Stephen Wolfram's remarkable book A New Kind of Science establishes that very complex systems can emerge from the repetition or iteration of simple principles. The experiments that confirm this result - at once basic and profound - are performed on computers, and they yield visual arrays of striking variety, of varying complexity, and of great beauty. Indeed - and this is really the point - fairly simple procedures can yield patterns of a complexity that is analogous to nature or the universe. Wolfram, I take it, establishes that it is possible that underlying nature and serving as its explanation, there could be a single systematic principle. This result is, we might say, Platonic, though it takes a form that Plato did not envision. More closely, we might say that Wolfram's result suggest that the universe could in some sense "consist of" formulae that are expressible in very simple mathematics. The ancient Pythagoreans, whose own views were secret and arcane, but who apparently held the universe to consist of numbers, or to be underlain by harmonies out of the range of our hearing, constructed a system similar to Wolfram's. And the Pythagoreans in turn had an influence on the later Greeks. In the Metaphysics, Aristotle describes the Pythagorean view as follows: "they thought that the principles of mathematics were the principles of all things. . . justice being such and such a modification of numbers, soul and reason being another. . . . Seeing, further, that the properrties and ratios of the musical consonances were expressible in numbers in numbers, and that indeed all other things seemed to be modelled in their nature upon numbers, they took numbers, they took numbers to be the whole of reality, the elements of numbers to be the elements of all existing things, and the whole of heaven to be a musical scale and a number."

This idea that the universe consists of ratios, or however it should ultimately be expressed, is not exactly Wolfram's idea, which takes computer modeling to supercede mathematics and ultimately to be much cleaner and clearer, indeed almost incredibly simple. Eventually, he hints, the basic principles of nature, such as e=mc2, could be replaced with simple rules for generating complex phenomena. But in the idea that reality ultimately reduces to a set of simple principles, the Pythagoreans have much in common with Wolfram's approach. And it is worth noting that the Pythagoreans' approach is basically what we would call religious and moral: that probably the notion that the universe was governed by numerically-expressible harmonies led to various hermetic practices of worship and that justice was held to be itself a harmony, which is the way, finally, that Plato and many other philosophers have conceived it.

Let us consider the simplest of the systems that Wolfram constructs, the systems that brought Wolfram to his initial discoveries: cellular automata. The simplest systems use a grid in which every square must be either black or white. They start with an initial condition - say a single black square - and then fill the grid according to a specific principle. It turns out the cellular automata of this sort can be exhaustively specified with eight rules, though often the principle can be formulated in a much simpler way in natural language.

Each of the eight rules specifies whether the center cell should be black or white in the next row down, given the colors of its neighbors in the present row. The "experiment" consists running this procedure over a certain number of steps, perhaps thousands. For example, if the rules specify that a cell should be black if either of its neighbors is black in the previous step, a simple triangular pattern is generated. But other rules yield more complex patterns, including "fractal"-type patterns that consist of intricately nested repetitive patterns in various sizes. Quite surprisingly, however, some sets of rules generate patterns which seem actually random: which over many thousands of steps never settle into a recognizable or graspable form. In fact, computer analyses of some of these patterns suggest that they really are random, that there is no pattern hidden in the form, though there are recognizable discreet structures which appear and disappear as the pattern proceeds.

The basic mode of analysis of these patterns, for Wolfram, is the eyeball. He sorts cellular automata into four categories which are based essentially simply on looking at trhe different sorts of forms. In class 1, the automaton quickly generates a uniform state: the grid turns all black or all white. In class 2, simple repeating structures are generated . Class 3 automata produce seemingly random structures, though discrete forms are also generated. In particular, triangles of various sizes appear. And in class 4 automata, randomness is replete with structure: the patterns resemble braided curtains or hanging macrame of extreme complexity: knots and tapestries that sweep downward in unpredictable ways. Class 4 automata are, for my money, the beautiful autmata, and they suggest that beauty can be understood informally as order amidst complexity, which is one of the traditional strategies for the definition of beauty going back to the Greeks.

Though there are some mathematically interesting differences in the classes of automata, the groupings are, as I say, informal, based essentially simply on how the structures look. But this is actually an important way of ordering them, or one might even say it is a scientific way of oirdering them. After all, our sense of the order of the universe is based on experience of it. And the universe displays a kind of extreme combination of randomness and pattern. This has led, for example, to one form of the debate over the existence of God. Those who use what is called the "argument from design" emphasize the degree of order that is displayed in the universe, which is indeed conspicuous. The organization of planetary systems, or organisms, for example, is obvious, and suggests to some folks that such systems are ordered intentionally. And also there may seem to be a kind of moral order in the universe, in which goodness is rewarded and evil punished, or in which vice destroys and virtue nurtures. On the other hand there is indeed great evil and disorder in the universe as well: there are apparently chaotic systems such as cloud formations, and of course the moral order seems anything but perfect, and often random. One's sense of what sort of place the universe is, one might say, effects one's beliefs about its source in randomness or intelligence. And it also effects one's ideas about its beauty: one can find beauty in order or in chaos, but more relevantly and commonly, one finds beauty in certain combinations of order and chaos. Perfectly ordered systems are merely boring, and perfectly chaotic systems are merely bewildering. But combined systems are arenas of desire in which one might intervene to impose order or attack it, in which one might introduce organizations or contribute to their disintegration. Such systems, we might say, are political: they can yield to transformations. In systems of extreme order, we long for disintegration, while in systems of extreme disorder we long for organization. Thus the sort of systems in which longing can be satisfied are systems of mixed order and disorder.

And the procedures by which we judge whether a system is well-ordered or not are essentially informal: they consist simply of experiencing the system in relation to ourselves rather than, for example, computer modeling. That is why Wolfram's ultimately informal taxonomy of cellular automata is important: because it corresponds to our experiences of order and disorder in the real world. And one compelling feature of cellular automata is that they can, shockingly, and on the basis of eight rules, yield systems that remind us of worlds, that create patterns of such intricate complexity - or combinations of order and chaos - that they resemble environments or natures. This of course suggests that underlying the beauty of the universe there is a single principle by which structures are generated and by which they peter out or disperse. In cellular automata where the initial state is random, and which grow in two dimensions rather than one, some rules generate snowflake-like structures of complexity and variety amid order. Other rules lead to uniform states, and still others to incredibly elaborate combinations of order and chaos that resemble worlds. Nor are such effects peculiar to cellular automata; Wolfram is able to generate similar results with a variety of systems. One rather shocking fact, informally expressed, is that more complex rules and more complex initial conditions do not necessarily yield more complex patterns over the long haul; indeed some of the most complex systems emerge from very simple rules and initial conditions. Or more accurately, it appears that extremely simple rules yield extremely simple patterns, whereas slightly more elaborate rules can yield arrays as complex as any that can be constructed. Plato would have loved this result, insofar as it establishes that there might be a coherence, a single form, that generates all experiencable structure, and that is the "Form" or the mathematical structure of the universe as a whole, and which is the principle of beauty. Wolfram's results, which after all consist only of computer models, cannot establish that the universe is this sort of Platonic or Pythagorean arena: but what I take them to establish is that this is possible. This is a very serious limitation, and one thing that should be obvious is that the universe may be produced by the iteration of simple rules, but it may be produced by very complex rules, or the idea of rules might itself be just the wrong metaphor or it might simply be false as an explanation of the universe. Even more seriously, and though Wolfram's results are extremely suggestive, as he builds complex forms like unto those of nature by the repetitive application of simple rules, it seems to me that there is simply no way in which the hypothesis could be clearly demonstrated to be true. In the absence of a dialogue with god, we will not be able to get a diagram of the rules on which the universe is based, and any such diagram will remain hypothetical, though the actual generation of forms might turn out to be extremely suggestive or compelling. Finally, though, the preference for the simplest rule that serves our explanatory purposes - a preference to which all of Wolfram's work is dedicated - is aesthetic rather than strictly empirical. The preference is a priori: something we bring to rather than derive from the data. It will allow clean, and clear and graspable explanations, so the aesthetic preference has cognitive effects. But it is not itself an empirical result. And we might say that finally, Wolfram's analysis is, like the Pythagoreans', moral, that it asserts a kind of balance and simplicity and harmony in the world that satisfies our thirst for justice. Thus, Wolfram's system is at once scientific, aesthetic, and moral: it connects the ultimate values of the true, the beautiful, and the good into a coherent foundation.

And the ferocity with which Wolfram has pursued simplicity- in complexity through experiments that sometimes generate billions of patterns are a testimony, finally, to the human longing for order in chaos, for a place of peace at the bottom of a chaotic experience.