation of scientific theories. To inquire and to represent also involves visualizing and picturing, leading to the proposal and establishment of a worldview. Visualization must include self-knowledge, the ability to understand ourselves as picturing, world-making selves.

7.1. Visual models and language

With our focus on the visual rather than the linguistic, we have evaluated the usage of a family of linguistic terms including similarity, resemblance, analogy, isomorphism, parallelism, and correspondence. The definition of a projection function for visual models has shown the remarkable difference between the technical usage of these words and their meaning in everyday usage and philosophical discussion. Specifically, efforts to explain 146 the meaning of the notions of similarity and resemblance by appealing to the mathematical meaning of isomorphism have been discredited. These differences have not been obvious in the studies of representational function of models in general because these investigations did not concentrate on visual models but on language. When W. V. O. Quine seeks to distinguish analytic from synthetic propositions, he finds that he has to explain analyticity in terms of synonymity and synonymity in terms of analyticity (Quine 1961, 29). He proposes that at some point in linguistic analysis one will be confronted with a group of notions or concepts which are interdefined. We are confronted with the same problem when we try to define analogy without using the concepts of (formal) similarity, or isomorphism, or parallelism; or trying to define one of these related concepts without having recourse to analogy. Quine concludes by proposing the notion of a web of belief, with "analytic" truths at the center and notions we could easily give up at the periphery. Any new knowledge claim, he states, will result in a rearrangement of this web; the severity of the upheaval will be determined by the amount of rearranging to be done (Quine 1961, 42). If analogy is one of the central concepts in our understanding not only of representation but a host of other notions, then any reformulation of this concept, any shift in our definition or usage of it will reverberate across the web of 147 our knowledge claims. This will occur especially in the case of knowledge claims about images and pictures, whose representational function cannot easily be explained by analogy. The shifts that I propose here may ultimately be reducible to linguistic reformulations, to a change from a language game based around the notions of analogy, parallelism or isomorphism towards a language-game centered on the notion of generative interaction. On this view, I will have proposed and developed a different philosophical perspective centered around a different language-game.

The notion of correspondence is a key concept for a large number of philosophical positions. The notion of correspondence is often used and applied with zest, and trumped up to explain an endless variety of relations holding between entities, seta, and structures. Words correspond to things; objects correspond to things in the world; logical structures correspond to states of affairs. Inside corresponds to outside; phenomena correspond to noumena, appearance corresponds to reality. The blind acceptance of correspondence rests on a simplistic equation of this concept with the mathematical concept of homomorphism. Correspondence must be treated like the notion of similarity: it makes sense in a formal, mathematical way. When we try to broaden the range of 148 correspondence, even slightly, by adding a little water to the wine, we dilute the concept to such an extent that it loses its explicative power and meaning. We use correspondence to indicate a type of relation holding between entities in different realms, such as language and reality, or perceived analogies of models and theories. But when we do this we are no longer using a formal and neutral relation: the insistence on correspondence becomes a value- judgement which determines a reciprocal mapping of one group of elements onto another. This value-judgement is of the same kind as the value-judgement on which we base analogies and similarities (see sections 2.1 and 6.1).

The objection will be made that correspondence plays a crucial role in the visual models examined here: signs and symbols on a map correspond to towns, rivers and highway numbers in the world, shapes and colors correspond to properties of turbulence data, and the Mandelbrot set corresponds to the boundary between different kinds of Julia sets. The user of a map is indeed likely to find that a red dot corresponds to a city of two million inhabitants. But projection cannot be reduced to correspondence; its dynamic of operation is radically different. A map is a projection of information onto a grid, a visualization of a state of the world determined by the projection method and a series 149 of manipulations. For every correspondence between symbol or sign and topographic information, there are an indefinite amount of differences: there are not exactly two million people in this city; this river has more bends; there are no names of towns and states etched into the mountain ranges. These specific examples may be circumvented by appealing to other maps with more accurate population readings, better scale ratio, or without names. But all the cognitive or philosophical links that are attributed to correspondence in this case can just as well be accounted by projection: we can situate and find information and our way on a map because it is a projection of information. The correspondences noted on the map are both a secondary feature and a consequence of projection. The same argument can be made even more forcefully about graphical imaging of turbulence or the Mandelbrot set. The fundamental feature of the visualization of turbulence is not that properties of a constructed shape correspond to the values of the square of the rate of spin of one of two million virtual nodal points, but that this shape is the projection of these values. Surely the shape of the visualized aspect of turbulence must correspond to the flow of turbulence in the real world? Unless one wants to stretch the meaning of correspondence to such an extent as to make it meaningless, the answer is no. What is the explanatory power of determining correspondences 150 between a shape and a physical aspect of the flow? The shape is the tortuous result of a projection of data, of measurements. The shape represents the empirical system. Representation is defined in terms of projection, projection is defined in terms of representation. Instead of positing unexplained correspondences between elements of the theory, of the visual model and of the empirical system (figure 7.1), we define the relations of projection and representation (figure 7.2):

THEORY VISUAL MODEL EMPIRICAL SYSTEM

correspondence

correspondence correspondence

Figure 7.1. Correspondences between theory and model.

THEORY VISUAL MODEL EMPIRICAL SYSTEM

projection

representation

Figure 7.2. Projection and representation. 151

There is no need to appeal to concepts without explanatory power such as analogy, similarity or correspondence to define representation in the case of generative visual models. The basis of the representational function of visual models is adequately explained as projection. We have used a study of images to show the unwarranted reliance on concepts borrowed from the theory of language. The reverse method is also possible: the concept of representation as projection can be applied to the relation between language and the world. Doing so would constitute a different project, however; I have limited myself to the study of images here.

7.2. Visualization and the visible

Two topological properties have extended our notions of projection and pictorial representation. The mathematical dimension of fractal images allows for a radically new topological measure of fractal shapes and images. The pictorial aspect of the visual model can be measured by straightforward mathematical formulae, extending the mathematical properties of images and in certain cases making the dimension of images crucial to the study of the empirical system the visual model represents. Constructing a picture by plotting it point by point on a grid differs radically from drafting or sketching an 152 image. The technique of plotting an image on a grid is not new; it was used to construct vanishing-point perspective in the renaissance, and is used extensively in map-making processes. Plotting images has received an enormous boost from computer graphics; computers are ideally suited for the large number of repetitive calculations needed to plot a detailed image. The many unexpected images which cannot be visualized without extensive plotting include the Mandelbrot set and Poincaré maps of strange attractors.

The construction of visual models is a scientific methodology for investigating empirical systems dependent on other scientific procedures, such as mathematical algorithms for simulation and geometric projection methods. Visualization combines three key elements: perception, geometrical projection, and creative draftsmanship. The many advantages of pictorial representation stimulate the demand for visual models. The basis of the representational function of visual models is projection consisting of formal geometrical transformations to turn data into images. The construction, interpretation and provision of meaning for the picture is a feature of our ability to create images and to respond to them culturally. Visualization requires an interplay of the cognitive faculty of perception, the mathematical faculty of geometrical projection, and the 153 aesthetic faculty involving the creation, appreciation and interpretation of images.

Images produced by projection are representations of the empirical system independently of what the image and the object of investigation "look like." In many cases, the phenomena have disappeared beyond the horizon of perception, and the visual model is the only representation of the empirical system that we have. Even if the system can be visually perceived, the visual model representing it need not be based on visual perception: some visual models of clouds may illustrate many properties of clouds, without in the least "looking like" a cloud. Conversely, a visual model may in fact look like an object in the world, without being a visual model of that object at all; this is the case of many fractal images. Fractal images of clouds do not tell us anything about clouds except for an approximation of their fractal dimension; but this dimension is first obtained by analyzing real clouds and then used in the production of simulated clouds. Since visual models are independent of the appearance of the objects they model, basing the representational function of visual models on resemblance is misleading. Our ability to construct a visual model and in doing so represent our world to ourselves should be independent of the knowledge we gain through perception of reality. 154

7.3. The construction of reality

Baudrillard has shown in his book Simulations that the simulacra which surround us in a postmodern world no longer refer us back to any original, any "real"; simulacra make up the "hyperreal" (1983, 2). In many sciences the notion of real as original has long since been given up, or at least has become a polite fiction for which there is little room in the mathematical descriptions of scientific theories. There are formal theories about "states of the world" or the empirical system, without any ontological claims for that reality. The entities under discussion have vanished behind the horizon of the observable, usually because of size or speed, and the study of these entities which have been observed but never seen has become the study of the traces they leave on the experimental apparatus. This does not mean that the objects of scientific study have become "hyperreal" in Baudrillard's sense. The "real" has always been a tricky notion, and the mistake has been to subscribe to a narrow and incomplete definition of the representational function of models or simulacra.

What is the source of pictures and images if they are not copies? Pictures are projected measurements from the 155 world, reality, phenomena, or an empirical system: they are defined by it and in turn define it. This does not make them copies of the world or of reality. Insofar as there is no original, or no accessible original, the representation is the reality. We have to acknowledge our responsibility for endowing constructed pictures with meaning. The production and use of generative visual models delineates the creation of a worldview, vindicating a constructivist position. Visual modelling is not only a means to make sense of the empirical system, or to understand the implications of our measurements; it is also a means to create a world. The data are measured, but the picture is produced, presented, created. Our worldview is a human construction; and it is independent of human perception. The shift in our understanding of the concept of representation consists in abandoning representation based on resemblance or analogies and choosing generative representation. The use of generative visual models means the loss of a representation of the world based on resemblance and the creation of a constructed worldview. By daring to lose the representation of the world we have found the conceptual tools to regain it.

There are two kinds of constructivism at stake here. We speak of the construction of the world through our 156 scientific endeavor and the concomitant creation of scientific objects; this constructivism is opposed to realism. We also speak of the visual aspect of the objects of science as constructed. Here visualizing means creating a world. What kind of world is being created and by whom? The ability to visualize data by projection methods hinged on the possibility of endowing the image with meaning, to reclaim it from the infinite number of possible images, to fixate the representation, to choose and claim it. Creating a worldview consists in endowing our constructed pictures with meaning, in defining them as that worldview.

The relentless pace of technological innovation, galvanized by the development and application of the computer, is not an alien influence on the history of science and thought. Our understanding of what an image is, of how it represents the outside world through our own theoretical constructions, and of how we interpret and judge visual models, is an understanding which has grown through scientific praxis. Technological advances are not the cause of the need to reevaluate the visualization process, but the means to do so. The fundamental metaphysical assumptions involved in constructing, interpreting and evaluating visual models involve philosophy, immediately and irrevocably. We have redefined our concept of the visual by examining the 157 construction and use of models in scientific practice, by conceptualizing the elusive notion of representation, and by redefining our concept of the visualizable. This study demonstrates and vindicates philosophy's entanglement in the production of scientific discourse. 158

Bibliography

Achinstein, Peter. 1964. Models, analogies, and theories. Philosophy of Science 31:328-49. ------. 1972. Models and analogies: a reply to Girill. Philosophy of Science 39:235-40. Akeroyd, F. Michael. 1990. An Oscillatory Model of the Growth of Scientific Knowledge. British Journal for Philosophy of Science 41:407-14. Arbib, Michael A., and M. B. Hesse. 1986. The Construction of Reality. Cambridge: Cambridge University Press. Arrell, Douglas. 1987. What Goodman should have said about representation. Journal of Aesthetics and Art Criticism 46:41-49. Barnsley, Michael F. 1988. Fractals everywhere. San Diego: Academic Press. ------. 1990. The desktop fractal design handbook. San Diego: Academic Press. Baudrillard, Jean. 1983. Simulations, trans. P. Foss, P. Patton, and P. Beitchman. New York: Semiotexte. Benammar, Karim. 1990. The function and limitations of musical notation. Unpublished M.A. Thesis. The Pennsylvania State University. Bergé, Pierre, Y. Pomeau, and Ch. Vidal. 1986. Order within Chaos: towards a deterministic approach to turbulence, trans. L. Tuckerman. New York: John Wiley. Berlinski, David. 1975. Mathematical models of the world. Synthese 31:211-27. Black, Max. 1962. Models and Metaphors. Ithaca: Cornell University Press. Blinder, David. 1986. In defense of pictorial mimesis. Journal of Aesthetics and Art Criticism 45:19-27. 159

Blum, A. S. 1991. A better style of art: illustrations for nineteenth-century American zoology. Princeton: Princeton University Press. Bradie, Michael. 1980. Models, metaphors and scientific realism. Nature and System 2:3-20. Brown, Harold I. 1977. Objective knowledge in science and the humanities. Diogenes Spring 77:85-102. Bunn, James H. 1981. The Dimensionality of Signs, Tools, and Models. Bloomington: Indiana University Press. Burrough, Peter. 1985. Fakes, facsimilies, and facts: fractal models of geophysical phenomena. Science and Uncertainty, S. Nash, ed. IBM United Kingdom. Bushkovitch, A.V. 1977. The concept of model in scientific theory. International Logic Review 8:24-31. Calvino, Italo. 1974. Invisible Cities, trans. W. Weaver. New York: Harcourt Brace Jovanovitch. Carloye, Jack C. 1971. An interpretation of scientific models involving analogies. Philosophy of Science 38:562-69. Cartwright, Nancy. 1983. How the Laws of Physics Lie. Oxford: Clarendon Press. ------. 1989. Nature's Capacities and their Measurement. Oxford: Clarendon Press. Chattopadhyaya, D.P. 1982. Models and Metaphors in Arts, Science and Physics. In Mind, Language and Necessity, ed. K.K. Banerjee. Atlantic Highlands, NJ: Humanities Press. Churchland, Paul M., and C. A. Hooker, eds. 1985. Images of Science. Chicago: University of Chicago Press. Crawford, T. Hugh. 1993. An interview with Bruno Latour. Configurations 1(2):247-269. Cushing, James T. 1989. The justification and selection of scientific theories. Synthese 78:1-24. Cusumano, Joseph P. and M. T. Shakardy. Forthcoming. An experimental study of bifurcation and chaos in a dynamically-buckled magneto-elastic system. Fourteenth Biennial ASME Conference in Mechanical Vibration and Noise. 160

Dambska, Izydora. 1969. Modele et objet de la connaissance. Revue Internationale de Philosophie 23:34-43. Downes, Stephen M. 1992. The importance of models in theorizing: a deflationary semantic view. In PSA 1992 ed. D. Hull, M. Forbes, and S. Okruhlik, 1:142-53. East Lansing, Mich.: Philosophy of Science Association.

Dyke, C. 1990. Strange attraction, curious liaison: Clio meets chaos. Philosophical Forum 21(4):369-92. Falconer, Kenneth. 1990. Fractal geometry. Chichester: John Wiley and Sons. Fan, L. T., D. Neogi, and M. Yashima. 1991. Elementary introduction to spatial and temporal fractals. New York: Springer-Verlag. Feder, Jens. 1988. Fractals. New York: Plenum. Feyerabend, Paul K. 1981a. Realism, Rationalism and Scientific Method (Philosophical Papers vol.1). Cambridge: Cambridge University Press. ------. 1981b. Problems of Empiricism (Philosophical Papers vol.2). Cambridge: Cambridge University Press. ------. 1988. Against Method, rev. ed. London: Verso. Fisher, Roland. 1981. Matter's mastermind: the model-making brain. Diogenes 116:18-39. Forge, John. 1983. Towards a theory of models in physical science. Philosophy Research Archives 8:321-38. Freudenthal, Hans, ed. 1961. The concept and the role of the model in mathematics and natural and social sciences. Dordrecht: Reidel. Frey, Gerhard. 1969. The use of the concepts 'isomorphic' and 'homomorphic' in epistemology and the theory of science. Ratio 11:1-13. Geertz, Clifford. 1983. Local Knowledge. New York: Basic Books. George, Wilma. 1982. Darwin. London: Fontana. Gibbard, Allan, and H. R. Varian. 1978. Economic models. The Journal of Philosophy 75(11):664-77. 161

Giere, Ronald N. 1985. Constructive realism. In Churchland and Hooker 1985. ------. 1988. Explaning Science: a cognitive approach. Chicago: University of Chicago Press. Gilbert, Scott F. 1991. Epigenetic Landscaping: Waddington's use of cell fate bifurcation diagrams. Biology and Philosophy 6:135-54. Gilbert, Scott F. and Wade, Michael J. 1988. Laboratory models, causal explanation and group selection. Biology and Philosophy 3:67-96. Girill, T.R. 1971. Formal models and Achinstein's "analogies". Philosophy of Science 38:96-104. ------. 1972. Analogies and models revisited. Philosophy of Science 39:241-44. Gleick, James. 1988. Chaos: Making a New Science. New York: Penguin. Goodman, Nelson. 1976. Languages of Art, 2nd ed. Indianapolis: Hackett. ------. 1978. Ways of Worldmaking. Indianapolis: Hackett. Griesemer, James R. 1991. Must scientific diagrams be eliminable? The case of path analysis. Biology and Philosophy 6:155-80. Guyon, Etienne, and H. E. Stanley, eds. 1991. Fractal forms. Amsterdam: Elsevier North-Holland. Hacking, Ian. 1983. Representing and Intervening. Cambridge: Cambridge University Press. Hall, Stephen S. 1992. Mapping the next millenium. New York: Random House. Harré, Rom. 1960. Metaphor, Model and mechanism. Proceedings of the Aristotelian Society. 60:101-22. ------. 1988a. Where Models and Analogies Really Count. International Studies in the Philosophy of Science 2(2):118- 33. ------. 1988b. Realism, Reference and Theory. Philosophy (Supp) 24:53-68. 162 Hatten, Robert S. 1982. Toward a Semiotic Model of Style in Music: Epistemological and Methodological Bases. Unpublished Ph.D. Dissertation, Indiana University. ------. 1993. Metaphor in music. Musical Signification, ed. Eero Tarasti. The Hague: Mouton. Hausman, Carl R. 1993. Metaphor and realism in science. Unpublished manuscript. Hayles, N. Katherine. 1990. Chaos bound. Ithaca: Cornell University Press. Hesse, Mary B. 1966. Models and analogies in science. South Bend, Ind.: University of Notre Dame Press. ------. 1967. Models and analogies in science. In The Encyclopedia of Philosophy, ed. P. Edwards, 1:9-15. New York: Routledge. Hoffmann, James R. 1990. How the models of chemistry vie. In PSA 1990 eds. A. Fine and J. Leplin, 1:405-19. East Lansing, Mich.: Philosophy of Science Association. Hoffman, James R., and P. A. Hofmann. 1992. Darcy's law and structural explanation in hydrology. In PSA 1992, eds. D. Hull, M. Forbes, and K. Okruhlik, 1:23-35. East Lansing, Mich.: Philosophy of Science Association. Holden, Arun V., ed. 1986. Chaos. Princeton: Princeton University Press. Holden, Arun V. and Muhammad. 1986. In Holden 1986. Horz, Heinz. 1978. Philosophical problems of modelling. Revolutionary World 27:28-36. Hunt, G. M. K. 1987. Determinism, predictability and chaos. Analysis 47:129-33. Johnson, Mark. 1981. Philosophical perspectives on metaphor. Minneapolis: University of Minnesota Press. Johnson, Mark and G. Lakoff. 1980. Metaphors We Live By. Chicago: University of Chicago Press. Kargon, Robert. 1969. Model and analogy in Victorian science: Maxwell's critique of the French physicists. Journal of the History of Ideas 30:423-36.

Kellert, Stephen H. 1993a. In the wake of chaos. Chicago: University of Chicago Press. 163

------. 1993b. Chaos Theory. In PSA 1992 ed. D. Hull, M. Forbes, and K. Okruhlik, vol. 2. East Lansing, Mich.: Philosophy of Science Association.

Kockelmans, Joseph J. 1987. On the problem of truth in the sciences. Proceedings of the American Philosophical Association Supp. 61:5-26. Kuhn, Thomas S. 1970. The Structure of Scientific Revolutions, 2nd ed. Chicago: University of Chicago Press. Krohn, Roger. 1991. Why are graphs so central in science? Biology and Philosophy 6:181-203. Latour, Bruno. 1987. The Enlightenment without the critique: a word on Michel Serres' philosophy. In Contemporary French Philosophy ed. Paul Griffith. Cambridge: Cambridge University Press. ------. 1990. Postmodern? No, Simply A-Modern! Steps Towards an Anthropology of Science. Studies in History and Philosophy of Science 21(1):145-71. Larkin, J. and H. Simon. 1987. Why a diagram is (sometimes) worth ten thousand words. Cognitive Science 11:65-100. Lauwerier, Hans. 1991. Fractals: endlessly repeated figures. Princeton: Princeton University Press. Leatherdale, W. H. 1974. The role of analogy, model and metaphor in science. Amsterdam: North-Holland. Lloyd, Elisabeth A. 1986. Thinking about models in evolutionary theory. Philosophica 37:87-100. ------. 1987. Confirmation of Ecological and Evolutionary Models. Biology and Philosophy 2:277-93. Lorenz, Edward. 1963. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20:130-41. Lynch, Michael. 1985. Discipline and the material form of images: an analysis of scientific visibility. Social Studies of Science 15:37-66. ------. 1988. The externalized retine: selection and mathematization in the visual documentation of objects in the life sciences. Human Studies 11:201-234.

------. 1991. Science in the age of mechanical reproduction: 164 moral and epistemic relations between diagrams and photographs. Biology and Philosophy 6:205-26. Lynch, Michael, and S. Woolgar. 1988. Introduction: Sociological orientations to representational practice in science. Human Studies 11:99-116. Maienschein, Janie. 1991. From presentation to representation in E. B. Wilson's The Cell. Biology and Philosophy 6:227-54. Makower, Joel, ed. 1986. The Map Catalog. New York: Random House. Mandelbrot, Benoit. 1982. The Fractal Geometry of Nature. New York: W. H. Freeman. Maynard, Patrick. 1972. Depiction, Vision, and Convention. American Philosophical Quaterly 9(3):243-50. McAllister, James W. 1989. Truth and beauty in scientific reasoning. Synthese 78:25-51. Mitchell, W. J. T. 1986. Iconology: Image, Text, Ideology. Chicago: University of Chicago Press. Monmonier, Mark. 1991. How to lie with maps. Chicago: University of Chicago Press. Moquin, Brian. 1993. Personal Communication. Nersessian, Nancy. 1988. Reasoning from imagery and analogy in scientific concept formation. In PSA 1988 eds. A. Fine and J. Leplin, 2:393-405. East Lansing, Mich.: Philosophy of Science Association. Nicolis, Gregoire, and Ilya Prigogine. 1989. Exploring Complexity. New York: W. H. Freeman. O'Hara, Robert J. 1988. Diagrammatic classification of birds, 1819-1901: views of the natural system in 19th- century british ornithology. Acta XIX Congressus Internationalis Ornithologici, ed. H. Ouellet. Ottawa: National Museum of Natural Sciences, 2746-2759, 1988. ------. 1991. Representations of the natural system in the nineteenth century. Biology and Philosophy 6:255-274. Oldershaw, Robert L. 1980. Hierarchical modelling in the sciences. Nature and System 2:189-98.

Peitgen, Heinz-Otto, and Richter. 1986. The beauty of 165 fractals. New York: Springer Verlag. Peitgen, Heinz-Otto, H. Jurgens and D. Saupe. 1992. Chaos and fractals. New York: Springer Verlag. Peetz, Dieter. 1987. Some current philosophical theories of pictorial representation. British Journal of Aesthetics 27:227-37. Pickover, Clifford A. 1991. Computers and the imagination. New York: St. Martin's Press. Porksen, Uwe. 1992. Visualisation: drawings, screen projections, graphs. Unpublished synopsis of lectures at the Pennsylvania State University. Post, H. R. 1971. Correspondence, invariance and heuristics: in praise of conservative induction. Studies in the History and Philosophy of Science 2(3):213-55. Prigogine, Ilya, and I. Stengers. 1984. Order out of Chaos. New York: Bantam. Quine, W. V. O. 1963. From a logial point of view. New York: Harper and Row. Ramsey, Jeffry L. 1992. Towards and expanded epistemology for approximations. In PSA 1992, ed. D. Hull, M. Forbes, and K. Okruhlik, 1:154-64. East Lansing, Mich.: Philosophy of Science Association. Redhead, Michael. 1980. Models in physics. British Journal for Philosophy of Science 31:145-63. Richardson, Jacques, ed. 1984. Models of Reality: Shaping Thought and Action. Mt. Airy, Md.: Lomond Books. Richardson, Robert C. 1986. Models and scientific explanation. Philosophica 37:59-72. Ritchin, Fred. 1990. In our own image: the coming revolution in photography. New York: Aperture. Robinson, Jenefer. 1979. Some remarks on Goodman's language theory of pictures. British Journal of Aesthetics 19(1):63-75. Root-Berstein, Robert S. 1985. Visual thinking: the art of imagining reality. Transactions of the American Philosophical Society 75:50-67.

Ruelle, David. 1991. Chance and chaos. Cambridge: Cambridge 166 University Press. Ruse, Michael. 1973a. The value of analogical models in science. Dialogue 12:246-53. ------. 1973b. The nature of scientific models: formal vs material analogy. Philosophy and the Social Sciences 3:63- 80. Sakabe, Megumi. 1985. "Modoki" - sur la tradition mimetique au Japon. Acta Institutionis Philosophiae et Aesthetica 3:95-105. Schier, Flint. 1986. Deeper into pictures: an essay on pictorial representation. Cambridge: Cambridge University Press. Schopf, Thomas J. M. 1972. Models in Paleobiology. San Fransisco: Freeman & Cooper. Schroeder, Manfred. 1991. Fractals, chaos, power laws: minutes from an infinite universe. New York: W. H. Freeman. Serres, Michel. 1969. Hermes I - La communication. Paris: Minuit. ------. 1972. Hermes II - L'interference. Paris: Minuit. ------. 1974. Hermes III - La traduction. Paris: Minuit. ------. 1977. Hermes IV - La distribution. Paris: Minuit. ------. 1980. Hermes V - Le passage du Nord-Ouest. Paris: Minuit. ------. 1982. Hermes: Literature, Science, Philosophy. Trans., eds. J.V.Harari and D.F.Bell. Baltimore: The Johns Hopkins University Press. ------. 1985. Les Cinq Sens Paris: Grasset. ------. 1987. L'anthropologie des sciences: un programme pour la philosophie? (interview). Etudes Philosophiques 14:147-71. ------. 1989. Detachment. G. James and R. Federman, trans. Athens: Ohio University Press. ------. 1992. Eclaircissements. Paris: Francois Bourin.

Sokolowski, Robert. 1977. Picturing. Review of Metaphysics 167

31:3-28. Stachowiak, Herbert. 1980. Der Modellbegriff in der Erkenntnistheorie. Zeitschrift fur Allgemeine Wissenschaftstheorie XI(1):53-68. Stone, Mark A. 1989. Chaos, prediction and Laplacean determinism. American Philosophical Quaterly 26(2):123-31. Suppes, Patrick. 1960. A comparison of the meaning and uses of models in mathematics and the empirical sciences. Synthese 12:287-301. ------. 1988. Representation theory and the analysis of structure. Philosophy and Nature 25:254-68. Swanson, J. W. 1966. On models. British Journal for Philosophy of Science 17(4):297-311. Taylor, Peter J., and Blum, Ann S. 1991a. Pictorial representation in biology. Biology and Philosophy 6:125- 34. ------. 1991b. Ecosystems as circuits: diagrams and the limits of physical analogies. Biology and Philosophy 6: 275- 94. Thom, Rene. 1975. D'un modele de la science a une science des modeles. Synthese 31:359-74. Tibbetts, Paul. 1988. Representation and the realist- constructivist controversy. Human Studies 11:117-32. Tiles, Mary. 1988. Scientific dreamspace: symbolic forms and scientific theories. International Studies in the Philosophy of Science 2(2):189-204. Tufte, Edward R. 1990. Envisioning Information. Cheshire, Conn.: Graphics Press. Van Fraassen, Bas C. 1980. The Scientific Image. Oxford: Clarendon Press. ------. 1993. PSA Presidential address. In PSA 1992, vol. 2, eds. D. Hull, M. Forbes, and K. Okruhlik. East Lansing, Mich.: Philosophy of Science Association. Voss, Richard. 1985. Random Fractal Forgeries: from mountains to music. In Science and Uncertainty, ed. S. Nash. London: IBM United Kingdom. 168

Waldrop, M. Mitchell. 1992. Complexity: the emerging science at the edge of order and chaos. New York: Simon & Schuster. Wittgenstein, Ludwig. 1958. Philosophical Investigations, trans. G. E. M. Anscombe. Oxford: Basil Blackwell. ------. 1974. On Certainty, trans. D. Paul and G. E. M. Anscombe. Oxford: Basil Blackwell. Xenakis, Iannis. 1971. Formalized Music. Bloomington: Indiana University Press. ------. 1985. Arts/Sciences: Alloys. New York: Pendragon Press. Vita

Karim Benammar was born on May 4, 1966, in Regensburg, Germany. He attended the European School in Bergen, Netherlands, where he graduated with a European Baccalaureate in 1984. In June 1987, he received a B.A. with First Class Honors in Philosophy from Sussex University, Brighton, Great-Britain. Karim Benammar came to The Pennsylvania State University in the Fall of 1987. During 1988-89, he took a leave of absence to travel through Asia, and taught English in Taiwan and Japan. He received his M.A. in May 1990. He was awarded a Fellowship from the Office for the Promotion of Interdisciplinary Studies during 1990-91, and studied electronic music and the system of classical musical notation. During 1991-92, he was awarded an Edwin Erle Sparks Fellowship. Karim Benammar has given papers at major conferences on the theory of truth, on Michel Serres and ecology, and on the idea of community in modern French thought. He has accepted a Fellowship from the Japanese Ministry of Education (Mombusho), and will be a research fellow at Kyoto University from 1993 to 1995; he intends to study contemporary Japanese philosophy.