We know it when we see it?

 Intuitive Specified Complexity: A User-Friendly Account

William Dembski

Even though this series is titled “Specified Complexity Made Simple,” there’s a limit to how much the concept of specified complexity may be simplified before it can no longer be adequately defined or explained. Accordingly, specified complexity, even when made simple, will still require the introduction of some basic mathematics, such as exponents and logarithms, as well as an informal discussion of information theory, especially Shannon and Kolmogorov information. I’ll get to that in the subsequent posts. 

At this early stage in the discussion, however, it seems wise to lay out specified complexity in a convenient non-technical way. That way, readers lacking mathematical and technical facility will still be able to grasp the gist of specified complexity. Here, I’ll present an intuitively accessible account of specified complexity. Just as all English speakers are familiar with the concept of prose even if they’ve never thought about how it differs from poetry, so too we are all familiar with specified complexity even if we haven’t carefully defined it or provided a precise formal mathematical account of it. 

In this post I’ll present a user-friendly account of specified complexity by means of intuitively compelling examples. Even though non-technical readers may be inclined to skip the rest of this series, I would nonetheless encourage all readers to dip into the subsequent posts, if only to persuade themselves that specified complexity has a sound rigorous basis to back up its underlying intuition. 

To Get the Ball Rolling…

Let’s consider an example by YouTube personality Dave Farina, known popularly as “Professor Dave.” In arguing against the use of small probability arguments to challenge Darwinian evolutionary theory, Farina offers the following example:

Let’s say 10 people are having a get-together, and they are curious as to what everyone’s birthday is. They go down the line. One person says June 13th, another says November 21st, and so forth. Each of them have a 1 in 365 chance of having that particular birthday. So, what is the probability that those 10 people in that room would have those 10 birthdays? Well, it’s 1 in 365 to the 10th power, or 1 in 4.2 times 10 to the 25, which is 42 trillion trillion. The odds are unthinkable, and yet there they are sitting in that room. So how can this be? Well, everyone has to have a birthday.

Farina’s use of the term “unthinkable” brings to mind Vizzini in The Princess Bride. Vizzini keeps uttering the word “inconceivable” in reaction to a man in black (Westley) steadily gaining ground on him and his henchmen. Finally, his fellow henchman Inigo Montoya remarks, “You keep using that word — I do not think it means what you think it means.”

Similarly, in contrast to Farina, an improbability of 1 in 42 trillion trillion is in fact quite thinkable. Right now you can do even better than this level of improbability. Get out a fair coin and toss it 100 times. That’ll take you a few minutes. You’ll witness an event unique in the history of coin tossing and one having a probability of 1 in 10 to the 30, or 1 in a million trillion trillion. 

The reason Farina’s improbability is quite thinkable is that the event to which it is tied is unspecified. As he puts it, “One person says June 13th, another says November 21st, and so forth.” The “and so forth” here is a giveaway that the event is unspecified. 

But now consider a variant of Farina’s example: Imagine that each of his ten people confirmed that his or her birthday was January 1. The probability would in this case again be 1 in 42 trillion trillion. But what’s different now is that the event is specified. How is it specified? It is specified in virtue of having a very short description, namely, “Everyone here was born New Year’s Day.” 

Nothing Surprising Here

The complexity in specified complexity refers to probability: the greater the complexity, the smaller the probability. There is a precise information-theoretic basis for this connection between probability and complexity that we’ll examine in the next post. Accordingly, because the joint probability of any ten birthdays is quite low, their complexity will be quite high. 

For things to get interesting with birthdays, complexity needs to be combined with specification. A specification is a salient pattern that we should not expect a highly complex event to match simply by chance. Clearly, a large group of people that all share the same birthday did not come together by chance. But what exactly is it that makes a pattern salient so that, in the presence of complexity, it becomes an instance of specified complexity and thereby defeats chance? 

That’s the whole point of specified complexity. Sheer complexity, as Farina’s example shows, cannot defeat chance. So too, the absence of complexity cannot defeat chance. For instance, if we learn that a single individual has a birthday on January 1, we wouldn’t regard anything as amiss or afoul. That event is simple, not complex, in the sense of probability. Leaving aside leap years and seasonal effects on birth rates, 1 out 365 people will on average have a birthday on January 1. With a worldwide population of 8 billion people, many people will have that birthday. 

Not by Chance

But a group of exactly 10 people all in the same room all having a birthday of January 1 is a different matter. We would not ascribe such a coincidence to chance. But why? Because the event is not just complex but also specified. And what makes a complex event also specified — or conforming to a specification — is that it has a short description. In fact, we define specifications as patterns with short descriptions.

Such a definition may seem counterintuitive, but it actually makes good sense of how we eliminate chance in practice. The fact is, any event (and by extension any object or structure produced by an event) is describable if we allow ourselves a long enough description. Any event, however improbable, can therefore be described. But most improbable events can’t be described simply. Improbable events with simple descriptions draw our attention and prod us to look for explanations other than chance.

Take Mount Rushmore. It could be described in detail as follows: for each cubic micrometer in a large cube that encloses the entire monument, register whether it contains rock or is empty of rock (treating partially filled cubic micrometers, let us stipulate, as empty). Mount Rushmore can be enclosed in a cube of under 50,000 cubic meters. Moreover, each cubic meter contains a million trillion micrometers. Accordingly, 50 billion trillion filled-or-empty cells could describe Mount Rushmore in detail. Thinking of each filled-or-empty cell as a bit then yields 50 billion trillion bits of information. That’s more information than contained in the entire World Wide Web (there are currently 2 billion websites globally). 

But of course, nobody attempts to describe Mount Rushmore that way. Instead, we describe it succinctly as “a giant rock formation that depicts the U.S. Presidents George Washington, Thomas Jefferson, Abraham Lincoln, and Theodore Roosevelt.” That’s a short description. At the same time, any rock formation the size of Mount Rushmore will be highly improbable or complex. Mount Rushmore is therefore both complex and specified. That’s why, even if we knew nothing about the history of Mount Rushmore’s construction, we would refuse to attribute it to the forces of chance (such as wind and erosion) and instead attribute it to design.

Take the Game of Poker

Consider a few more examples in this vein. There are 2,598,960 distinct possible poker hands, and so the probability of any poker hand is 1/2,598,960. Consider now two short descriptions, namely, “royal flush” and “single pair.” These descriptions have roughly the same description length. Yet there are only 4 ways of getting a royal flush and 1,098,240 ways of getting a single pair. This means the probability of getting a royal flush is 4/2,598,960 = .00000154 but the probability of getting a single pair is 1,098,240/2,598,960 = .423. A royal flush is therefore much more improbable than a single pair.

Suppose now that you are playing a game of poker and you come across these two hands, namely, a royal flush and a single pair. Which are you more apt to attribute to chance? Which are you more apt to attribute to cheating, and therefore to design? Clearly, a single pair would, by itself, not cause you to question chance. It is specified in virtue of its short description. But because it is highly probable, and therefore not complex, it would not count as an instance of specified complexity. 

Witnessing a royal flush, however, would elicit suspicion, if not an outright accusation of cheating (and therefore of design). Of course, given the sheer amount of poker played throughout the world, royal flushes will now and then appear by chance. But what raises suspicion that a given instance of a royal flush may not be the result of chance is its short description (a property it shares with “single pair”) combined with its complexity/improbability (a property it does not share with “single pair”). 

Let’s consider one further example, which seems to have become a favorite among readers of the recently released second edition of The Design Inference. In the chapter on specification, my co-author Winston Ewert and I consider a famous scene in the film The Empire Strikes Back, which we then contrast with a similar scene from another film that parodies it. Quoting from the chapter:

Darth Vader tells Luke Skywalker, “No, I am your father,” revealing himself to be Luke’s father. This is a short description of their relationship, and the relationship is surprising, at least in part because the relationship can be so briefly described. In contrast, consider the following line uttered by Dark Helmet to Lone Starr in Spaceballs, the Mel Brooks parody of Star Wars: “I am your father’s brother’s nephew’s cousin’s former room­mate.” The point of the joke is that the relationship is so compli­cated and contrived, and requires such a long description, that it evokes no suspicion and calls for no special explanation. With everybody on the planet connected by no more than “six degrees of separation,” some long description like this is bound to identify anyone.

In a universe of countless people, Darth Vader meeting Luke Skywalker is highly improbable or complex. Moreover, their relation of father to son, by being briefly described, is also specified. Their meeting therefore exhibits specified complexity and cannot be ascribed to chance. Dark Helmet meeting Lone Starr may likewise be highly improbable or complex. But given the convoluted description of their past relationship, their meeting represents an instance of unspecified complexity. If their meeting is due to design, it is for reasons other than their past relationship.

How Short Is Short Enough?

Before we move to a more formal treatment of specified complexity, we are well to ask how short is short enough for a description to count as a specification. How short should a description be so that combined with complexity it produces specified complexity? As it is, in the formal treatment of specified complexity, complexity and description length are both converted to bits, and then specified complexity can be defined as the difference of bits (the bits denoting complexity minus the bits denoting specification). 

When specified complexity is applied informally, however, we may calculate a probability (or associated complexity) but we usually don’t calculate a description length. Rather, as with the Star Wars/Spaceballs example, we make an intuitive judgment that one description is short and natural, the other long and contrived. Such intuitive judgments have, as we will see, a formal underpinning, but in practice we let ourselves be guided by intuitive specified complexity, treating it as a convincing way to distinguish merely improbable events from those that require further scrutiny.