land in serious trouble.

a c o m m on e r ro r

Another common error is to overlook the fact that the order

in which A and B are written matters and can be crucial.

Formally, it is not generally true that P(A g i v e n B) is equal

to P(B g i v e n A). Treating these two probabilities as the same

is often known as the prosecutor™s fallacy because it sometimes

arises in legal cases.4 Suppose that DNA found at the site of a

crime is compared with that obtained from a suspect and that

the two match. It may be claimed in court that the probability

of such a match occurring with the DNA of a randomly chosen

individual in the population is one in several million. The jury

is then invited to infer that the accused is guilty because what

has happened would be extremely unlikely if the accused were

not guilty. Let us express this in the formalism set out above.

The probability, which is said to have a value of one in several

million, may be written

P(match g i v e n accused not guilty).

If we (incorrectly) reverse the elements inside the brackets we

would interpret this as

Examples are not con¬ned to the ¬eld where the term was coined. One of

4

the earliest errors of this kind was due to Hoyle and Wickramasinghe and

before them, Le Comte du No¨ y, who used it to prove that God exists. In

u

the next chapter we shall see that because some occurrence in the physical

world has an incredibly small probability g i v e n the chance hypothesis, it

cannot be assumed that this was the probability of the chance hypothesis

g i v e n the occurrence and hence the probability of its complement (that

God was the cause) was very close to one. More details will be found in

in Bartholomew (1984), see especially chapter 3. Overman (1997) mentions

these and other examples and falls into the same trap.

God, Chance and Purpose

76

P(accused not guilty g i v e n match).

As this is negligibly small, the complementary event, that the

accused is guilty, seems overwhelmingly probable. The rea-

soning is fallacious but an over-zealous prosecutor and lay jury

can easily be persuaded that it is sound. It is true that there

are circumstances in which these two probabilities might hap-

pen to coincide but this is not usually the case. I shall not go

further into the technicalities of why the reasoning behind the

prosecutor™s fallacy is mistaken but one can get an indication

that something might be wrong by the following considera-

tion. Intuitively, one would expect that the judgement of guilt

ought to depend also on the probability of being not guilty

in the absence of any evidence at all “ but here this does not

come into the picture.

The prosecutor™s fallacy often arises when very small prob-

abilities are involved, as in the above example. Very small

probabilities also play a key role in judging whether bio-

logically “ and theologically “ interesting events could have

occurred. It is, therefore, particularly important to be alert for

examples of A and B being inadvertently reversed.

c h a pt e r 6

What can very small probabilities tell us?

It is often claimed that such things as the origin of life on earth and

the coincidental values of cosmic constants are so improbable on the

˜chance hypothesis™ that they must point to divine action. The correct-

ness of this conclusion depends on both correct calculation and valid

forms of inference. Most examples fail on both counts. Two common

mistakes in making probability calculations were identi¬ed in chapter

¬ve. In this chapter I explain and illustrate the three main approaches

to inference: signi¬cance testing, likelihood inference and Bayesian

inference.

w h at i s t h e a rg u m e n t a b o ut ?

It is tempting to argue that if something has a very small

probability we can behave as if it were impossible. When we

say ˜small™ here we are usually thinking of something that is

extremely small, like one in several millions. We run our lives

on this principle. Every day we are subject to a multitude of

tiny risks that we habitually ignore. We might be struck by

a meteorite, contract the ebola virus, ¬nd a two-headed coin

or forget our name. Life would come to a halt if we paused

to enumerate, much less prepare, for such possibilities. Even

the most determined hypochondriac would be hard put to

identify all the risks we run in matters of health. Surely, then,

we can safely reject the possibility of anything happening if

the probability is extremely small? Actually the position is

77

God, Chance and Purpose

78

more subtle and we must be a little more careful about what

we are saying.

Every time we play a card game such as bridge we are dealt

a hand of thirteen cards. The probability of being given any

particular hand is extremely small “ about one in 6.4 — 1011

and yet it undoubtedly has happened and we have the hand

before us as evidence of the fact. That very small probability

does not, however, give us grounds for doubting that the

pack was properly shuf¬‚ed. There are other situations like

this, where everything that could possibly happen has a very

small probability, so we know in advance that what will have

happened will be extremely unlikely. It is then certain that a

very rare event will occur! Does this mean that there is never

any ground to dismiss any hypothesis “ however absurd?

It is clear that there is a problem of inference under uncer-

tainty here with which we must get to grips. Before plunging

into this major topic, I pause to mention a few examples where

very small probabilities have been used as evidence for the

divine hand. In the more spectacular cases these have been

held to show that there must have been a Designer of the

universe. It is important, therefore, that we get our thinking

straight on such matters. Actually there is a prior question

which arises concerning all of these examples; not only must

we ask whether the inferences based on them are legitimate,

but whether the probabilities have been correctly calculated

in the ¬rst place. Most fail on both counts.

e x a m p l e s o f v e ry s m a l l p ro ba b i l i t i e s

A common line of argument, employed by many Christian

apologists, is that if we can ¬nd some event in nature which,

on chance alone, would have an extremely small probability,

then chance can be ruled out as an explanation. The usual

What can very small probabilities tell us? 79

alternative envisaged is that the event must have been caused

by the deity, or someone (or thing) operating in that role. It

is not hard to ¬nd such events and, even if their probabili-

ties cannot always be calculated exactly, enough can usually

be said about them to establish that they are, indeed, very

small. Some of these examples are longstanding and have

been suf¬ciently discredited not to need extended treatment

here. A number were given in God of Chance (Bartholomew

1984), and Overman (1997) reports ¬ve such calculations in his

section 3.7.

The earliest example appears to have been John

Arbuthnot™s discovery that male births exceeded female births

in London parishes in each of the eighty-two years from 1628

to 1710. Le Comte du No¨ y looked at the more fundamen-

u

tal question of the origin of life and considered the proba-

bility that it might have happened by the fortuitous coming

together of proteins to form an enzyme necessary for life

to exist. He quoted the extraordinarily small probability of

2.02 — (1/10)321 (Hick 1970, p. 15). This is an example of

a mistake in the calculation. It falls into the common error

of supposing that enzymes are formed by a kind of random

shaking together of their constituent parts. Hick, appealing

to Matson (1965), recognised this and rejected the calcula-

tion. No biologist supposes that enzymes were formed in such

a way and the calculation is not so much wrong as irrele-

vant. In effect this is another attempt to use the multiplica-

tion rule for independent probabilities, when the probabili-

ties involved are not independent. Ayala (2003)1 makes this

point very clearly (see, especially page 20). He returns to

This paper by Ayala is an excellent review of the ˜argument from design™

1

by a biologist and philosopher. Apart from dealing with the matter of small

probabilities the paper contains much that is relevant to the debate on

Intelligent Design, which is treated in the following chapter.

God, Chance and Purpose

80

this topic in his most recent book (Ayala 2007). In chapter 8,

especially on pages 153 and 154, Ayala points out that the

assumptions on which such calculations rest are simply wrong.

There would, perhaps, be no point in relating these episodes

were it not for the fact that the same error is continually being

repeated. In his attempt to show that life was ˜beyond the

reach of chance™, Denton (1985) uses essentially the same

argument, taking as an illustration the chance of forming

real words by the random assemblage of letters. This exam-

ple has been discussed in more detail in Bartholomew (1996,

pp. 173ff.)2 but it suf¬ces here to say that Denton™s calcu-

lation is based on an oversimpli¬ed model of what actually

happens in nature. In addition to Overman™s examples men-

tioned earlier, Hoyle and Wickramasinghe (1981) made the

same kind of error and this deserves special mention for two

reasons. First, it appears to have given currency to a vivid

picture which helps to grasp the extreme improbabilities of

the events in question. Hoyle compares the chance to that

of a Boeing 747 being assembled by a gale of wind blow-

ing through a scrap yard.3 This analogy is widely quoted,

misquoted (one example refers to a Rolls Royce!) and mis-

interpreted as any web search will quickly show. It was in

Hoyle™s book The Intelligent Universe (1983, p. 19) where it

appears as:

Ayala (2007) also uses the analogy of words formed by typing at random in

2

a section beginning on page 61, headed ˜A monkey™s tale™. He then proposes

a more realistic version on page 62, which is based on the same idea as

the example referred to here. This example ¬rst appeared in Bartholomew

(1988).

None of the quotations I have tracked down give a precise reference to

3

its origin. However, on the web site http://home.wxs.nl/∼gkorthof/

kortho46a.htm Gert Kortof says that the statement was ¬rst made in a

radio lecture given by Hoyle in 1982.

What can very small probabilities tell us? 81

A junkyard contains all the bits and pieces of a Boeing-747, dismem-

bered and in disarray. A whirlwind happens to blow through the yard.

What is the chance that after its passage a fully assembled 747, ready

to ¬‚y, will be found standing there?

Inevitably, for such a famous utterance, there is even doubt as

to whether Hoyle was the true originator. Whether or not the

probability Hoyle and Wickramasinghe calculated bears any

relation at all to the construction of an enzyme is very doubt-

ful, but its vividness certainly puts very small probabilities in

perspective.

Secondly, their argument is a ¬‚agrant example of the pros-

ecutor™s fallacy, or the illegitimate reversal of the A and B in

the probability notation above (see p. 75). For what Hoyle and

Wickramasinghe actually claimed to have calculated was the

probability that life would have arisen g i v e n that chance

alone was operating. This is not the same as the proba-

bility of chance g i v e n the occurrence of life, which is how

they interpret it. That being so impossibly small, Hoyle and

Wickramasinghe wrongly deduce that the only alternative

hypothesis they can see (God) must be virtually certain.

In more recent examples the errors in calculation are some-

times less obvious, usually because no explicit calculation is

made. Here I mention two very different examples. The ¬rst

lies behind Stephen J. Gould™s claim that if the ¬lm of evolu-

tion were to be rerun, its path would be very different and we,

in particular, would not be here to observe the fact! We shall

meet this example again in chapter 11 where its full signi¬cance

will become clear. Here I concentrate on it as an example of a

fallacious probability argument. Gould visualises evolution as

a tree of life. At its root are the original, primitive, forms from

which all life sprang. At each point in time at which some dif-

ferentiating event occurs, the tree branches. Evolution could

God, Chance and Purpose

82

have taken this path or that. The tree thus represents all possi-

ble paths which evolution could have taken. The present form

of the world is just one of many end-points at the very tip of one

of the branches. All of the other tips are possible end-points

which happen not to have been actualised. In reality our pic-

ture of a tree is limiting, in the sense that there are vastly more

branches than on any tree we are familiar with. The probabil-

ity of ending up at our particular tip is the probability of taking

our particular route through the tree. At each junction there

are several options, each with an associated probability. These

probabilities need not be individually small but, if successive

choices are assumed to be independent, the total probability

will be the product of all the constituent probabilities along

the path. Given the enormous number of these intermediate

stages, the product is bound to be very small indeed. From this

Gould™s conclusion follows “ or does it? His assumption was

tacit but wrong. It is highly likely that the branch taken at any

stage will depend on some of the choices that have been made

earlier and that other environmental factors will be called into

play to modify the simplistic scheme on which the indepen-

dence hypothesis depends. In fact there is no need to rely on

speculation on this point. As we shall see in chapter 11, there

is strong empirical evidence that the number of possible paths

is very much smaller than Gould allows. In that chapter we

shall look at Simon Conway Morris™ work on convergence in

evolution. This draws on the evidence of the Burgess shales

of British Columbia just as Gould™s work did. It shows how

facile it can be to make the casual assumption of independence

and then to build so much on it.

My second example is quite different and is usually dis-

cussed in relation to the anthropic principle.4 It relates to the

The anthropic principle was discussed from a statistical point of view in

4

Bartholomew (1988, pp. 140ff.), where some references are given. It has

What can very small probabilities tell us? 83

remarkable coincidence in the basic parameters of the universe

which seem to have been necessary for a world of sentient

beings to come into existence and to observe the fact! Any list

of such coincidences will include the following:

(i) If the strong nuclear force which holds together the parti-

cles in the nucleus of the atom were weaker by more than

2 per cent, the nucleus would not hold together and this

would leave hydrogen as the only element. If the force

were more than 1 per cent stronger, hydrogen would be

rare, as would elements heavier than iron.

(ii) The force of electromagnetism must be 1040 times

stronger than the force of gravity in order for life as

we know it to exist.

(iii) If the nuclear weak force were slightly larger there would

have been little or no helium produced by the big bang.

In combination, these and other coincidences of the same

kind add up to a remarkable collection of surprising facts about

the universe. It thus turns out that the values of many basic

constants have to be within very narrow limits for anything

like our world to have emerged. Probabilities are brought into

the picture by claiming that the chance of such coincidences is

so remote that we must assume that the values were ˜¬xed™ and

that the only way they could have been ¬xed was for there to be

a supreme being who would have done it. This is seen by many

been used as an argument for God™s existence by, for example, Monte¬ore

(1985), where the list of coincidences which he gives in chapter 3 is somewhat

different from that given here. Other discussions will be found in Holder

(2004) and Dowe (2005). Sharpe and Walgate (2002) think that the principle

has been poorly framed and they wish to replace it by saying ˜that the

universe must be as creative and fruitful as possible™. The most recent

discussion in a much broader context is in Davies (2006). An illuminating

but non-probabilistic discussion will be found in ˜Where is natural theology

today?™ by John Polkinghorne (2006). Dembski (1999, p. 11) regards these

coincidences as evidence of Intelligent Design.

God, Chance and Purpose

84

to be one of the most powerful arguments for the existence of

God “ and so it may be “ but the probabilistic grounds for that

conclusion are decidedly shaky. The argument consists of two

parts. First it is argued that the probability of any one param-

eter falling within the desired range must be in¬nitesimally

small. Secondly, the probability of them all falling within their

respective ranges, obtained by multiplying these very small

probabilities together, is fantastically small. Since this rules

out the possibility of chance, the only option remaining is that

the values were deliberately ¬xed.

Let us examine each step in the argument, beginning with

the second. There is no obvious reason for supposing that

the selections were made independently. Indeed, if there were

some deeper model showing how things came about, it could

well be that some of the parameter values were constrained

by others or even determined by them. The total probabil-

ity might then be much larger than the assumption of inde-

pendence suggests. But surely something might be salvaged

because there is so little freedom in the determination of each

parameter treated individually; this brings us back to the ¬rst

point. Why do we suppose it to be so unlikely that a given

parameter will fall in the required small interval? It seems

to depend on our intuition that, in some sense, all possible

values are equally likely. This is an application of the prin-

ciple of insuf¬cient reason which says that, in the absence of

any reason for preferring one value over another, all should

be treated as equally likely. The application of this principle

is fraught with problems and it does not take much ingenuity

to construct examples which show its absurdity. The best we

can do is to say that if it were the case that all values were equally

likely and also if the parameters were independent, t h e n it

would follow that the creation ˜set-up™, was, indeed extremely

unlikely if nothing but chance were operating. But there is no

What can very small probabilities tell us? 85

evidence whatsoever for either of these assumptions “ they

merely refer to our state of mind on the issues and not to any

objective feature of the real world.

It should now be clear that the various very small probabil-

ities on offer in support of theistic conclusions must be treated

with some reserve. But that still leaves us with the question of

how we should interpret them if they were valid.

i n f e r e n c e f ro m v e ry s m a l l p ro ba b i l i t i e s

The question is: if some event, or proposition, can be validly

shown to have an exceedingly small probability on the chance

hypothesis (whatever that might turn out to mean) on what

grounds can we reject the hypothesis? As noted already, this

is the question which Dembski seeks to answer in his support

of Intelligent Design but, as Dembski himself recognises, this

is not a new issue but one older even than modern statis-

tics. We shall look at Dembski™s arguments in the following

chapter but ¬rst, and without introducing the complications

of the contemporary debate, consider the problem from ¬rst

principles.

First, I reiterate the point made in relation to dealing a

bridge hand selected from a pack of cards, namely that a

small probability is not, of itself, a suf¬cient reason for reject-

ing the chance hypothesis. Secondly, there are three broad

approaches, intimately related to one another, which are in

current use among statisticians. I take each in turn.

I start with signi¬cance testing, which is part of the everyday

statistical routine “ widely used and misused. If we wish to

eliminate chance from the list of possible explanations, we

need to know what outcomes are possible if chance is the

explanation. In most cases all possible outcomes could occur,

so chance cannot be ruled out with certainty. The best we

God, Chance and Purpose

86

can hope for is to be nearly certain in our decision. The basic

idea of signi¬cance testing is to divide the possible outcomes

into two groups. For outcomes in one group we shall reject

the chance hypothesis; the rest will be regarded as consistent

with chance. If the outcomes in the rejection set taken together

have a very small probability, we shall be faced with only two

possibilities (a) the chance hypothesis is false or (b) a very

rare event has occurred. Given that very rare events are, as

their name indicates, very rare, one may legitimately prefer

(a). If we make a practice of rejecting the chance hypothesis

whenever the outcome falls in the rejection set we shall only

be wrong on a very small proportion of occasions when that

is, indeed, the case. The probability that we shall be wrong in

accepting the chance hypothesis on any particular occasion is

therefore very small.

In essence this is how a signi¬cance test works, but it leaves

open two questions (i) how small is ˜small™ and (ii) how should

we construct the rejection set?

This brings us to a second version of the approach, often

associated with the names of Neyman and Pearson, for which

I begin with the second of the above questions. Let us imagine

allocating possible outcomes to the rejection set one by one.

The best candidate for inclusion, surely, is the one which is

furthest from what the chance hypothesis predicts, or alter-

natively perhaps, the least likely. In practice, these are often

the same thing. In fact, in this case, we may de¬ne ˜distance ™

by probability, regarding those with smaller probability as

further away. We can then go on adding to the rejection set,

starting with the least probable and working our way up the

probability scale. But when should we stop? This is where

the answer to the ¬rst question comes in. We want to keep

the overall probability small, so we must stop before the col-

lective probability of the rejection set becomes too large. But

What can very small probabilities tell us? 87

there are considerations which pull us in opposite directions.

On the one hand we want to keep the probability very small

and this argues for stopping early. On the other, if we make

the net too small, we reduce the chance of catching anything.

For the moment I shall describe the usual way that statisticians

take out of this dilemma. This says that there are two sorts

of mistakes we can make and that we should try to make the

chance of both as small as possible. The ¬rst kind of mistake

is to reject the chance hypothesis when it is actually true. The

probability of this happening is called the ˜size of the rejec-

tion region™. The second kind of error is failing to reject the

chance hypothesis when we ought to “ that is when some other

hypothesis is true. This argues for making the rejection region

as large as possible, because that way we make the net larger

and so increase our chance of catching the alternative. The

reader may already be familiar with these two kinds of error

under the guise of false positives and false negatives. If, for

example, a cervical smear test yields a positive result when the

patient is free of disease, this is said to be a false positive (error

of Type I). If a patient with the condition yields a negative

result that is described as a false negative (error of Type II). In

both contexts any rule for deciding must be a compromise.

Let us see how this works out in two particular cases. As

noted earlier, John Arbuthnot claimed to have found evidence

for the divine hand in the records of births in London parishes.

In the eighty-two years from 1629to 1710there was an excess of

male births over female births in every one of those years. If the

chances of male and female births were equal (as Arbuthnot

supposed they would be if chance ruled), one would have

expected an excess of males in about half of the years and an

excess of females in the rest. This is not what he found. In

fact the deviation from the ˜chance™ expectation was so great

as to be incredible if sex was determined as if by the toss

God, Chance and Purpose

88

of a coin. Can we eliminate chance as an explanation in this

case? Arbuthnot thought we could and he based his case on

the extremely small probability of achieving such an extreme

departure from expectation if ˜chance ruled™. It would have

been remarkably prescient of him to have also considered

the probability of failing to detect a departure if one really

existed, but this was implicit in his deciding to include those

cases where the proportion of males, or females, was very

different from equality.

My second example is included partly because it has ¬gured

prominently in Dembski™s own writings, to which we come

in the next chapter. It concerns the case of Nicolas Caputo,

the one time county clerk of New Jersey county in the United

States. It is a well-established empirical fact that the position

on the ballot paper in an election in¬‚uences the chance of suc-

cess. The position is determined by the clerk and, in this case,

the Democrats had ¬rst position on forty out of forty-one

occasions. Naturally, the Republicans suspected foul play and

they ¬led a suit against Caputo in the New Jersey Supreme

Court. If the clerk had determined the position at random,

the two parties should have headed the list on about half of

the occasions. To have forty of forty-one outcomes in favour

of the party which Caputo supported seemed highly suspi-

cious. Such an outcome is certainly highly improbable on the

˜chance™ hypothesis “ that Caputo allocated the top position

at random. But what is the logic behind our natural inclina-

tion to infer that Caputo cheated? It cannot be simply that the

observed outcome “ forty Democrats and only one Republican

is highly improbable “ because that outcome has exactly the

same probability as any other outcome. For example, twenty

Democrats and twenty-one Republicans occurring alternately

in the sequence would have had exactly the same probability,

even though it represents the nearest to a ¬fty:¬fty split that

What can very small probabilities tell us? 89

one could have. Our intuition tells us that it is the combination

of Caputo™s known preference with the extreme outcome that

counts. According to the testing procedure outlined above,

we would choose the critical set so that it has both a very small

probability and includes those outcomes which are furthest

from what we would expect on the chance hypothesis that

Caputo played by the rules.

i n f e r e n c e to t h e b e s t e x p lanat i on

This is a second way of approaching the inference problem.5

I borrow the terminology from philosophers and only bring

statisticians into the picture at a later stage. This is a delib-

erate choice in order to bring out a fundamental idea which

is easily lost if I talk, at this stage, about the statistical term

˜likelihood™. Inference to the Best Explanation is the title of a

book by philosopher Peter Lipton (Lipton 2004), whose ideas

have been seen as applicable to theology. The term inference to

There is an interesting parallel between developments in matters of infer-

5

ence in philosophy and statistics which has not, as far as I am aware, been

remarked upon before. With very few exceptions, these have been entirely

independent. The basic idea of signi¬cance testing is that hypotheses are

rejected “ not accepted. A statistical hypothesis remains tenable until there

is suf¬cient evidence to reject it. In philosophy Karl Popper™s idea of fal-

si¬cation is essentially the same. The likelihood principle, introduced into

statistics in the 1920s, has surfaced in philosophy as inference to the best

explanation, which is discussed in this section. Bayesian inference, to which

we come later in the chapter, is becoming popular in philosophy after the

pioneering work of Richard Swinburne set out in The Existence of God

(Swinburne 2004). In his earlier work he referred to Bayesian inference as

Con¬rmation Theory. A recent example is Bovens and Hartmann (2003).

One important difference is that statisticians give probability a more exclu-

sive place. For example, the judgement of what is ˜best™ in Lipton™s work

does not have to be based on probability. There is an interesting ¬eld of

cross-disciplinary research awaiting someone here.

God, Chance and Purpose

90

the best explanation is almost self-explanatory. It is not usually

dif¬cult to conceive of several explanations for what we see

in the world around us. The question then is: which explana-

tion is the best? What do we mean by ˜best™ and how can we

decide which is best? This is where probability considerations

come in.

Suppose we return home one day and ¬nd a small parcel

on the doorstep. We ask ourselves how it came to be there.

One explanation is that a friend called and, since we were

not at home, left the parcel for us to ¬nd. An alternative is

that a carrier pigeon left it. Suppose, for the moment, that

no other explanation suggests itself to us. Which is the better

explanation? The ˜friend™ hypothesis is quite plausible and

possible; the pigeon hypothesis much less so. Although it is

just possible, it seems scarcely credible that a pigeon should

be dispatched with the parcel and choose to deposit it on a

strange doorstep. The appearance of the parcel seems much

more likely on the ˜friend™ rather than the ˜pigeon™ explanation.

In coming to this conclusion we have, informally, assessed

the probability of a parcel™s being deposited by a friend or

a pigeon and concluded that the friend provides the better

explanation. Let us set out the logic behind this conclusion

more carefully. There are two probabilities involved. First

there is the probability that a friend decided to deliver a parcel

and, ¬nding us out, deposited it on the doorstep. The event we

observe is the parcel. We might write the relevant probability

as P(parcel g i v e n friend). Similarly on the pigeon hypothesis

there is P(parcel g i v e n pigeon). We judge the former to

be much higher than the latter and hence prefer the friend

explanation because it makes what has happened much more

likely. In that sense the friend is the better explanation.

The procedure behind this method of inference is to take

each possible explanation in turn, to estimate how probable

What can very small probabilities tell us? 91

each makes what we have observed and then to opt for the

one which maximises that probability. In essence that is how

inference to the best explanation works in this context.

There is one very important point to notice about all of

this “ it merely helps us to choose between some given expla-

nations. There may be many other possible explanations which

we have not thought of. It does not, therefore, give us the ˜best™

explanation in any absolute sense but merely the best among

the explanations on offer. Secondly, notice that the probabili-

ties involved do not have to be large. It need not be very likely

that our friend would have left a parcel on our doorstep “

indeed, it may be a very unlikely thing to have happened. All

that matters is that it should be much more likely than that a

pigeon put it there. The relative probabilities are what count.

If the probabilities are so very low, there would be a powerful

incentive to look for other explanations which would make

what has happened more likely, but that is a separate issue.

The debate between science and religion, when pursued as

objectively as possible, involves choosing between explana-

tions. Two explanations for life in the universe are (a) that God

created it in some fashion and (b) that it resulted, ultimately,

from the random nature of collisions of atoms (as it has been

put). If we could calculate the probabilities involved, it might

be possible to use the principle to decide between them. Dowe

(2005),6 for example, uses the principle in relation to the ¬ne

tuning of the universe and the evidence which that provides

for theism.

This principle of choice has been one of the key ideas in the

theory of statistics for almost a century. It is also particularly

associated with the name of Sir Ronald Fisher and especially

with a major publication in 1922, although the idea went back

The discussion begins on p. 154 in the section on God as an explanation.

6

God, Chance and Purpose

92

to 1912 (see Fisher Box (1978), especially number 18 in the

list of Collected Papers). In statistics the principle is known as

maximum likelihood, where the term likelihood is used in a

special sense which I shall explain below. One advantage of

drawing attention to this connection is that it facilitates the

elucidation of the limitations of the principle.

Because the ideas involved are so important for inference

under uncertainty, we shall look at a further example con-

structed purely for illustrative purposes. This is still to over-

simplify, but it enables me to repeat the essential points in a

slightly more general form.

Suppose tomorrow™s weather can be classi¬ed as Sunny (S),

Fair (F) or Cloudy (C) and that nothing else is possible. We

are interested in explaining a day™s weather in terms of the

barometric pressure on the previous day. Weather forecast-

ers are interested in such questions though they, of course,

would take many more factors into account. To keep this as

simple as possible, let us merely suppose the pressure to be

recorded, at an agreed time, as either above (A) or below

(B) 1,000 mm of mercury, which is near the middle of the

normal range of barometric pressure. The forecaster is able to

estimate probabilities for tomorrow™s weather outcomes given

today™s barometric pressure. We write these probabilities as,

for example, P(S g i v e n B). This is shorthand for the prob-

ability that it is sunny (S) tomorrow, g i v e n that today™s

barometric pressure is below 1,000 mm (B). The probabilities

available to the forecaster can be set out systematically in a

table, as follows.

P(S g i v e n A) P(F g i v e n A) P(C g i v e n A)

P(S g i v e n B) P(F g i v e n B) P(C g i v e n B)

In the ¬rst row are the probabilities of tomorrow™s weather

g i v e n that the barometer is above 1,000 mm today; in the

What can very small probabilities tell us? 93

second row are the corresponding values if the pressure is

below 1,000 mm today. A forecaster would choose the row

representing today™s pressure and look for the option which

had the largest probability. To make this more de¬nite, suppose

the numbers came out as follows.

0.6 0.3 0.1

0.2 0.4 0.4

If the barometer today is above 1,000 mm, the ¬rst row is

relevant, and since sunny has the highest probability (0.6),

this is the most likely outcome. In the contrary case, fair and

cloudy have the same probabilities. Both are more likely than

sunny but, on this evidence, there is no basis for choosing

between them.

However, the real point of introducing this example is to

look at the problem the other way round. Imagine that a day

has passed and we now know what the weather is like today

but have no record of yesterday™s pressure. What now is the

best explanation we can give of today™s weather in terms of

the barometric pressure on the day before? In other words,

what do we think the pressure on the day before would have

been g i v e n that it is sunny, say, today?

To answer this question, the principle of inference to the

best explanation says we should look at the columns of the

table. In our example there are only two elements to look at:

P(S g i v e n A) and P(S g i v e n B) or, in the numerical exam-

ple, 0.6 and 0.2. The principle says that we should choose

A because it gives a higher probability to what has actually

happened than does B (0.6 as against 0.2). Similarly, if it was

cloudy today we would say that B was the most likely expla-

nation (because 0.4 is bigger than 0.1).

When we look at the probabilities by column we call them

likelihoods. Choosing the largest value in the column is thus

God, Chance and Purpose

94

maximising the likelihood. In statistics we talk about the princi-

ple of maximum likelihood; in philosophy, as inference to the

best explanation. For those with some knowledge of mathe-

matics, we use the term probability function when we consider

the probability as a function of its ¬rst argument, when the

second “ after the g i v e n “ is ¬xed. It is called the likeli-

hood function when it is considered as a function of the second

argument, with the ¬rst ¬xed.

We can put the matter in yet another way by saying that

when we use the probabilities to look forward in time we

are making forecasts; when we use them to look backward in

time we are making an inference to the best explanation. This

prompts a further observation before we leave this example.

The rows in the tables always add up to one because all the pos-

sible outcomes are included, so one or other must happen “

that is, the event that one or other happens is certain. The

columns do not add up to one because, when looking down

a column, it is only the relative values of the entries that

matter.

Likelihood inference was used in this fashion in

Bartholomew (1996) in relation to the existence of God. How-

ever, the likelihood principle is part of Bayes™ rule which has

been used extensively by the philosopher Richard Swinburne

and others. The problem with inference to the best explana-

tion is that it takes no account of anything we may know about

the hypothesis in advance. For example, in a country where

the barometric pressure was almost always high, the fact of

knowing that it was a sunny day would not greatly affect our

propensity to believe that the pressure had been high the day

before. The high prior probability of the pressure being high

is not greatly affected by the additional information that it is

sunny today, because the two usually go together anyhow. The

trouble with comparing likelihoods is that it does not take into

What can very small probabilities tell us? 95

account the relative probabilities of hypotheses at the out-

set. This is what the third approach to inference, Bayesian

inference, does for us.

bay e s i an i n f e r e n c e

I shall not go into the technicalities but, instead, give a further

example which illustrates how the idea works. Incidentally, this

example also demonstrates that if we use likelihood inference

on the question of God™s existence, it produces a rather odd

conclusion. This fact gives added impetus to the search for

something better.

The likelihood comparison always makes the ˜God hypoth-

esis™ more credible than any competing hypothesis. For sim-

plicity, suppose that we wish to compare just two hypotheses:

one that there exists a God capable of bringing into being what-

ever world he wishes, the other that there is no such God. On

the ¬rst hypothesis it is certain that the world would turn out

exactly as God intended and hence the conditional probability

of its happening thus must be one. On the alternative hypoth-

esis we may not be able to determine the probability precisely,

or even approximately, but it will certainly be less than one,

so the ˜God hypothesis™ is to be preferred. I suspect that few

atheists would be converted by this argument and it is perti-

nent to ask ˜why not?™ The whole thing smacks of sleight of

hand. It sounds like cheating if we contrive a hypothesis with

the sole purpose of making what has happened certain. If you

begin with the conviction that God™s existence is intrinsically

unlikely there ought to be some way of making sure that that

fact is taken into account. That is exactly what Bayes™ theorem

does.

Essentially it asks whether the likelihood is suf¬cient to

outweigh any prior odds against the hypothesis. Looked at in

God, Chance and Purpose

96

another way, it allows us to bring our own feelings, judgements

or prior evidence into the equation. Whether or not we should

be allowed to do this is an important question.

In order to get a rough idea of how the theorem works let

us introduce the idea of a likelihood ratio. This is a natural

and simple extension of the ideas already introduced. Let us

go back to the example about weather forecasting. There we

considered two probabilities, P(S g i v e n A) and P(S g i v e n

B). Their numerical values as used in the example were 0.6 and

0.2, respectively. I regarded A as the more likely explanation of

the sunny weather, represented by S, because its likelihood was

three times (=0.6/0.2) the likelihood of the alternative. This

was the likelihood ratio for comparing these two hypotheses.

The bigger it is, the more strongly inclined are we to accept

A as the explanation. However, if we had a justi¬able prior

preference for B we might ask how strong that preference

would have to be to tip the evidence provided by the likelihood

ratio the other way. Bayes™ theorem tells us that if the prior odds

on B were three to one, that would just balance the evidence

from the likelihood ratio the other way. The odds would have

to be less than three to one for the evidence of the likelihood

ratio to carry the day and more than three to one for it to go

the other way.

In general we have to compare the likelihood ratio to the

prior probability ratio. The product of the two determines

which way the conclusion goes.

We shall meet these ideas again in the next chapter when

we move on to the stage where Intelligent Design is debated.

c h a pt e r 7

Can Intelligent Design be established

scienti¬cally?

Intelligent Design has been proposed as a credible scienti¬c alterna-

tive to the theory of evolution as an explanation of life on earth. Its

justi¬cation depends on an extension of Fisherian signi¬cance testing

developed by William Dembski. It is shown, in this chapter, that there

is a fatal ¬‚aw in the logic of his method, which involves a circularity.

In order to construct a test to detect design and ˜eliminate™ chance,

one has to know how to recognise design in the ¬rst place. Dembski™s

calculation of the probability required to implement the method is also

shown to be erroneous.

w h at i s t h e a rg u m e n t a b o ut ?

Intelligent Design is at the centre of one of the ¬ercest debates

currently taking place in the science and religion ¬eld. Its

proponents claim that the scienti¬c establishment is set on

an atheistic course by refusing to countenance the possibility

that the world might contain evidence of design. All they ask

is that design should not be arbitrarily ruled out from the

start and that nature should be allowed to speak for itself; no

special privileges are asked for. The whole debate should then

take place within the bounds of science and according to its

principles of rationality.

The opponents will have none of this, claiming that Intel-

ligent Design makes no claims that can be tested empirically

97

God, Chance and Purpose

98

and, because it cannot be falsi¬ed, it is not science. Many see

it as crypto-creationism masquerading under the guise of sci-

ence. They suspect that it is a scienti¬c front for an ideol-

ogy whose aims are more sinister and which are carefully

concealed.

The United States of America is the birthplace and home of

Intelligent Design1 and it is out of the heady mix of a conser-

vative fundamentalism and the threat of religion™s trespassing

into education that the heat of the debate comes. If evolution

is ˜only a theory™ then why, it is argued, should not other

theories get ˜equal time™ in education? After all, should not

children be given the opportunity to make up their own minds

on such an important matter and should not their parents have

a say in what their children are taught? The proponents of

Intelligent Design, it is alleged, wear the clothes of liberals

pleading that all sides should be given a fair hearing, whereas,

from the other side, the scienti¬c establishment is presented

as a group of reactionaries seeking to control what is taught.

There is an enormous literature on this topic, much of it highly controversial.

1

It is dif¬cult to select a few articles as background reading. A broad, if

uncritical, survey will be found in O™Leary (2004) about a quarter of whose

book consists of notes, which re¬‚ect the author™s wide reading. O™Leary

is a journalist who makes no pretence of being an expert in science. Her

sense of fairness doubtless leads to her tendency to grant ˜equal time™ to

all sides of an argument. This means that minority viewpoints may appear,

to the uninitiated, to carry more weight than they ought. This is true,

for example, of the few creationists who have scienti¬c quali¬cations. The

far more numerous members of the scienti¬c establishment are treated with

less sympathy. A more academic treatment will be found in Peterson (2002).

The journal Perspectives on Science and Christian Faith (the journal of the

American Scienti¬c Af¬liation) has carried two extensive debates that re¬‚ect

the divisions on the subject among more conservative Christians: volume

54 (2002): 220“63 and volume 56 (2004): 266“98. Much of the technical

material is due to William Dembski and this will be referred to in the course

of the chapter.

Can Intelligent Design be established scienti¬cally? 99

This is not a private ¬ght con¬ned to one country but, since

it goes to the heart of what is true in both science and religion,

anyone may join in. Those who start as spectators may well

see more of the game (to change the metaphor) and so have

something to contribute.

The main thesis of the Intelligent Design movement runs

counter to the central argument of this book. Here I am arguing

that chance in the world should be seen as within the providence

of God. That is, chance is a necessary and desirable aspect of

natural and social processes which greatly enriches the poten-

tialities of the creation. Many, however, including Sproul,

Overman and Dembski, see things in exactly the opposite

way. To them, belief in the sovereignty of God requires that

God be in total control of every detail and that the presence

of chance rules out any possibility of design or of a Designer.

To such people, the fact that evolution by natural selection

involves chance in a fundamental way appears to rule out the

design and purpose without which the whole theistic edi¬ce

collapses.Defenceoftheismthusinvolvesshowingthatchance

is non-existent or, at best, is no more than a description of our

ignorance. It is allowed to have no ontological status at all.

The Intelligent Design movement is dedicated to showing

that the world, as we know it, simply could not have arisen

in the way that evolutionary theory claims. This is not to

say that evolution by natural selection could not have played

some part, but that it could have done so only in a secondary

manner. The broad picture could not, they argue, have come

about without the involvement of a Designer. In this chapter I

shall examine the claim that it can be rigorously demonstrated

that chance does not provide a suf¬cient explanation for what

has happened. I shall not be directly concerned with other

aspects of Dembski™s argument, in particular his claim that

information cannot be created by chance.

God, Chance and Purpose

100

Essentially, there are two matters to be decided. Is the logic

of the argument which, it is claimed, leads to the ˜design™

conclusion valid and, if it is, are the probability calculations

which it requires correct? The logic is effectively that of sig-

ni¬cance testing outlined in the last chapter. According to

the chief theoretician among the proponents of Intelligent

Design, William Dembski, the logic is an extension of Sir

Ronald Fisher™s theory of signi¬cance testing. Given Fisher™s

eminence as a statistician, it is appropriate that his voice should

be heard on a matter so close to the core of his subject. Prob-

ability calculations also come under the same heading, so I

shall examine how Dembski made his calculations.

William Dembski has single-mindedly pursued his goal of

establishing Intelligent Design as a credible alternative to evo-

lution in several major books and a host of other articles, books

and lectures. This publication trail started with Design Infer-

ence, in which he set out the basic logic of eliminating chance

as an explanation of how things developed. This was followed

by No Free Lunch and The Design Revolution (Dembski 1998,

2002 and 2004).2 The latter book is subtitled Answering the

Toughest Questions about Intelligent Design and is, perhaps,

the clearest exposition of his basic ideas for the non-technical

reader. He has also collaborated with Michael Ruse in editing

Debating Design; From Darwin to DNA (Dembski and Ruse

2004).

Much of Dembski™s argument is highly technical, and well

beyond the reach of anyone without a good preparation in

mathematics, probability theory and logic. This applies as

much to the material written for a general readership as to

In Zygon 34 (December 1999): 667“75, there was an essay review by Howard

2

J. van Till of both Dembski (1998) and Overman (1997). This is in substantial

agreement with the views expressed in this book. Van Till™s review was

followed by a rejoinder from Paul A. Nelson on pp. 677“82.

Can Intelligent Design be established scienti¬cally? 101

the avowedly technical monograph which started it all off

(Dembski 1998). In one sense this is highly commendable,

because the clarity and rigour which can be attained by this

means offers, at least, the prospect of establishing the ideas

on a secure scienti¬c foundation, so that the debate can take

place within the scienti¬c community.3 However, this fact

poses a serious dilemma for anyone who wishes to engage

with him. If the case against Intelligent Design is made at too

high a level, it will pass over the heads of many of those who

most need to question it. If it is too elementary, it will fail

to treat the opposition seriously enough. One must also bear

in mind the psychology of the readership. A highly technical

treatment can have two opposite effects. On the one hand

there is the tendency, on the part of some, to put undue trust

in mathematical arguments, thinking that anything which is

beyond their reach is also beyond question and almost certainly

correct! On the other hand, others may dismiss it instantly as

they dismiss all such material, on the grounds that what cannot

be expressed in simple everyday language can be ignored as

esoteric nonsense. Neither view is correct in this case. The

extensive theoretical treatment cannot be so easily dismissed,

but neither should it be swallowed whole.

d e m b s k i ™ s a rg u m e n t

To do justice to the subtleties of Dembski™s arguments we

shall have to take things fairly slowly, but the reader may be

The two opposite reactions mentioned in this paragraph will be familiar to

3

anyone who, like the author, has attempted to explain technical “ especially

mathematical “ matters to lay audiences. As noted in the preface, the problem

is acute in a book such as this. It often is a case of being ˜damned if you do

and damned if you don™t™.

God, Chance and Purpose

102

grateful for a simple statement at the outset of the essence of

the situation.

Theuniverseisverylargeandveryold,sotherehasnotbeen

either enough time or enough space for some exceptionally

rare events to occur. Roughly speaking, Dembski claims to

be able to calculate the probability of the rarest event one

could have expected to happen ˜by chance™ somewhere at some

time. It is simply not reasonable to expect any chance event

with smaller probability to have occurred at all. Hence if we

can ¬nd existing biological entities, say, whose probability of

formation by chance is less than that critical bound, we can

infer that they could not have arisen by chance. Hence they

could not have arisen by evolution if that process is driven by

chance. Dembski claims that at least one such entity exists “

the bacterial ¬‚agellum “ and that suf¬ces to establish design

and, necessarily, a Designer.

It is important to notice that the ˜design™ conclusion is

reached by a process of elimination. According to Demb-

ski there are only two other possible explanations: natural law

or chance. Once these are eliminated, logic compels us to fall

back on design. It is not often noted that the entity must not

only be designed but also brought into being.4 There must,

therefore, be a Maker as well as a Designer. Since natural law

can be regarded as a special, but degenerate, case of chance

the main thing is to eliminate chance. That is exactly what a

Fisherian signi¬cance test was designed to do.

t h e e l i m i nat i on o f c h an c e

It can never be possible to eliminate the chance explanation

absolutely. The best we can do is to ensure that our probability

Howard van Till is an exception. Van Till distinguishes ˜the mind-like action

4

of designing from the hand-like action of actualising . . . what had ¬rst been

designed™ (2003, p. 128).

Can Intelligent Design be established scienti¬cally? 103

of being wrong, when we claim to have eliminated chance, is

so small that it can be neglected. Dembski believes that Fisher

essentially solved this problem but that his procedure had two

gaps which can be closed. When this is done, the way is clear

to reach the goal.

I shall begin by recapitulating the basic idea of a test of

signi¬cance, which starts from the assumption that chance is

the explanation and then seeks to demonstrate that what has

happened is not consistent with that hypothesis. This time,

however, I shall use what is, perhaps, the simplest possible

kind of example which still facilitates comparison with what

Dembski actually does. We have already seen this example,

discussed by John Arbuthnot in chapter 6 on sex determina-

tion. Here, as he did, we suppose that the probability of a

female birth is exactly 0.5, independently of all other births.

In other words, it is just as if sex was determined by coin

tossing.

Suppose we have a sample of eight births. The ¬rst step

is to list all possible outcomes. One possibility is that they all

turn out to be male, which we might write as MMMMMMMM;

another would be MFMMFFMM, and so on. Altogether there

are 28 = 256possibilities. The next step is to calculate the prob-

ability of each outcome. Because of the simple assumptions we

have made they all have the same probability of 1/256. The

¬nal step is to construct a rejection set such that any occur-

rence in that set will lead us to reject the hypothesis. Since we

do not wish to reject the hypothesis when it is actually true, the

probability of falling in this set should be small. One possible

way to do this would be to make our rejection set consist of

the two outcomes MMMMMMMM and FFFFFFFF, that is: all

male or all female. It seems clear that if one of these goes in,

the other should too, because they represent equivalent depar-

tures from what one would expect “ roughly equal numbers

of males and females. The probability associated with this set

God, Chance and Purpose

104

of two outcomes is 1/128, which is not particularly small but

it will serve for purposes of illustration if we treat it as ˜small™.

If we now adopt the rule that we will reject the chance

hypothesis whenever we observe all males or all females in a set

of eight births, we shall wrongly reject the hypothesis one time

in 128. Is this a sensible procedure? It certainly ensures that

we shall rarely reject the chance hypothesis when it is, in fact,

true but that would be the case for any set of two outcomes we

might happen to select for the rejection set. What is required,

it seems, is a rejection set which has both small probability

and which ˜catches™ those outcomes which are indicative of

non-randomness or, in Dembski™s terminology, design. At

this point there is some divergence between Dembski and the

traditional statistical approach, as represented by Fisher. It

will be instructive to look at these two approaches in turn.

The Fisherian would want to include in the rejection set

those outcomes which were ˜furthest™ from ˜chance™, in some

sense, that is, from what the hypothesis under test predicts. If

the probability of a male birth is really 0.5 we would expect

around four males in every eight births. An all-male or an all-

female outcome would be the most extreme and these would

be the ¬rst candidates for inclusion in the rejection set. Next

would come those with only one male or female, then those

with two, and so on. The process would stop when the prob-

ability of the rejection set reached the value we had chosen

as the small probability that we had set as the risk we were

prepared to run of wrongly rejecting the chance hypothesis.

To the end of his life Fisher thought that his procedure just

described captured the essence of the way that scientists work,

though he strongly objected to the idea of rejection ˜rules™.

He preferred to quote the signi¬cance level, which was the

size of the smallest set which just included the observed sam-

ple. Nevertheless, the distinction is not important for present

Can Intelligent Design be established scienti¬cally? 105

purposes. Other, perhaps most, statisticians, came to think

that more explicit account should be taken of the alternative

hypotheses which one was aiming to detect. This was done

implicitly in the sex ratio example by constructing the rejec-

tion region starting with those samples whose proportion of

males, and hence females, was furthest from 0.5. Neyman and

Pearson extended the theory to one which bears their name,

by arguing that the rejection set should be determined so as to

˜catch™ those outcomes which were indicative of the alterna-

tives envisaged. Thus, for example, if one were only interested

in detecting a tendency for males to be more common, then

one would only include outcomes where male births predom-

inated.

It is in choosing the critical region that Dembski™s aim is

different. He is looking for outcomes which show evidence

of design, so his critical region needs to be made up of those

outcomes which bear evidence of being non-random. One

example of such an outcome would be MFMFMFMF. As male

and female births are precisely equal, this outcome would not

be allocated to the critical region in the Fisherian approach.

This difference directs our attention to the fact that Dembski

is actually testing a different hypothesis. In our example, the

hypothesis concerned the value of the probability “ whether

or not it was 0.5. The question we were asking was: are the out-

comes consistent with births being determined at random and

with equal probability, in other words, just as in coin tossing?

Dembski is not concerned with the value of the probability

but with the randomness, or otherwise, of the series. It is

not altogether clear from his writing whether Dembski has

noticed this distinction. He does, however, recognise that the

Fisherian scheme needs to be developed in two respects to meet

his needs. First he notes that one has to decide what counts

as ˜small™ in ¬xing the signi¬cance level. Dembski claims to

God, Chance and Purpose

106

have an answer to this question and we shall return to it below.

The other point, which is germane to the discussion of how to

select the rejection region, is that Dembski wishes to eliminate

all chance hypotheses not, as in our example, just the one with

probability 0.5.

Although it is not totally clear to me how Dembski thinks

this should be handled, it is implicit in much of his discussion.

Essentially he wishes to include in the rejection set all those

outcomes which show unmistakable evidence of design. He

calls this property, speci¬ed complexity. An interesting way of

approaching this is through the work of Gregory Chaitin and

his way of measuring non-randomness, which I have already

discussed in chapter 4. Outcomes exhibiting speci¬ed com-

plexity will score highly on a measure of non-randomness and

so will be candidates for inclusion. Of all possible outcomes it

is known that almost all of them appear to be random, so the

proportion which show some pattern form a vanishingly small

set which must, inevitably, have small probability. Dembski™s

approach is slightly different in that he sometimes appears to

use ˜small probability™ as a proxy for ˜speci¬ed complexity™.

This is plausible if one thinks that anything which is designed is

bound to be virtually impossible to construct by chance alone

and hence must have an exceedingly small probability. Con-

structing a rejection region by ¬rst including outcomes with

the smallest probabilities will thus ensure that we only reject

the chance hypothesis in favour of something which has spec-

i¬ed complexity. However, while it is true that any outcome

exhibiting speci¬ed complexity will have small probability,

the converse is not necessarily true.

All of these ideas are put to the test when we come to

consider particular examples, and for Dembski, that means the

bacterial ¬‚agellum. But ¬rst we must return to the question of

what is meant by ˜small™ in this context.

Can Intelligent Design be established scienti¬cally? 107

t h e u n i v e r sa l p ro ba b i l i t y b o u n d :

h ow s m a l l i s s m a l l ?

According to Dembski, one of the defects of Fisherian signif-

icance testing is that it does not say what is meant by ˜small™

when choosing a signi¬cance level. He provides an answer to

this question in what he calls the universal probability bound.

The idea is very simple; the calculation less so. The idea is

that the universe is simply not old enough, or big enough, for

some events to have materialised anywhere at any time. To put

a ¬gure on this requires a calculation of how many events of

speci¬ed complexity could have occurred. This leads to what

Dembski calls the universal probability bound of 1/10150. I

shall not go into the details of his calculation but it depends,

for example, on the number of elementary particles in the

universe (1080), the rate at which changes can take place, and

so on.

It is worth pausing to re¬‚ect on the extreme smallness of the

probability that I have just been talking about. The number

of elementary particles in the universe is, itself, unimaginably

large. It is dif¬cult enough to imagine the number of stars in

the universe but this dif¬culty is compounded by the fact that

every single star is composed of a vast number of elementary

particles. Even when all these are added up we are still many

orders of magnitude short of the number 10150. The only point

of raising this issue is to turn the spotlight onto the importance

ofgettingthecalculationright,ifwhatwearegoingtodonextis

compare our calculated probability with some in¬nitesimally

small bound.

The calculation is not straightforward. Although we shall

not go into details, there are a number of pitfalls in mak-

ing such calculations, which Dembski avoids, even though he

has to invent a whole new terminology to express what he is

God, Chance and Purpose

108

about. To begin with, Dembski ¬nds it necessary to introduce

the notion of what he calls probabilistic resources. This has to

do with the fact that there may have been many opportunities

at many places for a particular event to occur. So the question

we ought to be asking is not whether that event occurs exactly

once, but at least once. Another dif¬culty is that there is simply

not always enough information to make an exact calculation. It

is sensible, however, to err on the safe side, so Dembski™s ¬nal

answer is not, therefore, an exact ¬gure but a lower bound.

This means that the true ¬gure cannot be smaller than this

bound, but may be higher. So if, when we make a compari-

son with another probability, that probability turns out to be

smaller than the bound, it will certainly be smaller than the true

¬gure.

There is one more important aspect of Dembski™s argument

to which we should pay attention. He proposes that the rejec-

tion set should consist of the speci¬cally complex outcomes. At

this point we run up against the fact that Dembski sometimes

appears to regard a rejection set as consisting of one outcome

but this is not strictly true. Just as he introduces the idea of

probabilistic resources so he introduces structurally complex

resources. The former allows for the fact that an event which

has only a very small probability of occurring at a particular

time and place will have a much larger probability if it can

occur at many places and times. Similarly, the latter allows for

the fact that there may not be just one structurally complex

outcome but a number. The probability of observing at least

one of them is, therefore, larger than that of exactly one. In

effect this means that we have a rejection set consisting of sev-

eral outcomes just as I supposed in describing the Fisherian

signi¬cance test.

If we let this collection of speci¬cally complex outcomes

constitute the rejection set, we will achieve two objectives at

Can Intelligent Design be established scienti¬cally? 109

once. First, since the set of speci¬cally complex outcomes is

very small, its size (probability) will also be very small, thus

meeting the ¬rst requirement of a test of signi¬cance. Secondly,

if we reject the chance hypothesis whenever the outcome falls

in this set, we shall never make an error of Type II (false

negative). This is because every element in the rejection set is

certainly indicative of design, by de¬nition. That is, we may

ascribe design to the outcome in such cases without any risk

of making a mistake. This is exactly what Dembski wishes to

achieve.

A moment™s re¬‚ection will show that there is something a

little odd about this line of reasoning. It says that we should

reject the chance hypothesis whenever the outcome exhibits

speci¬c complexity. In doing so, we shall certainly be cor-

rect if design is, in fact, present and our chance of wrongly

rejecting the chance hypothesis will be very small (the size

of the rejection set). However, one may legitimately ask why

we need all this technical apparatus if we know already that

certain outcomes exhibit design. The conclusion is, indeed, a

tautology. It says that if something bears unmistakable evi-

dence of design, then it has been designed! The nature of what

Dembski is doing, and its absurdity, will be even more obvious

when we set it in the context of what he calls ˜comparative ™

methods below. First, I digress to point out the other ¬‚aw in

Dembski™s argument.

t h e p ro ba b i l i t y o f t h e bac t e r i a l f lag e l lu m

Although Dembski spends a great deal of time developing a

version of Fisherian signi¬cance testing designed to eliminate

chance, the main application is to one particular case where

the theory is not much in evidence. This concerns a remark-

able biological structure attached to the bacterium Escherichia

God, Chance and Purpose

110

coli,5 which drives it in the manner of a propeller. The ques-

tion is whether this construction could have been assembled

by chance or whether its presence must be attributed to design.

Dembski makes a rough calculation of the probability that this

structure could have come about by chance and arrives at the

exceedingly small value of 1/10263 . How on earth, one may

wonder, could anyone ever arrive at such a seemingly pre-

cise ¬gure? Inevitably there have to be some approximations

along the way, but he chooses them so as to err on the safe side.

However, there is no need to stay on the details because the

whole enterprise is seriously ¬‚awed. Howard van Till (2003)

has put his ¬nger on the source of the problem. His criticism

is that Dembski™s probability calculation in no way relates

to the way in which the ¬‚agellum might conceivably have

been formed. Dembski treats it as what van Till calls a discrete

combinatorial object. Essentially, Dembski counts the number

of ways in which the ingredients of the ¬‚agellum could be

brought together and assembled into a structure. The bland,

and false, assumption that all of these structures are equally

likely to have arisen then yields the probability.

It is dif¬cult to understand how such an elementary mistake

can have been made by someone so mathematically sophisti-

cated. Possibly it stems from confusion about what is meant

by ˜pure chance™. There are many examples in the literature of

similar combinatorial calculations which purport to show that

such things as the origin of life must have been exceedingly

small. This has already been noted in chapter 6 in relation

The case of the bacterial ¬‚agellum dominates the literature, almost as though

5

it were the only suf¬ciently complicated biological organism. Later in this

paragraph we come to van Till™s discussion of its probability, which was the

main purpose of the paper quoted in note 4 above. In strict logic, of course,

only one case is needed to establish the conclusion that some things are too

complicated to have evolved.

Can Intelligent Design be established scienti¬cally? 111

to the work of du No¨ y and to Hoyle and Wickramasinghe,

u

amongothers.Asnotedinthelastchapter,nobiologisthasever

supposed that such complicated entities can be assembled as a

result of some cosmic shuf¬‚ing system. Indeed, the main point

of Dawkins™ book Climbing Mount Improbable (Dawkins 2006

[1996]) is to demonstrate that complicated structures which it

would be virtually impossible to assemble as discrete combi-

natorial objects could be constructed in a series of small steps

which, taken together, might have a much larger probability

(see ˜Chance in evolution™ in chapter 11, below). According

to evolutionary theory the growth in complexity would have

taken place sequentially over immense periods of time. What is

needed is a model of how this might have happened before we

can begin to make any meaningful calculations. To produce an

argument, as Dembski does, that the ¬‚agellum could not have

been formed by an ˜all-at-once™ coming together and random

assembly of the ingredients is hardly more than a statement of

the blindingly obvious. The inference that Dembski wishes to

make thus fails, even if his universal probability is accepted.

t h e pa r a d ox

We now return to the logic of Dembski™s argument. Because

the fallacy is so fundamental, I shall repeat what was said

above but in a slightly different way.

Dembski has always seen his approach as standing squarely

in the Fisherian tradition, in which no account needs to be

taken of alternative hypotheses. At ¬rst sight this seems to

be a reasonable position to adopt, because any alternative

hypothesis would have to be speci¬ed probabilistically and it

is the express purpose of the exercise to eliminate all chance

hypotheses. It is thus somewhat ironic that Dembski™s logic

can be set out quite simply within the framework of the

God, Chance and Purpose

112

Neyman“Pearson approach to inference. The clarity which

we gain thereby also serves to underline the essentially tauto-

logical character of the formalism.

Let us think of a situation, like the coin-tossing exercise,

in which there are very many possible outcomes, each having

very small probability (in Dembski™s terminology these are

complex). Some of these outcomes will be what Dembski calls

speci¬cally complex. These outcomes exhibit some kind of

pattern which bears the hallmark of design “ let us leave aside

for the moment the question of whether or not ˜pattern™ can

be adequately de¬ned. The essence of the Neyman“Pearson

approach to statistical inference is to choose the rejection set

to include those outcomes which are most likely to have arisen

under some alternative hypothesis. In this case the alternative

is that the outcomes are the result of design. The characteristic

of a designed outcome is that it exhibits speci¬ed complexity.

The rejection set should therefore consist of all those out-

comes.

Now let us consider the consequences of what we have

done. The likelihood of wrongly rejecting the chance hypoth-

esis is very small because speci¬ed outcomes have very small

probability. The probability of correctly rejecting the chance

hypothesis is one (that is, certain) because all outcomes in the

rejection set are certainly the result of design (that is why they

were selected). In other words, we have maximised the chance

of detecting design when it is present. We thus seem to have

a foolproof method of detecting design whose logic has been

made clearer by setting it in the Neyman“Pearson framework

(which Dembski seems to be hardly aware of ). So where is the

catch? The problem is that, in order to construct the rejection

set, we have to be able to identify those outcomes which are

the result of design. If we know that already, why do we need

the test in the ¬rst place?

Can Intelligent Design be established scienti¬cally? 113

One possible response is to say that we only identify design

indirectly through the very small probability which it assigns

tosomeoutcomes.Thiswouldsuggestthattherejectionregion

should be formed of those outcomes which have the smallest

probabilities and leave, in particular, those which are less than

the universal probability bound. In that case we are entitled to

ask why we need the formalism at all. If the rule to follow is to

reject the chance hypothesis whenever the outcome observed

has probability that is so small that it could not have arisen in

a universe as large or old as the one we inhabit, is that not a

suf¬cient ground of itself?

d e m b s k i ™ s c r i t i c i s m s o f c o m pa r at i v e m et h o d s

Dembski is highly critical of what he calls comparative meth-

ods and his reasons are set out in chapter 33 of Dembski

(2004).6 A comparative method is any method which involves

the comparison of the chance hypothesis with an alternative.

Such a method involves selecting one from several possibil-

ities and is thus concerned with the relative rather than the

absolute credibility of hypotheses. At ¬rst sight this is a sur-

prising position to take because there clearly is an alternative

in mind “ that the complexity we observe is the work of a

Designer. However, this alternative clearly has a different

status in Dembski™s mind, presumably because it is not speci-

¬ed probabilistically. There are three comparative methods

in common use which I have already reviewed in chapter 6.

The ¬rst is the Neyman“Pearson approach, which uses the

alternative hypotheses to select the rejection set; the second

Dembski mentions (2004, p. 242) a conference at Calvin College in May

6

2001 on Design Reasoning, at which he spoke. Timothy and Linda McGrew

and Robin Collins are reported as putting Bayesian arguments. In particular

these critics objected to the notion of speci¬cation.

God, Chance and Purpose

114

is the likelihood method, or inference to the best explanation

approach; and the third is the Bayesian method, in which the

alternatives have speci¬ed prior probabilities. Dembski seems

hardly aware of the ¬rst two approaches and concentrates his

¬re on the Bayesian threat. Possibly this is because his own

treatment has been challenged from that quarter and this is

where much current philosophical interest in uncertain infer-

ence lies.

I agree with Dembski™s strictures on the use of Bayesian

inference in this particular context primarily because the intro-

duction of prior probabilities makes the choice too subjective.

In an important sense it begs the question because we have to

decide, in advance, the strength of our prior belief in the exis-

tence, or not, of a designer. Bayesian inference tells us how our

beliefs should be changed by evidence, not how they should be

formed in the ¬rst place. What Dembski seems to have over-

looked is that his method is, in fact, a comparative method and

that it can be seen as such by setting it within the framework

of the Neyman“Pearson theory as demonstrated in the last

section. By viewing it in that way we saw that its tautological

character was made clear and hence the whole inferential edi-

¬ce collapses. Given, in addition, that Dembski™s probability

calculation is certainly incorrect I conclude that Intelligent

Design has not been established scienti¬cally.

is intelligent design science?

Much of the debate over Intelligent Design has not been on

its statistical underpinning but on the more general question

of whether or not it is science. This usually turns on whether

it makes empirical statements which can be tested empiri-

cally. Although this is true, in my view it approaches the

problem from the wrong direction. To make my point it is

Can Intelligent Design be established scienti¬cally? 115

important to distinguish ˜science™ from ˜scienti¬c method™.

Scienti¬c method is the means by which science as a body

of knowledge is built up. Dembski has proposed a method by

which, he claims, knowledge is validly acquired. The question

then is: is this method valid? That is, does it yield veri¬able

facts about the real world? As I noted at the outset, two ques-

tions have to be answered: is the logic sound and is the method

correctly applied? To the ¬rst question my answer is that the

logic is not sound, because the extension proposed to Fisherian

signi¬cance testing is not adequate in itself and also because

almost all statisticians ¬nd the original Fisher method incom-