March 2, 2008 — -- Since writing my book "Irreligion" and some of my recent Who's Counting columns, I've received a large number of e-mails from subscribers to creation science (who have recently christened themselves intelligent design theorists). Some of the notes have been polite, some vituperative, but almost all question "how order and complexity can arise out of nothing."

Since they can imagine no way for this to happen, they conclude there must be an intelligent designer, a God. (They leave aside the prior question of how He arose.)

My canned answer to them about biological order talks a bit about evolution, but they often dismiss that source of order for religious reasons or because of a misunderstanding of the second law of thermodynamics.

(See Complexity and Intelligent Design for my Who's Counting discussion of biological and economic order and complexity arising out of very simple programs.)

Because the seemingly inexplicable arising of order seems to be so critical to so many, however, I've decided to list here a few other sources for naturally occurring order in physics, math, and biology. Of course, order, complexity, entropy, randomness and related notions are clearly and utterly impossible to describe and disentangle in a column like this, but the examples below from "Irreligion" hint at some of the abstract ideas relevant to the arising of what has been called "order for free."

Let me begin by noting that even about the seemingly completely disordered, we can always say something. No universe could be completely random at all levels of analysis.

In physics, this idea is illustrated by the kinetic theory of gases. There an assumption of disorder on one formal level of analysis, the random movement of gas molecules, leads to a kind of order on a higher level, the relations among variables such as temperature, pressure and volume known as the gas laws. The law-like relations follow from the lower-level randomness and a few other minimal assumptions. (This bit of physics does not mean that life has evolved simply by chance, a common mischaracterization of evolution.)

In addition to the various laws of large numbers studied in statistics, a notion that manifests a different aspect of this idea is statistician Persi Diaconis' remark that if you look at a big enough population long enough, then "almost any damn thing will happen."

A more profound version of this line of thought can be traced back to British mathematician Frank Ramsey, who proved a strange theorem. It stated that if you have a sufficiently large set of geometric points and every pair of them is connected by either a red line or a green line (but not by both), then no matter how you color the lines, there will always be a large subset of the original set with a special property. Either every pair of the subset's members will be connected by a red line or every pair of the subset's members will be connected by a green line.

If, for example, you want to be certain of having at least three points all connected by red lines or at least three points all connected by green lines, you will need at least six points. (The answer is not as obvious as it may seem, but the proof isn't difficult.)

For you to be certain that you will have four points, every pair of which is connected by a red line, or four points, every pair of which is connected by a green line, you will need 18 points, and for you to be certain that there will be five points with this property, you will need -- it's not known exactly - between 43 and 55. With enough points, you will inevitably find unicolored islands of order as big as you want, no matter how you color the lines.

A whole mathematical subdiscipline has grown up devoted to proving theorems of this same general form: How big does a set have to be so that there will always be some subset of a given size that it will constitute an island of order of some sort?

Ramsey-type theorems may even be part of the explanation (along, of course, with Diaconis' dictum) for some of the equidistant letter sequences that constitute the bible codes. Any sufficiently long sequence of symbols, especially one written in the restricted vocabulary of ancient Hebrew, is going to contain subsequences that appear meaningful.

Finally, of more direct relevance to evolution and the origin of living complexity is the work of Stuart Kauffman. In his book, "At Home in the Universe," he discusses what he has termed the aforementioned notion of "order for free."

Motivated by the idea of hundreds of genes in a genome turning on and off other genes and the order and pattern that nevertheless exist, Kauffman asks us to consider a large collection of 10,000 light bulbs, each bulb having inputs from two other bulbs in the collection.

Assume that you connect these bulbs at random, that a clock ticks off one-second intervals, and that at each tick each bulb either goes on or off according to some arbitrarily selected rule. For some bulbs, the rule might be to go off at any instant unless both inputs are on the previous instant. For others it might be to go on at any instant if at least one of the inputs is off the previous instant. Given the random connections and random assignment of rules, it would be natural to expect the collection of bulbs to flicker chaotically with no apparent pattern.

What happens, however, is that very soon one observes order for free, more or less stable cycles of light configurations, different ones for different initial conditions. Kauffman proposes that some phenomenon of this sort supplements or accentuates the effects of natural selection.

Although there is certainly no need for yet another argument against the seemingly ineradicable silliness of "creation science," these light bulb experiments and the unexpected order that occurs so naturally in them do seem to provide one.

In any case, order for free and apparent complexity greater than we might naively expect are no basis for believing in God as traditionally defined. Of course, we can always redefine God to be an inevitable island of order or some sort of emergent mathematical entity. If we do that, the above considerations can be taken as indicating that such a pattern will necessarily exist, but that's hardly what people mean by God.

*John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as of the just-released "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up " His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.*