Last summer I read the book A New Kind of Science, by Stephen Wolfram (Stephen Wolfram. 2002. A New Kind of Science. Champaign Il: Wolfram Media, Inc). It is a long book, if you exclude the preface and index there are 1197 pages. (However it isn’t quite as bad as it seems, the main text is only 846 pages and there are many black and white diagrams).
In this article I will attempt to explain the basic ideas, but I can only give a rough overview of the book.
The book is primarily about models used in fields such as Biology, Physics and Astronomy and new way of producing models that Wolfram discovered. To understand the book, you need to understand what is a model is, and that is best done with examples. Scientific models first appeared centuries ago in the form of mechanical devices and mathematical equations used to predict the motion of the planets. Early models were not very accurate; but they improved over time. Astronomers now use models based on Isaac Newton’s equations. Turning these equations into a prediction is not straightforward. Getting an exact solution is possible, but only under highly artificial conditions. For example, you can solve the equations if for a universe with exactly two objects. No one knows how to get an exact solution for any reasonable set of conditions.
An obvious question is: if this is so, how do we know that Mars will be in opposition in August 2003 or Venus will transit the sun in 2004? The answer resolves around the word exact. While we can’t get exact solutions, we can get approximate solutions (using so called perturbation methods), and these approximation solutions are very close. The difference is extremely small over short time frames (say a few years), but there is a slight error that gets larger over time. After a few billion years this error is very large; it is not possible to use this type of model to make predictions over such long time spans.
If we want to model an entire galaxy, typically we don’t use perturbation methods. It is very difficult to follow the trajectory of each star in a large galaxy. Instead astronomers often use a statistical approach; they determine the average velocity of stars. There is a set of equations (the Jeans equations) that computes the average velocity if you know the average density. This relationship can be used in reverse to predict the mass within a galaxy.
When astronomers first used this approach to calculate the mass of galaxies, they obtained a mass that didn’t agree with the mass obtained by counting stars. This leads to the “missing mass” problem. This missing mass has persisted even as astronomers have improved their techniques. Astronomers generally assume there is some mass that we can’t detect (so called “dark matter”) which accounts for this difference.
(A set of equations Albert Einstein developed give more accurate results particularly in large gravitational fields. These equations are much harder to use, which is why Newton’s equations are normally used).
While models like these have been very successful, they are not perfect. Many scientific models require approximation techniques; statistical methods often are based on assumptions that may or may not be correct. Can we build models that don’t use approximation or statistics? Yes we can. To understand how, we need to look at something called a cellular automata (abbreviated as CA).
What is a CA? Some of you may have heard of the “Game of Life,” a computer program that produced interesting patterns. It was incorporated into computer programs beginning in the late 1970’s. A few years later it found its way into screen savers. You can understand how it works by considering a chessboard. Begin by placing chess pieces in a random pattern. Then look at each square one by one. Count the number of pieces on the 8 squares adjacent to a particular square. If this count is zero, 1, 4, 5, 6, 7 or 8 call it “0”. If this count is 2 and the central square is empty call it “0.” Otherwise call it “1.” Collect these zeros and ones for all 64 squares, produce a new pattern, repeat the process. This is tedious if you try it by hand, but it is easy to program a computer to do it.
Wolfram wasn’t that interested in the Game of Life, but he looked at a collection of 256 other CAs, which he named rule 0, rule 1 and so on up to rule 255. Rule 0 always forms a uniform sheet of black pixels. Rule 4 usually forms a series of black lines (the exact location of the lines will vary). Rule 22 usually produces complex patterns, but in certain cases it produces a highly symmetrical pattern (a fractal called the Sierpinski Gasket). Rule 30 always produces very complex patterns; these patterns include many triangles of varying sizes, but otherwise the output looks totally random. This randomness is quite unexpected.
Wolfram examined a number of other systems. While these other systems are not CAs by a strict definition, they were similar. Wolfram lumped CAs and these similar systems under the label “simple programs.” Those of you who more mathematically included should realize that this term covers a range of possibilities, but excludes anything based on a differential equation or any other type of continuous equation.
Even after examining many simple programs over a period of many years, Wolfram found they all belong to one of four categories. Some like rule 0 produced a regular pattern. Some like 4 produced lines. Some like rule 22 produced fractals and some like rule 30 produce what appears to be randomness.
You might be thinking, what does this have to do with scientific models? These systems make patterns, but you can also think of them as performing a computation. They take a set of input (also called initial conditions), apply a set of rules and produce a set of output. If the output helps us understand a scientific phenomenon, it can be considered a model. Wolfram found simple programs that produced complex patterns, like rule 30, could be used to construct models.
Wolfram produced a number of models based on simple programs in various areas of Biology, Economics and Physics and other areas. Wolfram argues that conventional approaches do not handle complexity very well; that’s when models based on simple programs offer an alternative. He claims that it is difficult to verify the results of the standard approach when applied to a complex phenomena. In such cases it can be difficult or impossible to prove that the results of the model actually agree with what the original equation suggests. However this is not an issue with models based on one of Wolfram’s simple programs. (Wolfram devotes many pages to complexity, but a complete discussion is beyond the scope of this article).
So what’s the verdict? Is the technique Wolfram proposes worthwhile?
Wolfram described a wide range of phenomena including fluid flow, the shape of mollusc shells and development of snowflakes using simple programs. Most of the models seemed reasonable. His attempt to apply network systems (a type of simple program) to quantum mechanics was less convincing. (Wolfram admitted as much in an interview he gave to ABC News reporter Robert Krulwich).
To really answer the question we must evaluate some CA models. How do you evaluate a model? You need to ask several questions. 1) Does it make predictions that agree with reality? 2) Does it predict previously unknown phenomena? 3) Is it easier to use than competing models?
We don’t have enough models to answer these questions. Wolfram supplied a number of models, but if the technique is useful, more will appear. I suspect this will happen. In five or ten years if a number of models based on Wolfram’s simple programs have been produced, we will be a better position to judge how they work in general.
However I don’t think CAs will ever completely displace more conventional approaches that have worked well in many areas. There is no reason to discard approaches that work.
In conclusion, Wolfram’s book is full of worthy ideas. However they could have been expressed in much shorter book. The main text is easy to read even if you have a limited scientific background, but it gets a little tedious at times. Some of the notes assume specialized knowledge, however it is not necessary to read the notes. Only time will tell whether Wolfram’s ideas will become part of mainstream science.