The math described on this web page is Stephen Wolfram's, but the subsequent opinions about theology and society are mine. I am writing this web page because I am tired of hearing science attacked and undermined by people who do not understand or respect it.

One formulation of "intelligent design" can be summarized as follows:

	(1)	The world around us is filled with both order and complexity -- "ordered complexity," if you will; and
	(2)	such ordered complexity could not have arisen by accident.
Therefore
	(3)	there is, or was, some intelligence who designed this world.

If (3) turns out to be true, we may or may not someday have proof of it. (Some people say we already have proof, but I mean proof that is strong enough to convince everyone.) But if (3) turns out to be false, we still will never have definitive proof of it. Indeed, even if we someday finish filling in all the details in a scientific explanation of the universe, so detailed and so supported by evidence that it is indisputable, and even if it turns out that god doesn't show up anywhere in that explanation, we will still be unable to rule out the possibility that some creator built this universe but -- for reasons known only to him -- intentionally designed it in such a way as to hide his own part in its creation. He could have put the fossils under the mountains, and faked the age of everything so expertly that he could fool our carbon-dating and red-shift measurements. (Indeed, maybe he created us all this morning, and gave us fake memories of what happened yesterday and earlier.) Some have suggested that he is now sitting in a coffeeshop somewhere, reading newspaper accounts of the school board trials, and chuckling over how he fooled us. So I am not trying to disprove (3); I know I can't. This web page is concerned only with showing that (2) is not valid reasoning, and does not constitute a proof of (3).

I assert that statement (2) is mathematically just plain wrong -- and I assert that the wrongness of (2) is not merely opinion, but fact. The people who believe (2) are being rather unimaginative, but in addition to imagination I can offer evidence. Stephen Wolfram's book shows -- with mathematical examples -- how complex systems can arise from simple ones. I will sketch some of the ideas below.

A cellular automaton is a sort of self-propelled computer program. You specify the initial conditions and the transformation rules, and then you tell it to "go," and the program proceeds to do things on its own. Probably the best known example of a cellular automaton is the "Game of Life," devised by mathematician John Horton Conway in 1970 and popularized by mathematics journalist Martin Gardner in the same year. You can see an illustration from that game, to the left of this paragraph. The program operates on a two-dimensional array of squares. Each square is either black (alive) or white (dead, or empty), at each instant of time. The status of cell X at time n+1 is determined by the status of X and its immediate neighbors at time n.
      The rules are simple enough that they could easily be programmed into even the rather rudimentary computers that were available in 1970. Investigating the patterns of "Life" became a recreational pastime of many programmers. Nowadays, if you're interested in seeing more of this game, you don't have to be a computer programmer; you can just download one of the programs that someone else has already written for your personal computer.

Stephen Wolfram made himself famous (among scientists, at least) with his creation of Mathematica, a computer program widely used by scientists since 1988. (See illustration at right.) The program continues to improve with new editions. It (and other programs like it, though Mathematica is the leader in this genre) have made a new kind of mathematics possible: Mathematics has always been theorems and proofs, but now it is also experiments. With just a few minutes' work, a mathematician can now generate dozens or hundreds of examples and look at pictures of the results; this makes it easier to discover conceptual principles that underlie the examples and are shared by them.

But for many years Wolfram also worked on mathematical research, using such computer experiments. He learned in 1982 that complexity in mathematical systems can arise from very simple origins. He investigated this idea extensively for many years. He finally published some of his findings in 2002, in an enormous book titled A New Kind of Science. It was a science book which received mixed reviews from the scientific community -- not because there was anything wrong with Wolfram's science, but because of the slightly unconventional way that he published it: as a large, popular, expository, self-published book, rather than as a specialized, arcane article in a peer-reviewed journal.
      Wolfram did not publish the book for profit -- you can read the book for free online if you simply register. If you want a printed copy, don't bother printing it out yourself -- your costs won't be much less than what Wolfram is charging for a very nicely bound copy.

Each picture is generated by an extremely simple rule -- one that could easily occur "by accident" in nature, as it were -- but many of the pictures show incredible ordered complexity. For instance, a small portion of picture number 30 is shown at the right. You can see that its right half is very disordered, but some order is apparent on the left half. And the whole picture is very complicated; yet it is generated by a very simple rule.

In each picture, the top row has only one black square. In rows below that, each cell's behavior (black or white) is determined by the behavior of the three cells above it (directly above, diagonally left, and diagonally right). There are 2x2x2 = 8 possible behaviors of those three cells: black-black-black,  black-black-white, and so on. For each of those 8 combinations, the rule must specify either a black result or a white result in the new row. Thus, to specify a rule, we must specify 8 yes-or-no choices. The number of rules of this sort is therefore 2x2x2x2x2x2x2x2 = 256. The book's website includes a very nice one-page summary of this idea, including some pictures that illustrate rule 30, rule 90, and rule 254.

Another way to get complicated pictures from simple rules is to use points on a plane instead of black and white squares. Fractals have been popular among mathematicians since their study was made possible by modern computers around the mid-1970's; some of these immensely complicated pictures are generated by very simple rules. For instance, the Mandelbrot set, shown at right, consists of the set of points c in the complex plane for which the iteratively defined sequence

        z₀ = 0,     z_n+1 = z_n² + c

The works described above are experiments conducted on computers, but the ideas are mathematical. The certainty of mathematics is greater than that of any of the other sciences. In a chemistry or physics experiment, there is always the possibility that the outcome is influenced by some phenomena that have not yet been observed -- e.g., that our instruments aren't sensitive enough, or our perception is not unbiased enough. But a problem in mathematics is finite; a mathematics problem has only the ingredients that are put into the problem by the mathematician. There are no hidden influences, and so we can be completely certain of the outcome.

Wolfram's book does mention some of the similarities between mathematically generated patterns and patterns in nature. Wolfram also mentions intelligent design very briefly, both with regard to complexity and with regard to purpose. But for the most part he avoids the controversy. After all, his book is one of science, not sociology or politics.

But I will face that controversy right here, because I am tired of hearing people who understand nothing of science claiming that they have an equally justifiable explanation of things. The advocates of intelligent design are simply wrong when they say that ordered complexity cannot occur by accident. Wolfram's mathematics shows in examples that ordered complexity does occur by accident.

Let me emphasize that I am not asserting the nonexistence of God. If you want to believe in a God, I won't dispute it; I just hope that ordered complexity isn't the foundation of your belief.

Most of us prefer to believe in whatever is the simplest explanation that fits all the facts. Scientists often call this a principle of science, but most non-scientists follow this rule too. However, what is "simplest" is a somewhat subjective matter. For instance, what makes an automobile go? It is "the internal combustion engine" if you have studied and understand such things, or if you at least have some respect for the people who built the car or who know how to repair it. But it is "magic" if you have neither that understanding nor that respect. Most people in our society do respect their automobile mechanics. I'm sorry that so many people don't have the same respect for biology researchers, who trained for much longer.

If you want to teach creationism or "intelligent design" in science classes in schools, I think you should explain it to students this way:

Some people prefer to believe in creationism or "intelligent design" because they find that explanation simpler than the secular view that is prevalent in the scientific community -- though that preference reflects their own lack of understanding of the ideas that the scientific community has produced. One of the goals of this science class is to explain evolution so that you do understand it.

Moreover, despite overwhelming evidence (perhaps because they have not looked at the evidence), some people continue to believe that the universe cannot be explained without the presence of a creator. One of the goals of this science course is to show you that it can be explained without that additional assumption.

By the way, the Wolfram examples only show complexity building upward, but biological evolution has other mechanisms available too, involving the removal of structures. Don Lindsay, a computer scientist, has posted a particularly clear web page explaining some of these mechanisms.

Still another point of confusion may stem from the general public's poor understanding of probability. (The lottery is a tax levied on people who do not know math.) I think that most scientists would agree that relatively few planets are suitable for the origination of life -- i.e., if you pick a planet at random, there's very little chance that it will be a planet where life can develop. However, that does not mean that our being on such a planet is unlikely. To understand the question, you have to turn it around and ask it the other way. Given that we did wake up one day (or one millenium) and start asking questions, what are the odds that we did it on one of the few planets capable of sustaining life? Very high, obviously. Or here is still another way to put the question: There are billions of planets in the universe. No two are identical, but we could classify them into certain "types" according to whether they have similar chemical compositions, axial tilts, distance from their parent stars, and so on. Is there a certain "type" of planet on which life is highly likely to evolve? Most astrobiologists would say yes. If you're interested in this sort of question, you might want to look at Wiki's discussion of the Drake equation.

Why does it matter? What is the harm in introducing a bit of theology in a science class? Well, science and theology operate on entirely different principles, and belong in different places.

• The "scientific method" operates on these principles: Question everything and everyone, even your teachers. A document's great age does not make it authoritative; there is no authority except repeatable experimental evidence and pure reason. Moreover, one must constantly be on guard against biases (accidental or otherwise) in the experiments or implicit assumptions in the reasoning. No answer is ever final -- an established tradition can be overthrown instantly if its experimental evidence is shown to have overlooked something or to be faulty in some other respect -- but until such an overthrow, proper respect must be given to "theories" that are supported by overwhelming evidence.

• Theology, on the other hand, operates on rather different principles. I'm not sure what they are -- I'm no expert on theology -- but authority and tradition appear to play bigger roles, and experiment plays a much smaller role, as far as I can see.

In addition to purely philosophical reasons, there are also practical reasons. As historian Edward J. Larson has explained, the evolutionary viewpoint has been confirmed by its measurable productivity: Its insights have led to discoveries in medicine, biotechnology, etc. -- i.e., it has improved our lives. Its lessons should not be denied to our students, nor obscured by unrelated notions.