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Is the Universe a Computer?

1.

Everyone knows that electronic computers have enormously helped the work of science. Some scientists have had a grander vision of the importance of the computer. They expect that it will change our view of science itself, of what it is that scientific theories are supposed to accomplish, and of the kinds of theories that might achieve these goals.

I have never shared this vision. For me, the modern computer is only a faster, cheaper, and more reliable version of the teams of clerical workers (then called “computers”) that were programmed at Los Alamos during World War II to do numerical calculations. But neither I nor most of the other theoretical physicists of my generation learned as students to use electronic computers. That skill was mostly needed for number crunching by experimentalists who had to process huge quantities of numerical data, and by theorists who worked on problems like stellar structure or bomb design. Computers generally weren’t needed by theorists like me, whose work aimed at inventing theories and calculating what these theories predict only in the easiest cases.

Still, from time to time I have needed to find the numerical solution of a differential equation,1 and with some embarrassment I would have to recruit a colleague or a graduate student to do the job for me. It was therefore a happy day for me when I learned to use a program called Mathematica, written for personal computers under the direction of Stephen Wolfram. All one had to do was to type out the equations to be solved in the prescribed code, press shift-enter, and, presto, the answer would pop up on the monitor screen. The Mathematica user’s manual now sits on my desk, so fat and heavy that it does double duty as a bookend for the other books I keep close at hand.

Now Wolfram has written another book that is almost as heavy as the Mathematica user’s manual, and that has attracted much attention in the press. A New Kind of Science describes a radical vision of the future of science, based on Wolfram’s long love affair with computers. The book’s publisher, Wolfram Media, announces “a whole new way of looking at the operation of our universe” and “a series of dramatic discoveries never before made public.” Wolfram claims to offer a revolution in the nature of science, again and again distancing his work from what he calls traditional science, with remarks like “If traditional science was our only guide, then at this point we would probably be quite stuck.” He stakes his claim in the first few lines of the book: “Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation….”

Usually I put books that make claims like these on the crackpot shelf of my office bookcase. In the case of Wolfram’s book, that would be a mistake. Wolfram is smart, winner of a MacArthur Fellowship at age twenty-two, and the progenitor of the invaluable Mathematica, and he has lots of stimulating things to say about computers and science. I don’t think that his book comes close to meeting his goals or justifying his claims, but if it is a failure it is an interesting one.

The central theme of the book is easily stated. It is that many simple rules can lead to complex behavior. The example that is used repeatedly to illustrate this theme is a favorite toy of complexity theorists known as the cellular automaton, so I will have to say a bit about what cellular automata are.

Take a piece of white graph paper that has been cross-hatched into little squares. These are the “cells.” Blacken one or more of the cells in the top row, chosen any way you like, leaving all the others white. This is your input. Now blacken some cells in the second row, according to some fixed rule that tells you to make any cell black or leave it white depending on the colors of its three neighboring cells in the first row (that is, the cells in the first row that are either immediately above the cell in the second row or one cell over to the right or left.) Then use the same rule, whatever it is, automatically to color each cell in the third row according to the colors of its three neighboring cells in the second row, and keep going automatically in the same way to the rows below. The coloring rule used in this way is an elementary cellular automaton.

This may seem like a solitaire variation on tick-tack-toe, only not as exciting. Indeed, most of the 256 possible2 elementary cellular automata of this sort are pretty boring. For instance, consider rule 254, which dictates that a cell is made black if the cell immediately above it, or above it and one space over to the left or right, is black, and otherwise it is left white. Whatever the input pattern of black cells in the top row, the black cells will spread in the rows below, eventually filling out an expanding black triangle, so that the cells in any given column will all be black once you get to a low-enough row.

But wait. Wolfram’s prize automaton is number 110 in his list of 256. Rule 110 dictates that a cell in one row is left white if the three neighboring cells in the row above are all black or all white or black-white-white, and otherwise it is made black. Figure A shows the result of applying this rule twenty times with a very simple input, in which just one cell is made black in the top row. Not much happening here. Wolfram programmed a computer to run this automaton, and he ran it for millions of steps. After a few hundred steps something surprising happened: the rule began to produce a remarkably rich structure, neither regular nor completely random. The result after 700 steps is shown in Figure B. A pattern of black cells spreads to the left, with a foamy strip furthest to the left, then a periodic alternation of regions of greater and lesser density of black cells which moves to the right, followed by a jumble of black and white cells. It is a dramatic demonstration of Wolfram’s conclusion, that even quite simple rules and inputs can produce complex behavior.

Wolfram is not the first to have worked with cellular automata. They had been studied for decades by a group headed by Edward Fredkin at MIT, following the ground-breaking work of John Von Neumann and Stanislas Ulam in the 1950s. Wolfram is also not the first to have seen complexity coming out of simple rules in automata or elsewhere. Around 1970 the Princeton mathematician John Horton Conway invented “The Game of Life,” a two-dimensional cellular automaton in which cells are blackened according to a rule depending on the colors of all the surrounding cells, not just the cells in the row above. Running the game produces a variety of proliferating structures reminiscent of microorganisms seen under a microscope. For a while the Game of Life was dangerously addictive for graduate students in physics. A decade later another mathematician, Benoit Mandelbrot, the inventor of fractals, gave a simple algebraic prescription for constructing the famous Mandelbrot set, a connected two-dimensional figure that shows an unbelievable richness of complex detail when examined at smaller and smaller scales.

There are also well-known examples of complexity emerging from simple rules in the real world. Suppose that a uniform stream of air is flowing in a wind tunnel past some simply shaped obstacle, like a smooth solid ball. If the air speed is sufficiently low then the air flows in a simple smooth pattern over the surface of the ball. Aerodynamicists call this laminar flow. If the air speed is increased beyond a certain point, vortices of air appear behind the ball, eventually forming a regular trail of vortices called a “Von Karman street.” Then as the air speed is increased further the regularity of the pattern of vortices is lost, and the flow begins to be turbulent. The air flow is then truly complex, yet it emerges from the simple differential equations of aerodynamics and the simple setup of wind flowing past a ball.

What Wolfram has done that seems to be new is to study a huge number of simple automata of all types, looking specifically for those that produce complex structures. There are cellular automata with more than two colors, or with coloring rules like the Game of Life that change the colors of cells in more than one row at a time, or with cells in more than two dimensions. Beyond cellular automata, there are also automata with extra features like memory, including the Turing machine, about which more later. From his explorations of these various automata, Wolfram has found that the patterns they produce fall into four classes. Some are very simple, like the spreading black triangle in the rule 254 elementary cellular automaton that I mentioned first. Other patterns are repetitive, such as nested patterns that repeat themselves endlessly at larger and larger scales. Still others seem entirely random. Most interesting are automata of the fourth class, of which rule 110 is a paradigm. These automata produce truly complex patterns, neither repetitive nor fully random, with complicated structures appearing here and there in an unpredictable way.

So what does this do for science? The answer depends on why one is interested in complexity, and that depends in turn on why one is interested in science.

2.

Some complex phenomena are studied by scientists because the phenomena themselves are interesting. They may be important to technology, like the turbulent flow of air past an airplane, or directly relevant to our own lives, like memory, or just so lovely or strange that we can’t help being interested in them, like snowflakes. Unfortunately, as far as I can tell, there is not one real-world complex phenomenon that has been convincingly explained by Wolfram’s computer experiments.

Take snowflakes. Wolfram has found cellular automata in which each step corresponds to the gain or loss of water molecules on the circumference of a growing snowflake. After adding a few hundred molecules some of these automata produce patterns that do look like real snowflakes. The trouble is that real snowflakes don’t contain a few hundred water molecules, but more than a thousand billion billion molecules. If Wolfram knows what pattern his cellular automaton would produce if it ran long enough to add that many water molecules, he does not say so.

Or take complex systems in biology, like the human nervous or immune systems. Wolfram proposes that the complexity of such systems is not built up gradually in a complicated evolutionary history, but is rather a consequence of some unknown simple rules, more or less in the way that the complex behavior of the pattern produced by cellular automaton 110 is a consequence of its simple rules. Maybe so, but there is no evidence for this. In any case, even if Wolfram’s speculation were correct it would not mean that the complexity of biological systems has little to do with Darwinian evolution, as Wolfram contends. We would still have to ask why organisms obey some simple rules and not other rules, and the only possible answer would be that natural selection favors those rules that generate the kind of complex systems that improve reproductive fitness.

  1. 1

    A differential equation gives a relation between the value of some varying quantity and the rate at which that quantity is changing, and perhaps the rate at which that rate is changing, and so on. The numerical solution of a differential equation is a table of values of the varying quantity, that to a good approximation satisfy both the differential equation and some given conditions on the initial values of this quantity and of its rates of change.

  2. 2

    The automaton must tell you the color of a cell in one row for each of the 2 x 2 x 2 = 8 possible color patterns of the three neighboring cells in the row above, and the number of ways of making these eight independent decisions between two colors is 28 = 256. In the same way, if there were 3 possible colors, then the number of coloring decisions that would have to be specified by an elementary cellular automaton would be 3 x 3 x 3 = 27, and the number of automata (calculated using Mathematica) would be 327 = 7625597484987.

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