Mathematics has long had a place in the arsenal of anti-evolutionism, but in recent years it has become especially prominent. This is understandable because mathematics affords the possibility of an “in-principle” argument against evolution. If you can carry out a calculation to show that evolution is impossible or can appeal to an abstract mathematical principle that rules it out, then all the circumstantial evidence in the world will not save the theory. If the numbers do not add up, then the theory is wrong—regardless of any inferences you draw from paleontology, anatomy, genetics, embryology, and so on.
It is like representing a defendant at a criminal trial. One approach is to challenge each piece of evidence against your client individually in the hope of creating reasonable doubt. But a far better approach, if it is available, is to show that your client simply cannot be guilty. If he has an ironclad alibi or is just physically incapable of having committed the crime, then he is not guilty regardless of any evidence against him.
For the past fifteen years, the anti-evolutionist literature has been dominated by mathematical arguments, typically drawn from the fields of probability theory, information theory, and combinatorial search. These arguments are often presented with copious mathematical jargon and notation, which can make them difficult to parse for people not immersed in the relevant fields. By making liberal use of mathematical symbols, Greek letters, and subscripts, it is easy to create the illusion that something profound has been said. But when the arguments are shorn of their technical pretensions and presented in readily comprehensible language, it is never difficult to spot the fatal conceptual flaws at their core. Scientists are right to scoff at them, just as they rightly scoff at the anti-evolution arguments drawn from other branches of science.
What follows will provide a quick guide to the main themes of anti-evolutionary mathematics and explain in plain terms why these arguments fail.
Probability
Leaving aside religious concerns, evolution appears implausible to some because it seems to imply that something very unlikely has occurred. Evolution says that natural forces have led organisms to become more complex over time, but our everyday experience is that things break down unless energy and intelligence is expended to maintain them. Since probability is the branch of mathematics that quantifies our intuitions as to what is likely and what is not, it is natural that it should appear in anti-evolutionist literature.
You are probably familiar with elementary applications of probability. For example, when we flip a fair coin, we say that heads and tails each have a probability of 1/2. The fraction’s denominator indicates that there are two possible outcomes (heads or tails), and we assume that each is equally likely to appear. If we flip a coin twice, then each of the four outcomes (HH, HT, TH, TT) has a probability of 1/4, which is the same as 1/2 times 1/2. There are eight sequences of three Hs and Ts (HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT). Since each of these sequences is as likely as any other, the probability of any one of them is 1/8, which is the same as 1/2 times 1/2 times 1/2. Generalizing from these examples, if we toss a coin x times, then the probability of any specific sequence is 1/2 multiplied by itself x times. If x is large, then this will be a very small number.
A classic anti-evolutionist argument now goes like this: A gene can be viewed as a sequence of DNA bases, typically represented by the letters A, C, G, and T. A gene can therefore be viewed as a sequence of letters, just as the outcomes of multiple coin tosses can be viewed as a sequence of Hs and Ts. If a specific gene is, say, 100 bases long, then the probability of getting just that sequence by random chance is 1/4 multiplied by itself 100 times. This is such a small number, the argument continues, that it could not possibly have been the outcome of a naturalistic process and must instead have arisen through intelligent design. Sometimes the argument refers to proteins (which can be viewed as sequences of amino acids) instead of genes, and sometimes additional biological details are invoked that in no way affect the logic of the argument. In various forms, this argument has long been a mainstay of creationist literature. For examples, have a look at Morris and Parker (1987, 97–100), Roth (1998, 69–70), and Foster (1999).
However, this argument is premised on the notion that genes and proteins evolve through a process analogous to tossing a coin multiple times. This is untrue because there is nothing analogous to natural selection when you are tossing coins. Natural selection is a non-random process, and this fundamentally affects the probability of evolving a particular gene.
To see why, suppose we toss 100 coins in the hopes of obtaining 100 heads. One approach is to throw all 100 coins at once, repeatedly, until all 100 happen to land heads at the same time. Of course, this is exceedingly unlikely to occur. An alternative approach is to flip all 100 coins, leave the ones that landed heads as they are, and then toss again only those that landed tails. We continue in this manner until all 100 coins show heads, which, under this procedure, will happen before too long. The creationist argument assumes that evolution must proceed in a manner comparable to the first approach, when really it has far more in common with the second.
The failure to consider the role of natural selection in evolution is really such a crass blunder that scientists rightly consider the persistence of such arguments among anti-evolutionists evidence of their fundamental lack of good faith. Modern proponents of intelligent design (ID) are usually too sophisticated to make such an error. Instead, they present a superficially more sophisticated probability-based argument. Their idea is best illustrated by example.
Let us return to coin-tossing. Suppose we toss a coin 100 times, thereby producing a chaotic jumble of heads and tails. It was very unlikely that just that sequence would appear, but we do not suspect trickery. After all, something had to happen. But now suppose we obtained 100 Hs or a perfect alternation of Hs and Ts. Now we probably would suspect trickery of some kind. Such sequences are not only improbable but also match a recognizable pattern. ID proponents argue that it is the combination of improbability and matching a pattern that makes them suspect that something other than chance or purely natural processes are at work. They use the phrase “complex, specified information” to capture this idea. In this context, “complex” just means “improbable,” and “specified” means “matches a pattern.”
As applied to biology, the argument goes like this: Consider a complex, biological adaptation such as the flagellum used by some bacteria to propel themselves through liquid. The flagellum is a machine constructed from numerous individual proteins working in concert. Finding this exact functional arrangement of proteins is extremely unlikely to happen by chance. Moreover, they continue, the structure of the flagellum is strongly analogous to the sort of outboard motor we might use to propel a boat. Therefore, the flagellum exhibits both complexity and specificity, and it therefore must be the product of intelligent design.
The most prominent defender of this argument is ID proponent William Dembski. He has presented it in numerous books and essays, most notably in his book No Free Lunch (Dembski 2002). Dembski presents his argument with a copious amount of mathematical notation and jargon. However, while his argument contributes some superficial sophistication, it is ultimately no improvement over what we have seen.
Let us address the issue of “specificity” first. There is a danger that saying that a flagellum looks like an outboard motor is comparable to saying that a fluffy, cumulus cloud looks like a dragon. We need to be able to distinguish the design-suggesting patterns from the ones we impose on nature through excessive imagination. Biologists say that natural selection produces functional structures as a matter of course. Thus, when we see a resemblance between a flagellum and a motor, are we seeing a design-suggesting pattern, or are we seeing something that is readily explained by natural selection?
In simplistic examples such as coin tossing, we have extensive background knowledge to help us with this problem. We know what usually happens when coins are tossed, and that enables us to distinguish design-suggesting patterns from what normally happens. Likewise for another example ID proponents use to illustrate specificity: Mt. Rushmore. We know what mountains look like when people do not carve faces into them, and this allows us to recognize Mt. Rushmore as the product of intelligent design. This sort of background knowledge is precisely what we lack in the case of evolution. When it comes to biological adaptations, we have no base of experience for distinguishing design-suggesting patterns from the ones explicable by natural processes.
The argument likewise founders on the question of complexity. According to ID proponents, establishing complexity requires carrying out a probability calculation, but we have no means for carrying out such a computation in this context. The evolutionary process is affected by so many variables that there is no hope of quantifying them for the purposes of evaluating such a probability.
In summary, any anti-evolutionist argument based on probability theory can simply be dismissed out of hand. There is no way to carry out a meaningful calculation, and adding “specificity” to the mix does nothing to improve the argument.
Information and Combinatorial Search
A different approach to anti-evolutionary mathematics invokes information theory instead of probability. The basic argument is this: Genes encode meaningful information, and such information is always the hallmark of intelligent design. Natural processes only lead to weathering, erosion, and the breakdown of functional structures. In particular, genetic mutations, which are said to lie at the heart of evolutionary theory, can only lead to losses of information. Therefore, complex genetic information cannot be the result of a naturalistic process. For examples of this sort of argument, have a look at the book by Gitt (2001) or the paper by McIntosh (2009, 375).
By speaking of increases or decreases in information, we are presupposing we have some way of quantifying information. (How else can we determine that some process leaves us with more or less information than we started with?) This is where the mathematics comes in. Mathematicians have devised several methods for measuring information content, the most well-known being the system developed by Claude Shannon in 1948. Roughly, Shannon’s idea was that information content is closely related to probability; we learn more information from low-probability events than from high-probability events. His system provides the tools for quantifying this intuition, and these tools ultimately lead to a fruitful mathematical theory of information.
Anti-evolutionists frequently refer to the technical minutiae of information theory in their discourse, but such references do nothing to help their argument. Known genetic mechanisms are fully adequate to explain the growth of genetic information through the evolutionary process. One especially important mechanism plays out when a gene duplicates itself, thereby leaving the organism with two copies of a gene that previously occurred only once. The second copy is then free to acquire mutations without harming the organism because the first gene is there to carry out the original function.
This process is analogous to increasing the information stored on a computer by inserting a thumb drive. Inserting the drive is comparable to the duplication event; it increases the information storage capacity of the computer just as the duplication increases the storage capacity of the genome. When new files are placed on the drive, the quantity of information stored on the computer has increased, just as mutations in the duplicate copy lead to new genetic information.
Therefore, any claim to the effect that genetic mutations can only degrade information can be dismissed as simply empirically false. Moreover, we should note that this claim is logically impossible. After all, if the mutation from A to B represents a loss of information, then the reverse mutation from B back to A must represent a gain of information.
Modern ID proponents have tried a different approach to these sorts of arguments, invoking the branch of mathematics known as combinatorial search. We can illustrate their main idea with an analogy.
You have probably seen word search puzzles in which a small number of meaningful words are concealed in a large array of meaningless letters. If you have ever tried to solve such puzzles, you know that it is very inefficient, and not a whole lot of fun, to examine every letter in all eight directions looking for words. Instead, you quickly hit on the trick of looking for pairs or triples of letters at the same time. You can also look for unusual letter combinations that will stand out against the background. For example, if your word has a Q in it, then you need to look for QU pairs. The point is that if you search for small targets in large spaces, then you need some kind of edge. Blind search is not going to cut it.
There is a sense in which the evolutionary process can also be viewed as a search problem. The small targets are the various complex adaptations used by organisms to survive. The large space is the collection of all theoretically possible gene sequences, most of which do not code for anything useful. Scientists claim that natural selection provides the needed edge for evolution to find the small targets in the large space. The set of all possible gene sequences is incredibly vast, but this is irrelevant because natural selection shifts the probability distribution dramatically toward the functional sequences and away from the nonfunctional sequences.
In the language of combinatorial search, the “edge” to which I refer is called an “algorithm.” Researchers in this area have devised a variety of algorithms for tackling different sorts of search problems. In practical applications drawn from physics and engineering, the researcher always uses background knowledge about the specific problem to be solved to select the most appropriate search algorithm. It would be nice if there were an all-purpose algorithm that would be effective regardless of the specific properties of the space to be searched.
And this is where ID proponents see a point of attack. There is a collection of specific mathematical results, referred to as “No Free Lunch” theorems, that prove that there is no all-purpose algorithm. If an algorithm is effective for some search problems, it is necessarily ineffective on others. And if that is the case, then whatever success natural selection has as a search algorithm can result only from intelligent tailoring of the algorithm to the problem. Just as mathematicians use their intelligence to fit their search algorithms to specific problems, so too, the argument runs, must a greater intelligence have made the laws of nature such as to allow Darwinian evolution to operate. This line of attack was first presented by Dembski, again in his book No Free Lunch. It has since been elaborated upon in the more recent book by Marks, Dembski, and Ewert (2017).
There is much to be said against this argument. The No Free Lunch theorems are legitimate and important mathematical results, but as with all such theorems there is a wealth of technical detail to wade through before applying them to specific physical cases. Mathematicians and philosophers have cogently argued that evolution simply is not a search process in the precise sense addressed by the theorems. Developing this line of response requires more attention to minutiae than I can provide in a short article, but I would refer the interested reader to the book by Sarkar (2007, chapter 5) or the paper by Perakh (2005) for careful developments of this argument.
For our purposes, the main flaw lies elsewhere. Even if we accept everything Dembski and his coauthors are saying about these theorems, this whole line of attack simply amounts to nothing. Most of us did not need difficult mathematical theorems to understand that Darwinian evolution can work only if nature has certain properties. The search problem confronted by evolution arises ultimately from the laws of physics, but it is well outside biology’s domain to wonder why those laws are as they are. Dembski and his cohorts argue that the fundamental constants of the universe encode information of a sort that can arise only from an intelligent source, but they have no more basis for this claim than they did for the comparable claim about genetic sequences.
The Futility of Anti-Evolutionary Mathematics
I have addressed a few of the principal errors in anti-evolutionary mathematics, but there is far more wrong with these arguments than I can present here. For their part, the anti-evolutionists have presented many variations on the basic themes I have discussed, and it would require writing at book length to address every claim individually. Let me close, then, by noting that we do not need to address every individual nuance of mathematical anti-evolutionism to be deeply skeptical of the entire enterprise.
This article began by saying that mathematics affords the possibility of an “in-principle” argument against evolution. But is it plausible to think that any such argument could possibly succeed?
The basic ingredients of evolutionary theory are simple, empirical facts. Genes really do mutate, sometimes leading to new functionalities, and natural selection really can string together several such mutations into directional, evolutionary change. On a small scale, this has all been demonstrated and observed. But if evolution can produce small changes over short time periods, what abstract mathematical principle is going to rule out larger changes over longer time periods? If there is a mechanism for small change, then how can any calculation or abstract theorem rule out simply repeating the process to produce large change?
Everyone agrees that complex adaptations require a special sort of explanation. Scientists argue that actual biological systems show copious evidence of having resulted through evolution by natural selection. Anti-evolutionists reject this claim, but the ensuing debate, such as it is, has nothing to do with mathematics. This makes you wonder why anti-evolutionists insist on padding their work with so much irrelevant and erroneous mathematical formalism. The answer is that their literature has far more to do with propaganda than it does with serious argument. Mathematics is unique in its ability to bamboozle a lay audience, making it well suited to their purposes. But for all its superficial sophistication, anti-evolutionary mathematics is not even successful at raising interesting questions about evolution.
References
Dembski, W.A. 2002. No Free Lunch: Why Specified Complexity Cannot Be Purchased without Intelligence. Lanham, MD: Rowman & Littlefield.
Foster, D. 1999. Proving God exists. The Saturday Evening Post (Nov.–Dec.): 59–61, 78, 80–81, 84.
Gitt, W. 2001. In the Beginning Was Information. Bielefeld, Germany: Christliche Literatur-Verbreitung.
Marks II, R.J., W.A. Dembski, and W. Ewert. 2017. Introduction to Evolutionary Informatics. Singapore: World Scientific.
McIntosh, A.C. 2009. Information and entropy—top-down or bottom-up development in living systems? International Journal of Design & Nature Ecodynamics 4(4): 351–385.
Morris, H.M., and G.E. Parker. 1987. What Is Creation Science? Revised and expanded edition. El Cajon, CA: Master Books.
Perakh, M. 2005. There is a free lunch after all: William Dembski’s wrong answers to irrelevant questions. In M. Young and T. Edis, eds. Why Intelligent Design Fails: A Scientific Critique of the New Creationism. New Brunswick, NJ: Rutgers University Press.
Roth, A.A. 1998. Origins: Linking Science and Scripture. Silver Spring, MD: Review and Herald Publishing Association.
Sarkar, S. 2007. Doubting Darwin: Creationist Designs on Evolution. Malden, MA: Blackwell Publishing.