Review: John Allen Paulos. 2007. Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up. New York, NY: Hill and Wang. 176 pp.
This short book by John Allen Paulos presents twelve arguments for taking religion seriously and offers brief refutations of each of them. These are not meant to be definitive; rather, they are skeptical musings, intending to point out in a few words the holes in the arguments. It is not intended for theologians. It is more a book to be left around, hoping that someone will pick it up and absorb the skeptical attitude that it promotes.
Irreligion is quite unusual in that it combines religion with mathematics and humor. It might not be the right book to give to your elderly aunt or uncle who goes to church on Sunday and has no concept that religious ideas can be intellectual play. For such a person the short book Atheism: A Very Short Introduction by Julian Baggini might be the thing. It is earnest and polite and even has references to other books. Paulos doesn’t even bother with references.
Perhaps the right audience for Paulos’s book is a young person who already has something of a skeptical attitude about other matters, but who needs exposure to such ideas in the religious context. This young person need not be an expert on mathematics, as the mathematical ideas are introduced in a gentle way. It must be admitted that many of them are only tangentially relevant to the central theological point under discussion, but in places–such as the Bible Code discussion–they are at the heart of the issue.
Since many of the arguments are familiar to anyone with any acquaintance with the philosophy of religion, there is no point to going through all of them in this review. Instead, I will concentrate on a few points with more substantial mathematical content. So it will be perhaps heavier reading than the book, but not as wide-ranging and funny. (Too bad. Buy the book.)
The book begins with four classical arguments. These are the first cause argument, the argument from design, an appeal to the anthropic principle, and the ontological argument. Along with design arguments the author considers some creationist arguments. A favorite is that a particular biological outcome depends on a very long sequence of mutations, and that the probability of such a sequence is extraordinarily tiny.
It is not just creationists who are confused by such issues. A way of clarifying the situation is to distinguish between outcomes of an experiment and events that can happen as a result of such outcomes. An “outcome” is a complete and precise specification of how an experiment could result. An “event” is something that can happen (or not happen) as a result of the experiment.
Consider the experiment of tossing a coin three times. Each toss can come up heads, indicated by H, or tails, indicated by T. One possible outcome might be THH. There are a total of eight such outcomes. On the other hand, an event might be something like getting T on the second toss. This event would occur for outcomes HTH, TTH, HTT, TTT. Another event might be obtaining exactly two H’s. The second one would occur for outcomes HHT, HTH, THH.
Once you know the outcome of the experiment, you know whether or not the event happened. With the outcome THH there is no tail on the second toss, so the first event (getting T on the second toss) does not happen. With the same outcome there are exactly two heads, so the second event (obtaining two H’s) does happen.
As an aside, some people will enjoy the challenge of figuring out the number of events associated with the coin tossing experiment (three tosses). I will spoil their pleasure right now: the answer is 256. It comes out this way if you count two trivial events, one that happens no matter what, and one that can never happen, by definition. Perhaps most of these 256 events are not particularly natural or interesting, but surely some are of intense interest to gamblers. That leaves another challenge, to figure out how the number 256 arises. (It should not be so hard.)
Probability theory assigns probability to events. That is, you ask a sensible question about the experiment, and you ask for the chance that when the experiment is concluded that the answer is yes. On the other hand, the conclusion of the experiment is a particular outcome.
The sticky point is that each outcome also corresponds to a very special kind of event. This is the event that is said to happen only with this designated outcome and with none other. In most realistic situations, the probability of this kind of outcome-event is very, very small. In ten tosses of a fair coin, the probability of each pattern is already only (½)^10, which is 1/1024, less than one in a thousand. For twenty tosses, the probability (½)^20 is already less than one in a million. These are not the kinds of events that occur in typical probability calculations. After all, when you are using probability for prediction, you need to calculate the probability of an event that has a natural specification before the experiment is done.
Now a warning: the three paragraphs that follow are more mathematical than the rest of this review. Skip this part if you wish. However, some readers may enjoy seeing how computing probabilities of unlikely events can help to assess evidence. Paulos only hints at this part of the story, but it helps complete the picture.
The apparent paradox is that as experiments accumulate more and more evidence, then the outcome-events become even less and less probable. So why do we feel that we are getting useful information? One answer was suggested by the eighteenth-century British mathematician Thomas Bayes. Consider two theories competing to explain the evidence. Each one is reasonably plausible before the evidence is considered. Look at the probability of the evidence given the first theory and the probability of the evidence given the second theory. These may each be such small numbers as to be almost meaningless. But their ratios help decide which theory to believe. And ratios of very small numbers can be huge, decisively tipping the choice one way or the other. There are legitimate criticisms of Bayes’s method, and some statisticians prefer to speak in somewhat different terms of “likelihood ratio.” However formulated, it is quite reasonable evidence-based reasoning.
It is worth an example. Alice is a geneticist with a theory that says a certain genetic marker should occur ¾ of the time, in the long run. Bob is a another geneticist with a competing theory; according to him, the marker should occur ¼ of the time, again in the long run. The budget is limited, so the experiment consists of only 20 observations. The outcome is a certain pattern in which the marker does or does not occur. So it is something like the coin toss with twenty tosses. In fact, the presence of the marker may be indicated by H and the absence by T, and this makes an outcome look even more similar to that of the coin toss experiment. The difference is that the probability of the H marker is ¾ for each observation on Alice’s theory, or else the probability of the H marker is ¼ for each observation in Bob’s version. (By contrast, the probability of heads is ½ for each toss of the coin).
The experiment is conducted, and the outcome is HHTTH HHHTH HHHTT HTHHH. The marker showed up 14 times. This is an experimental number. What does it mean? It looks like Alice might be the one who is right. Let’s see. On her theory the outcome-event would have probability (¾)^14 * (¼)^6, a very small number. On Bob’s theory it would have probability (¼)^14 * (¾)^6, another tiny number. The particular outcome is quite unexpected on either theory. But the ratio of Alice’s number to Bob’s number (after everything cancels out) is the same as the ratio of 3^14 to 3^6, which is 3^8, quite an impressive tilt in favor of Alice. Given that the evidence came out exactly the way that it did, her theory looks much more reasonable as an explanation of it. For this kind of reasoning ratios of small numbers count, not the small numbers themselves. This is small comfort to the creationist who only cares about those irrelevant small numbers.
The next part of the book treats four general types of subjective arguments: coincidence, prophesy, subjectivity, and interventions. Some of the same probability issues arise in these contexts. Paulos writes this about coincidence:
As I’ve written elsewhere, the most amazing coincidence imaginable would be the complete absence of all coincidences. The above litany is intended to illustrate that there are an indeterminate number of ways for such events to come about, even though the probability of any one of them is tiny. And, as with creationists’ probabilistic arguments, after such events occur people glom onto their tiny probability and neglect to ask the more pertinent question: How likely is something vaguely like this to occur (p. 58).
This is again the same issue. In order to make an honest predictive use of probability, do not look at outcomes, but at events. Each event of interest should be specified without prior knowledge of the outcome, so that it cannot be tailored to order. Quick summary: no cheating.
The final part of the book presents four arguments that have a psychological flavor; Paulos calls these redefinition, cognitive tendency, universality (morality), and gambling. Pascal’s wager, in the gambling chapter, has a mathematical aspect. Blaise Pascal was a seventeenth-century French mathematician, physicist, and religious philosopher. He presented his case in an eloquent and somewhat confusing way, and of course in French. Here is how Paulos presents it in plainer words:
In the case of Pascal’s wager we can perform similar calculations to determine the expected values of the two choices (to believe or not to believe). Each of these expected values depends on the probability of God’s existence and the payoffs associated with the two possibilities: yes, He does, or no, He doesn’t. If we multiply whatever huge numerical payoff we put on endless heavenly bliss by even a tiny probability, we obtain a product that trumps all other factors, and gambling prudence dictates that we should believe (or at least try hard to do so) (p. 134).
This argument has holes that could be discovered by a child with the motivation to think about it critically, and Paulos dismisses it easily. There is also a large scholarly literature; see the Pascal’s wager entry in the online Stanford Encyclopedia of Philosophy. Thus it is troubling that the recent book What’s So Great About Christianity by Dinesh D’Souza makes vital use of the argument. Its role there is to fill a gap in reasoning, when D’Souza wants to pass from a prime-mover God to a personal God. Sometimes even educated people fall for this stuff.
Paulos’s book has an index that displays a wide range of mathematical ideas, described almost always very briefly. In addition to Bayes’s theorem, there are Boolean satisfiability, random branchings, cellular automata, the Gödel incompleteness theorem, Ramsey theory, Turing machines, and so on. It may take more courage to write about such things than about God and morality.
The book is frankly critical of religion, but the author writes from his personal outlook and has a cheerful and positive view. If you read it, look for insights and humor, not systematic exposition. My favorite quip is the surrealists’ two-word argument for the existence of God. Their argument: Pipe cleaner. Indeed, a glance at “La trahison des images” might be a good departure for arguments for the existence of God.
In conclusion, Irreligion is not one of the battleship tomes on philosophy of religion. It is a sporty day-sailor, ready to take advantage of the afternoon breeze. Let the battleship stand off; it may never be needed.
Copyright ©2008 William Faris. The electronic version is copyright ©2008 by Internet Infidels, Inc. with the written permission of William Faris. All rights reserved.