same, however long since the last one passed. Whether we have

been waiting for one minute or half an hour makes no differ-

ence to our prediction. More exactly, this can be expressed

mathematically to tell us what the frequency distribution of

inter-arrival times should be if this were true. It turns out to

be what is called the exponential distribution. It looks like the

frequency distribution sketched in ¬gure 3.1.

This distribution has its highest frequency near time zero

and declines steadily as the waiting time increases.3 Short

Students are often confused by what seems to be an implied contradiction

3

about the distribution of waiting times. The frequency does decline as stated,

Order out of chaos 35

Fig. 3.1 The form of the exponential frequency distribution

waiting times are, therefore, much more common, relatively

speaking, than large ones. This distribution occurs whenever

the condition for randomness is met, so we might legitimately

refer to it as a law. It is a regular and universal occurrence

under speci¬ed conditions “ and that is what laws usually are “

but it is a law generated by randomness, which seems rather

odd because we tend to think of laws as determining what

will necessarily happen. Here we suppose that there are no

determining factors, and yet we have a law.

But this is not the end of the story, because this exponential

law is not actually what is usually known as Poisson™s law. The

latter arises if we describe the process in another way. Instead

of observing the times between events we could count how

many events occur in ¬xed intervals of time. Standing at the bus

stop we could count how many buses arrive in successive ten

minute intervals, say. The result would be a string of numbers

which might be something like the following:

yet the expected waiting time is constant no matter how long one has waited.

The ¬rst relates to someone newly arrived, the second to someone who has

already waited some time. When you join a queue the chance of having

to wait time t certainly decreases as t increases. But once you have been in

the queue for a length of time t your position is no different from that of

someone who has just joined.

God, Chance and Purpose

36

3, 0, 1, 3, 2, 0, 0, 2, . . .

These do not look very lawful, but if we collect them together

a pattern begins to emerge. In a long sequence the successive

proportions will display a characteristic pattern known as the

Poisson distribution or Poisson™s law. In earlier times it was

sometimes referred to as the Law of Small Numbers (in con-

trast to the Law of Large Numbers) because it showed that,

even with rare events, there may be stable patterns.

I do not need to go into the mathematical details beyond

saying that the proportion of 3s, say, is simply related to the

proportion of 2s, and so on. The former is obtained by multi-

plying by the average and dividing by 3 (or whatever the cur-

rent number is). There are simple tests which can be applied

to see whether the proportions do conform to this pattern, and

it is that pattern which constitutes the law.

In reality it is, as we have seen, just another way of describ-

ingrandomnessintime.Thereisnosenseinwhichoccurrences

have to conform to the law. It is merely a description of what

happens when there is no law.

At this point the reader may still be slightly misled by the bus

example that I have used to illustrate these ideas. The objection

may be raised that there are, in fact, reasons why buses arrive

when they do. The fact that we are unable to predict them is

because we lack the necessary knowledge. This is correct, and

it illustrates the limitation of the illustration and also hints at

serious theological issues. Must there be a reason, as in the bus

example, or could it be there is no reason at all.

This brings us to the example of the emission of radioactive

particles. These do seem to conform to the requirements of

randomness in time with little prospect of our being able to

look behind the phenomenon and discover what is going on. It

is in such circumstances that some theologians have invoked

Order out of chaos 37

Fig. 3.2 The form of the normal frequency distribution

God as the controller. After all, if every event must have a

cause,andifnoothercausecanbedetected,theonlyalternative

seems to be God.4 This is a matter to which we must return

but the point being made here is quite simple. Even when we

specify a process which seems to be totally random, patterns

emerge which display all the characteristics we associate with

the idea of law. This must have profound implications for what

we mean when we speak of the lawfulness of the world and

what it means to attribute those laws to God.

t h e n o r m a l law

The oldest and best known of all statistical laws is the so-called

Normal Law or, re¬‚ecting its origin, the Normal Law of Error.

This goes back to Gauss but is now known to be so funda-

mental and widespread that its historical origins are incidental.

The Normal Law5 refers to the shape of a frequency distribu-

tion often described as bell-shaped. Figure 3.2 illustrates the

shape.

This would be the view of all those who take a broadly Calvinistic position.

4

The Normal Distribution is often referred to as the Bell Curve, a term made

5

popular by the somewhat notorious book with that title by Richard Herrn-

stein and Charles Murray (Free Press Paperbacks, 1994) on intelligence and

class structure in American life. Ironically the name is somewhat tangential

to the main message of that book.

God, Chance and Purpose

38

A quantity which is normally distributed will have a fre-

quency distribution like that shown above, where most values

are clustered in the middle of the range and then tail off towards

the extremes. It occurs widely in nature “ human heights are

one example. Is this because it was built into the creation as

something to which a wide range of phenomena would be

required to conform? The answer is, almost certainly, no. It

may be thought of as another example of order out of chaos.

It simply describes one of the regularities of randomness. In

other words it takes a world of chance to produce this law.

A simple example will help to illustrate one way in which

the law comes about. As always in this book, I shall ignore the

technical conditions which must be imposed if the mathematics

operating behind the scenes is to deliver the results. For present

purposes this is of no practical signi¬cance and enables us to

penetrate to the essential idea with greater ease.

Imagine that we are interested in the age distribution of the

members of a human society. Because they are too numerous

to question individually, we select a random sample. (A fuller

discussion of random sampling is given in chapter 10.) A ran-

dom sample is one chosen so that all possible samples have an

equal chance of being selected. In other words it is the fairest

possible method of selection “ no sample has any advantage

over any other. If the sample is fairly small “ under a hundred,

say “ it will not be too dif¬cult a job to ascertain all the ages.

Suppose we do this and ¬nd the average. Now imagine this

to be repeated many times on different, independent, samples

with the average being computed on each occasion. Surpris-

ingly the frequency distribution will have a form very close to

the Normal Law. The remarkable thing is that we can say this

without any knowledge whatsoever of the population age distri-

bution. The result must, therefore, be due to the method of

drawing the samples and have little to do with the underlying

Order out of chaos 39

distribution itself. It is thus a product of the sampling mech-

anism, and human choice had nothing to do with it nor did it

depend on any other feature of the world. Order, in the shape

of the distribution, has come out of the randomness of the

sampling procedure. It tells us nothing about any Creator but

more about our own ingenuity “ or does it?

To take this example a little further, I return to the emission

of particles from a radioactive source. If we observe many

emissions we shall ¬nd that the frequency distribution of the

time to the next emission is highly skewed. As already noted,

short intervals will be the most common and the frequencies

will decline as the duration increases. This is nothing like the

normal distribution, but the normal is not far away. Suppose

we repeatedly form the average of, say, twenty emission times

and look at their distribution. It will turn out that this is quite

close to normal. Once again, this has little to do with the

waiting-time distribution but neither, on this occasion, can

we attribute it to the randomness of sampling, for we did not

sample the intervals but took them as they came. This time

the randomness lies in the variation in waiting time and the

independence between successive emissions.

The surprising thing here is that the form of the waiting-

time distribution has very little to do with the result. It would

have been much the same whatever that distribution had been.

The unpredictability of the individual emission turns into the

predictability of the ¬nal distribution, with minimal conditions

on what the initial distribution is. It does not even matter

whether all the distributions are the same nor is it vital that

they be totally independent.

The ubiquity of the normal distribution arises not so much

because it is built into the structure of the world but rather that

it is imposed onto the world by the way we choose to observe

it. It may, therefore, tell us very little about the intentions of

God, Chance and Purpose

40

the Creator. But it does tell us that order and randomness are

so closely tied up together that we can hardly speak of one

without the other. Total disorder seems impossible. Order is

discernible at the aggregate level in even the most chaotic

micro-level happenings.

dy na m i c o r d e r

Order can be more subtle than that considered so far. It can

refer to stable patterns which develop and persist over time.

Stuart Kauffman6 has explored this kind of order over many

years in an effort to discover how biological complexity might

haveemerged.Ishalllaterdismissthecommonsimplisticargu-

ments about the origin of life which imagine all the ingredients

shaken up together and linking themselves together at random.

This kind of process bears no relation to any kind of plausible

biological process. Kauffman™s approach may be crude but it

is far more realistic in that it allows the interaction among the

ingredients to play their part. Kauffman™s aim is not to model

biological processes in minute detail “ that would be quite

Stuart Kauffman is a theoretical biologist who has championed the view

6

that the self-organisation of the relevant molecules may have created the

complexity necessary for life to appear. This has attracted the opposition

of Overman because it would mean that life would have appeared spon-

taneously, without the need for any Designer. Kauffman, himself, sees his

own approach as meeting the objections of those who claim that life could

not have arisen ˜by chance™. For example he draws attention to the extremely

small probabilities for the origin of life given by Hoyle and Wickramasinghe

1981 (see Kauffman 1993, p. 23) as showing that such a random assembly

was impossible. Kauffman sees such calculations as irrelevant. If we reject

special creation, this leaves the way clear for his own approach. There are

other, chemical, arguments for believing that life might arise spontaneously

in the conditions obtaining on the primeval earth. Recent examples are

reported on page 12 of New Scientist, 18 January 2003 and page 16 of 16

December 2006.

Order out of chaos 41

impossible “ but to capture the essential qualitative features

of this system. This may depend on the broad characteristics

of the system rather than its details. He constructs a model

consisting of elements linked together in a random fashion

and which exert an in¬‚uence on one another via links. In real-

ity the elements might be genes or enzymes but, at this stage,

it is the broad qualitative properties of such systems that I

wish to study. Many results for systems of this kind are given

in Kauffman (1993) or, at a more popular level, in Kauffman

(1995).

For illustrative purposes Kauffman imagines the elements

as being electric light bulbs which can be on or off. At any

one time, therefore, there will be a display of lights formed by

those bulbs which happen to be on.

The pattern now changes in the following manner. To take

a particular case, let us suppose that each bulb is connected to

three other bulbs, selected at random. The state of the initial

bulb at the next point in time is supposed to depend on those of

the three bulbs to which it has been connected. There has to be

a rule saying how this works. One possibility is that the initial

bulb will only light if all those to which it has been connected

are also alight. What has been done for one bulb is done for

all, so that we end up with a network of connections, such

that the state of any bulb at the next step is determined by the

three to which it was connected. Such a system is set off by

having an initial set of bulbs illuminated. Successive patterns

will then be generated by the on/off rules. A network of this

kind is called a Boolean7 net because the two sorts of rules

correspond to the and or or rules of Boolean logic. One or

George Boole (1815“64) was a British mathematician who is remembered

7

as the originator of Boolean algebra, which is central to computers which

work with binary numbers.

God, Chance and Purpose

42

other of these rules is attached to each bulb for determining

whether it is switched on or off.

The question now arises: what will happen if such a sys-

tem is allowed to run inde¬nitely? The ¬rst thing is to note

that there will usually be a very large number of possible pat-

terns. For example if there are 100 bulbs there would be 2100

possible patterns, which is an extremely large number. More

realistic examples with, say, 1,000 bulbs, would be capable of

generating an unimaginably large number of possible states.

To run through the full gamut, even at the rate of one million

per second, would take longer than the time the universe has

been in existence! This might lead one to expect a seemingly

endless succession of meaningless patterns.

The surprising and remarkable thing is that this may not

happen. In some circumstances the system may reach a rela-

tively small set of states around which it cycles for ever. This

small set may have only a few hundred patterns which is neg-

ligibly small compared with the original set. It turns out that

the critical numbers in determining the behaviour of the sys-

tem are the number of bulbs and the number of other bulbs to

which each is linked. We used three as an illustration. It turns

out that if there is only one connection, the behaviour is rather

uninteresting, whereas if it is large, greater than four or ¬ve,

say, the behaviour is chaotic and there are unlikely to be any

simple patterns. It seems that two connections are capable of

producing more interesting behaviour, sandwiched between

the monotony of just one connection and the chaos of many

connections. It is on this borderland of chaos and order that

the interesting things happen. There is just suf¬cient order

to produce simple patterns which persist, but not so much

as to render the result chaotic. Order on the edge of chaos

seems to be a key characteristic necessary for the emergence

of interesting phenomena.

Order out of chaos 43

The patterns here are not static. The order lies in the

repeatability of a small set of states through which the

system continually cycles. The order, remember, arises from

a random speci¬cation of the connections. It does not require

any detailed design work to produce the patterns. These

persist in spite of the randomness in the structure. It can

also be shown that other minor variations of the rules gov-

erning the behaviour leave the broad characteristics un-

changed.

i t ™ s a s m a l l wo r l d

There are about six billion people in the world so it would seem

obvious that almost all of them are so remote from us that we

could have nothing in common. Yet there are so many sur-

prising occurrences “ as when we meet two people in different

spheres who turn out to know one another, and we comment

˜it™s a small world™. This turns out to be truer than we might

think when we start to wonder how we might measure our

closeness to other people. There is a considerable literature

on this and related phenomena to which Barab´ si (2003) and

a

Buchanan (2002) provide a useful point of entry.

There are some well-known examples of the phenomenon.

Two widely discussed cases relate to the media and mathemat-

ics. Kevin Bacon acted in ¬lms. Other actors who appeared

with him in a particular ¬lm were obviously close to him in

a certain sense. People who did not appear in any ¬lms with

him, but who appeared with someone who had appeared with

Kevin Bacon, are less close. Let us say that they are distance

2 from Bacon. Those who came no closer than acting with

those with number 2, may be said to have distance 3, and so

on. What is the distribution of these distances, or, what is a

typical distance? It seems that the typical distance is around

God, Chance and Purpose

44

6 which, given the number of actors and ¬lms seems to be a

remarkably small number.

Mathematicians and many others whose work involves

mathematics have what is known as an Erd¨ s number. Paul

o

Erd¨ s was a proli¬c mathematician who collaborated with

o

many other mathematicians. One™s closeness to Erd¨ s can be

o

measured by one™s Erd¨ s number. Those who co-authored

o 8

a paper with Erd¨ s have an Erd¨ s number 1; there are about

o o

509 of them. Those who collaborated with a collaborator,

but not with Erd¨ s himself, have an Erd¨ s number 2; there

o o

are currently about 6984 of these and their names have been

published. Almost every mathematician has a small Erd¨ s o

number “ usually in single ¬gures. Again, this seems rather

remarkable given the vast size of the published literature in

mathematics.

It turns out that this phenomenon is widespread and in

totally different ¬elds, ranging from power grids to the study

of infectious diseases. Is there something going on that we have

missed or is this another example of order arising out of chaos?

The chaos in this case is the haphazard collection of links which

are formed between pairs of individuals. ˜Haphazard™ seems to

be the right word to use here because it conveys a sense of lack

of purpose. There may, indeed, be purpose in the formation

of individual links, but it is the overall pattern with which we

are concerned. It does not need anyone to oversee the overall

process. Purely local connections of a haphazard nature are

suf¬cient for the ˜small-world™ structure to emerge.

The study of small-world phenomena appears to have

begun with an experiment conducted by Stanley Milgram9

The Erd¨ s Number Project has a website: http://www.oakland.edu/enp,

o

8

from which this and similar information can be obtained.

Stanley Milgram (1933“84) was a psychologist who served at Yale Univer-

9

sity, Harvard University and City University of New York. While at Yale

Order out of chaos 45

in 1967. His work was concerned with how far any of us might

be from some public personage. We might ask whether we can

¬nd links through personal acquaintances which stretch from

us to the said personage. Although many of the details of the

experiment might be criticised, it established the hypothesis

that the average number of such links was about six. If one

wanted to establish contact, for example, with the President

of the United States by going through ˜someone, who knew

someone, who knew someone . . .™ it should not therefore need

too many steps to do so.

o r d e r i n n et wo r k s

The small-world phenomenon is one example of what happens

in networks created by linking individuals in a haphazard way.

Networks10 are everywhere and have become such a familiar

feature of our world that the word has passed into the language

he conducted experiments on obedience to authority which attracted a good

deal of criticism on the grounds that they were unethical. Also at Yale, in

1967, he conducted his ˜small-world experiment™. This was an empirical

investigation in which he sent letters to sixty randomly chosen people in

Omaha and Nebraska. Recipients were asked to hand the letters to some

one they thought might be able to reach the target “ a stockbroker in

Sharon, Massachusetts. The results were somewhat ambiguous and were

subject to a good deal of criticism but it was a truly pioneering study and

has proved a point of reference for much work since on the small-world

phenomenon.

There has been a great deal of interest in networks in the last few years. This

10

has resulted from the discovery that many of the networks of real life share a

common architecture. Much of this work has been done by physicists who

have seen parallels between the physical systems with which they work

and networks arising in the social sciences. Since about the year 2000 there

have been several books published which aim to make the results obtained

available to the wider public. Some, such as Strogatz (2003) and Barab´ si a

(2003), are authored by active researchers in the ¬eld. Others, such as

Buchanan (2002), are by experienced popularisers who are able to take a

God, Chance and Purpose

46

as a verb. In spite of their diverse origins, networks display

many common features remarkable for their simplicity. They

are a striking example of order out of disorder. Even highly

random networks exhibit order.

The simplest example is appropriately called the random

net. This appears to have had its origin in the study of neural

nets, where the interest was in such things as paths through

the network along which communication could take place.

A clutch of papers was published in and around the 1950s in

the Journal of Mathematical Biophysics by Anatol Rapoport11

and others. It was quickly recognised that the results also had

applications in the theory of epidemics, where the nodes are

people and the links are paths of infection. This work was

summarised in chapter 10 of Bartholomew (1982) (though

rather more detail is given in the ¬rst two editions of that

book, 1967 and 1973).

Rather surprisingly this work was not noticed by the math-

ematical community until a new impetus was given to it when

Erd¨ s and R´ nyi weighed in with a series of eight papers

o e

on random graph theory beginning in 1959. (The details will

be found at the top of page 245 in Barab´ si (2003).) This,

a

together with the growing power and accessibility of comput-

ers, opened the way for the rapid contemporary development

of the ¬eld. An interesting example of the application of a

random net is given by Kauffman in his book Investigations

(2000, pp. 35ff.). Kauffman™s interest is in the chemistry behind

broad but informed view. On 13 April 2002 the New Scientist published an

article by David Cohen entitled ˜All the world™s a net™.

Anatol Rapoport (1911“2007) was born in Russia but emigrated to the

11

United States in 1922. He has worked on a wide range of topics but the

focus was on the mathematics of decision making. From 1970 he was based

at the University of Toronto where he held chairs in Psychology and

Mathematics and in Peace and Con¬‚ict Studies. It is surprising that his

work on nets was unknown to Erd¨ s and his associates.

o

Order out of chaos 47

the origin of life but what he calls ˜the magic™ comes across

without going into the background.

Kauffman supposes that we have a large number of buttons

scattered across a hard wooden ¬‚oor. We begin by picking two

buttons at random and joining them with a coloured thread.

Next we repeat the process, joining another pair of buttons.

It could happen that one or both of those picked the second

time could include one of the original pair. This process is

repeated again and again. Every so often we pause and pick

up a single button chosen at random. It may turn out to be an

isolated button or it may be linked to others which will also

be lifted because of the threads which tie them together. In

the early stages these linked clusters will usually be small but

as time goes on they will get larger. One way of charting the

progress of this clustering is to see how the size of the largest

connected cluster increases as the number of threads (con-

nections) increases (expressed as a proportion of the number

of buttons). The magic occurs when this proportion comes

close to a half. Below a half the size of the largest cluster will

be small, but as it passes through a half it increases rapidly

and soon comes close to the maximum possible. In chemical

terms this is a phase change. We move from a situation in which

groups are small to one in which connectivity is high. This

happens without any ˜design™. The connections are purely

random and yet this haphazard sort of linking produces a situ-

ation which, to some eyes, might appear to express purpose or

design.

It is worth re¬‚ecting on this example a little longer to see

what these random connections have done for us. Suppose,

instead, that we initially lay the buttons in a straight line. If we

now connect neighbouring pairs we shall eventually create a

single cluster and the number of links will be one fewer than

the number of buttons. Expressed as a proportion, the ratio of

God, Chance and Purpose

48

links to buttons is very close to one “ much higher than was

obtained using random connections. The clusters obtained by

the two methods are likely to be very different and for some

purposes one or the other might be preferred. The point is

that it does not take many links, randomly assigned, to create

something very close to a single structure.

Networks can be created in all sorts of ways and the features

of interest may vary greatly. What does not vary is the presence

of patterns or regularities which seem to appear in spite of the

chaotic way in which things happen.

Epidemics provide another example. Infections such as

in¬‚uenza spread in human populations largely through con-

tacts. Who meets whom may be fairly random as in business,

shopping, education, and so on. Contact, in individual cases,

may be planned or unplanned yet the spread may be very

rapid. Diseases such as AIDS spread through sexual contacts.

Rumours spread in a manner rather similar to infections. They

are passed from friend to friend and often, it seems, at light-

ning speed. A suf¬ciently hot piece of news will spread quickly

enough without anyone going to the trouble of engineering

the spread. This fact is well known by those who wish to

counter rumours. To counter such a rapid spread, broadcast-

ing in some form which reaches everyone almost immediately

is essential.

There is a very extensive scienti¬c literature on how epi-

demics, whether of infection or rumour, spread. Not surpris-

ingly, some things about such processes are predictable and

constant in spite of the fact that they are driven by purposeless

activity. In the simplest examples, anything spread in a given

population will eventually reach everyone if enough time is

allowed. This is because in a freely mixing population every-

one will meet everyone else at least once and possibly several

times. But not many epidemics are like this; usually there is

Order out of chaos 49

some self-limiting factor. People who become infected, for

example, will withdraw from the population to recover. This

is voluntary isolation but, in serious cases, such as SARS,12

they will be removed from the exposed population. Rumours,

similarly, often die out before the epidemic has run its course.

Passing on a piece of salacious gossip ceases to be attractive

if potential hearers have already heard. This may cause the

spreader to give up.

Models have been constructed to quantify some of these

intuitively based observations. Under suitable conditions it is

possible to predict the proportion of those who have heard

or been infected at any stage of the epidemic. The growth

curve also has a characteristic shape, rising slowly at ¬rst,

then more rapidly before ¬nally levelling off. This shape also

depends remarkably little on the precise details of how the

spreading takes place. What is interesting for present purposes

is that haphazard and seemingly unguided processes prove to

have more pattern than we might have guessed. Randomness

achieves easily that which, by design, might have been very

dif¬cult.

In all of this work it is supposed that the population of nodes

(e.g. people) is ¬xed. In practice many networks are constantly

growing; the World Wide Web is, perhaps, the prime example.

It is found that such networks differ in some important respects

from the random net. For example, in a random net the number

of incoming paths to a node will vary but not by much. The

actual numbers will have a distribution fairly tightly packed

around the average. Growing networks are not like this. They

have a number of what are called hubs which are at the centre

of large numbers of links. To account for this Barab´ si anda

his colleagues proposed a model leading to what they called

Severe Acute Respiratory Syndrome.

12

God, Chance and Purpose

50

scale-free networks. They supposed that as each new node

appeared it established links with already existing nodes. The

choice of destination could be purely at random or the prob-

ability could be proportional to the number of links already

established. It is intuitively clear that nodes which appeared

near the beginning of the process will tend to have more

links, simply because they have been around longer and so

have had more opportunity. The effect of growing in this

way is to create a hierarchy of hubs, some with very large

numbers of links. The distribution of the number of links per

node now becomes highly skewed, with large numbers hav-

ing very few links and a few with a very large number. This

pattern is preserved as the system grows and hence accounts

for the description scale-free. For our purposes it is important

to note that a large measure of randomness is still present in

the formation of the net but the resulting pattern, though still

simple, is very different from that of a random net. It is also

interesting to note, in the context of this book, that one of

the examples Barab´ si uses is the spread of early Christian-

a

ity through the activities of St Paul, who created a network

as he travelled between centres of population in the ancient

world.

The number of links per node emerges from these analyses

as a key descriptor of a network. Once we get away from the

random net, the distribution of that number is, as we have

seen, highly skewed. But more can be said about it than that.

The long upper tail of that distribution can be approximated

by what is known as a power law. Such distributions have

been known for a long time in other contexts of which the best

known is, perhaps, the Pareto Law of Income Distributions.

They are yet another example of order out of disorder. Much

of that work is brought together in chapter 7 of Bartholomew

(1982).

Order out of chaos 51

a p p rox i m at e o r d e r

Appearances can be deceptive. Order can appear, persist for a

time and then disappear. I describe a simple example merely

to illustrate the possibilities. In a sense, the example is like a

sequence of independent tosses of a coin with the difference

that the probability of a head at any toss changes from time

to time depending on the outcomes of previous tosses. The

experiment goes as follows. Toss a fair coin 100 times, say, and

compute the proportion of heads. Make a further 100 tosses

but this time make the probability of a head be equal to the

proportion of heads in the previous 100 tosses. Repeat this

procedure and at each stage let the probability of a head be

equal to the proportion of heads in the previous set of 100

tosses. What will happen in the long run to the probability?

Intuition might lead us to expect it to remain around 0.5. We

expect to get about 50 per cent of heads the ¬rst time round

that should lead to around the same ¬gure next time, and so

on. In the very long term this argument will let us down.

Sooner or later all of the 100 tosses will be all heads or all

tails. When that happens the future is ¬xed, because if we

estimate the probability to be 0 or 1 we shall get nothing but

all heads or all tails for ever. Nevertheless, in the medium

term, the probability is likely to remain around 0.5 for some

time, thus exhibiting a degree of order which, though giving

the appearance of constancy, is doomed to disappear. Thus

order, when it arises, is not necessarily something which is

permanent.

sync

Christiaan Huygens invented the pendulum clock in about

1655. Ten years later he suffered a slight indisposition and

God, Chance and Purpose

52

was in a room with two such clocks when he noticed ˜a mar-

vellous thing™ about which he wrote to his friend, Sir Robert

Moray.13 He noticed that two similar clocks, hanging together

in the same room, were beating in time. This was an early

and very simple example of synchronisation. This might have

happened because the clocks were set going together and,

being very accurate timekeepers, had kept in step ever since.

But Huygens found that, however they were started, they

would always synchronise in about half an hour. There must

have been some means by which the action of one clock

in¬‚uenced the other in a manner which brought them into

harmony.

Pendulum clocks are not the only things which interact in

this way. Synchronisation is a very widespread phenomenon

and includes some spectacular examples. Fire¬‚ies, as their

name implies, produce ¬‚ashes of light and these occur at reg-

ular intervals. If one encountered a large number of ¬‚ies, all

¬‚ashing, one would expect to see a chaotic mixing of the indi-

vidual ¬‚ashes but this is not what happens. In Southeast Asia

and elsewhere thousands of ¬re¬‚ies gather along river banks

and provide a spectacular display of synchronised ¬‚ashing.

Sometimes this type of phenomenon can be harmful, as in

epilepsy, when the synchronised activity of many brain cells

can lead to convulsions. At other times it can be essential as

when many pacemaker cells in the heart combine to produce

a regular heartbeat. This kind of phenomenon is remarkably

widespread, which suggests that there is a common pattern in

what is going on which might be susceptible to mathematical

analysis. This turns out to be the case and the topic is the

subject of a fascinating account by Steven Strogatz, one of

Further details of this example and some further background will be found

13

in Strogatz (2003, chapter 9).

Order out of chaos 53

the principal researchers in this ¬eld. (See Strogatz (2003) for

example.)

This phenomenon is certainly an example of spontaneous

order arising from disorder. It is not necessarily a case of

order arising from chance, so some justi¬cation is required

for its inclusion here. The essential point is that there is no

overarching control which produces the order represented by

the regularly beating heart. There is no central timekeeper or

conductor orchestrating the behaviour of ¬re¬‚ies. The pat-

tern arises from local interactions which bear no obvious rela-

tionship to the global pattern. The clocks will beat in time

however much or little their starting positions differ. It would

make no difference if the starting positions were determined

randomly and that means that this aspect does not have to

be designed or planned. Viewed theologically, the regularities

can be achieved without any detailed planning and hence there

needs to be no detailed control at the micro level. Spontaneous

synchronisation can be achieved almost for nothing.

c on c lu d i n g r e m a r k s

I have ranged widely over many ¬elds to show that order often

arises as a consequence of disorder. The ˜laws™ I have consid-

ered might speak to the uninitiated of the direct intervention of

a God, at every stage, to ensure that things turned out exactly

as he desired. In fact the beautiful shape of something like the

Normal Distribution might be regarded as eloquent testimony

to the nature of the Designer. But as Mark Twain memorably

remarked, ˜it ain™t necessarily so™. Lawful behaviour in the

shape of statistical laws seems to emerge naturally from an

underlying chaos. Indeed one might go so far as to say that

chaos is a precondition of the order which, to some, speaks so

eloquently of the divine mind.

God, Chance and Purpose

54

The full theological implications of all this are profound but

a discussion of them must wait until more pieces of the jigsaw

are in place. For the present, I note that the involvement of

God in the creation may be much more subtle than theologians,

rushing to premature closure of the science“religion debate,

are prepared to allow.

c h a pt e r 4

Chaos out of order

The transition from chaos to order is not a one-way process. Just

as order can result from chaos, so can chaos result from order. In

this chapter I describe three ways in which this may happen. These

are: accidents, pseudo-random numbers and mathematical chaos. This

transition, also, has to be taken into account by our theology. If these

were the only sources of uncertainty in the world there would be no

need to invoke pure chance or to explain how it might be consistent

with divine purpose.

d i s o r d e r g e n e r at e d b y o r d e r

Our theological assertions about what God can or cannot do

in the world depend very much on what kind of place we

believe the world to be. In the last chapter we encountered

the somewhat disconcerting fact that much of the order and

lawfulness which we so readily attribute directly to God has

its roots in disorder. But this is only half of the story “ a great

deal of disorder attends the regularities that are all around us.

In this chapter, therefore, we shall look at the other side of the

coin as a prelude to seeing whether the apparently paradox-

ical situation which faces us can be resolved. For example, if

the purposefulness of God is to be discerned anywhere, one

would expect it to be in the regularity and predictability of the

aggregate. But, if this is the case, how does God engineer it?

55

God, Chance and Purpose

56

On the other hand, if the motions of individual molecules in

a gas are to be thought of as purposeless and undirected, how

can we attribute purpose to the aggregate behaviour which is

built on this irregularity?

This will demand a reassessment of what it means to say

that happenings in the world reveal the purposeful activity of

God “ or the lack of it. Here we shall look at three manifesta-

tions of chaos out of order: ¬rst, accidents, which in retrospect

can often be seen as determined by antecedent events but which

constantly surprise us; secondly, pseudo-random numbers “

this may seem to be an esoteric branch of mathematics but, in

reality, it is very big business in applied science of both the nat-

ural and social varieties; ¬nally, chaos theory, which, in a more

technical sense, has come into great prominence in recent years

and has sometimes been described as the ˜third great revolu-

tion in physics this [the twentieth] century™. Some theologians

have seen chaos theory as a lifeline offering a means for God to

exercise control without disturbing the orderliness of nature.

To me this claim seems premature and unconvincing.

ac c i d e n ts an d c o i n c i d e n c e s

Accidents often appear to be unintended and unpredictable

happenings. The phrase ˜accidents will happen™ expresses both

their inevitability in the long run and the uncertainty sur-

rounding particular occurrences “ as does the remark about

˜accidents waiting to happen™. Speaking of ˜happy accidents™

reminds us that it is not the consequences of the happening

that de¬ne an accident. The essential idea behind the word is

of something that is not planned or intended. There are no

obvious causative factors and, though there may well be rea-

sons for accidents, they are largely hidden from us. In other

words, they bear all the signs of being chance events. Yet they

can happen in systems that are fully determined.

Chaos out of order 57

Coincidences are closely related to accidents. Again there

is no suggestion that they are unnatural in any sense; two

events just happen to have occurred at the same time and

place. The distinctive thing about them is the juxtaposition of

two unrelated events which take on a new meaning because of

their coincidence.

The striking of the earth by a large asteroid may, except

perhaps in the last stages of its approach, have all the appear-

ances of an accident in the sense that it was unintended and

largely unpredictable. But we know that heavenly bodies are

subject to known gravitational forces and, given their trajecto-

ries over a period, collisions could be predicted with certainty.

The whole set-up may well be deterministic but our ignorance

of the contributing paths of causation would lead to our being

just as surprised as we would have been by some truly ran-

dom happening. From the perspective of chapter 2 on ˜What

is chance?™ we simply do not have enough information to pre-

dict with certainty what is going to happen; we are therefore

uncertain.

At a more mundane level, accidents also commonly arise

from the coincidence of two independent paths in space and

time. In speaking of paths we are borrowing terminology

which is familiar enough in two-dimensional geometry but

may seem odd in this context. Just as we can plot a path

on a two-dimensional map, so we can imagine doing it in

more dimensions than we can actually visualise. With three

space dimensions and one time dimension we have a four-

dimensional space but we can stretch our language to speak of

a path in the same way as in fewer dimensions where we can

visualise what is going on. When two causal paths coincide at a

particular time and place they may trigger the event we call an

accident. To an all-seeing eye the accident could be foreseen

and would come as no surprise. We sometimes ¬nd ourselves

in this position. We may shout a warning to someone who

God, Chance and Purpose

58

cannot see what is coming. To them it comes as a surprise

because they do not have enough information to make the

prediction that is a certainty to us.

An accident could, of course, be a truly random event trig-

gered by some purely chance happening as, for example, the

emission of a radioactive particle. Usually, however, it will

only appear to be random because of our ignorance of the

processes which give rise to it. All that we need to establish at

the moment is that, even in a fully deterministic system, events

happen that have, to us, all the characteristics of chance events.

For all practical purposes accidents and coincidences are just

like pure-chance happenings, but the theological issues they

raise are quite different.

At one level accidents pose no serious theological questions.

If from God™s perspective all is known to him in advance, then

his sovereignty is unchallenged. At another level, of course,

accidents raise profound questions as to why God should allow

themtohappen.Inturn,thisleadsontothequestionofwhether

the system could have been designed to avoid all such unde-

sirable happenings. Given the highly complex and interacting

nature of the creation, it is not at all obvious that it would be

possible to ˜design out™ all unwanted outcomes. The highly

interconnected character of the world may mean that one can-

not have some desired outcome without having others which,

so to speak, necessarily go along with it. This is a problem for

those who demand total sovereignty for God at every level. It

may be less of a problem for those who are prepared to allow

the elasticity which a modicum of uncertainty provides.

p s e u d o - r an d o m n u m b e r s

Pseudo-random numbers are big business. The production of

such numbers by computers “ even pocket calculators “ is a

Chaos out of order 59

largely unnoticed yet universal feature of contemporary soci-

ety. Basic scienti¬c research and the selection of winners in

competitions both call for a ready supply of random numbers.

But what are pseudo-random numbers, why do we need them,

and what bearing does their generation have on the present

question of chaos out of order? In passing, it should be noticed

that genuinely random numbers can be produced using phys-

ical processes, such as radioactive decay. If this is so and if the

real thing is available one might ask why we should bother with

something which is second best. The answer is that pseudo-

random numbers are almost as good and do not require any

physical apparatus with all its costs and inconvenience.

It may help to begin by looking at some familiar things

which serve to introduce the general idea. In some sense we

are talking about what might be called ˜contrived™ accidents.

Think ¬rst of the tables of ¬gures published by National Sta-

tistical Of¬ces. Narrowing it down, imagine we are looking

at the populations of large cities in a big country such as the

United States. One such ¬gure might be 1,374,216. Not all

of the digits have equal importance. The ¬rst 1 is the most

important. It immediately gives us an idea of whether this is

a big city or not; in this instance it is in the million or more

bracket. The next most signi¬cant digit is the 3 in the second

position. Knowing that the city population was in the region

of 1.3 million would probably tell us most of what we want

to know and it would certainly ¬x the size in relation to most

other major cities. As we move along the sequence the digits

become progressively less signi¬cant. Another way of putting

the matter is to say that the information, conveyed by these

successive digits, diminishes as we move along the sequence.

The ¬nal 6 tells us very little indeed about the size of this city,

or about its size in relation to other cities. In fact, it may not

be accurate given the uncertainties of counting such things. If

God, Chance and Purpose

60

we were to collect together the last digits of the populations of

many cities they would be almost devoid of meaning and so

would be rather like random numbers. But we very well know

that they are not strictly random. They have been arrived at

by a process of counting well-de¬ned objects (human beings,

no less). It would not be too far-fetched to describe collections

of digits arrived at in this way as pseudo-random digits on the

grounds that while they are certainly not random, nevertheless

they look very much as if they were.

Any method which involves using some well-de¬ned arith-

metical process and which then discards the most meaningful

parts is a way of producing pseudo-randomness. One tradi-

tional method, long used by children, is the use of counting-

out rhymes to decide who shall be it in a game. A method is

required which is accepted as fair by the participants in the

sense that, over the long run, it does not appear to favour any

particular individual.

There are several variations of the method but the usual

procedure is to count around the circle of children, counting

one word to each child. The child on whom the last word

falls is selected. In principle it would be perfectly possible to

work out in advance who would be selected. To do this we

would divide the number of words in the rhyme by the number

of children in the group. We would then disregard the inte-

ger part of the answer and keep the remainder. This number

determines which child would be selected. If the remainder

was four it would be easy to spot the fourth member of the cir-

cle “ much easier than doing the full mental calculation. The

method works because it is not immediately obvious what

the remainder will be, because people do not carry in their

heads the numbers of words in all, or any, of the many rhymes

there are. It is much easier to remember the words than their

number. The remainder is a pseudo-randomly selected digit.

Chaos out of order 61

For the large-scale uses of random numbers these sim-

ple methods would be far too slow, their yields would be

inadequate and there would be serious doubts about whether

they were suf¬ciently random. These dif¬culties are over-

come by using what are called random-number generators.

Many of these are based on much the same principle as the

counting-out rhyme. One does a calculation, such as a division

sum, and then retains the least signi¬cant part of the answer.

Nowadays most computers incorporate random-number gen-

erators. One characteristic of many such generators is that, if

you go on long enough, the output will start repeating itself.

This is one sign that the numbers are not genuinely random

but this fact is of little practical importance if the number we

need is very small in relation to the cycle length.

It is worth observing at this stage that coin tossing is also

pseudo-random. It does not produce a sequence of digits from

0 to 9 but it does produce the simplest kind of numbers we

can have “ that is binary numbers. If we code a head as 1 and

a tail as 0, a sequence of tosses will be a string of 0s and 1s.

Whether or not a coin falls heads or tails depends on what is

left over when the number of complete rotations is ¬nished.

This, in turn, depends on the impulse and spin imparted to the

coin by the tosser™s hand. Whether or not it falls heads or tails

depends on which side of the vertical the coin is when it hits the

ground on its return. The binary outcome is thus determined

by a remainder to the number of tosses in much the same

way as are pseudo-random numbers. This comparison shows

just how close to being genuinely random a pseudo-random

sequence can be. In turn, this raises the question of whether

there are degrees of randomness and whether they would tell

us anything of theological relevance.

There are two entirely distinct ways of approaching this

question. The traditional statistical approach is to ask what

God, Chance and Purpose

62

properties a random series should have and then to examine a

candidate series to see whether it has them. In this approach

there is no such thing as ˜a random number™, in the singu-

lar. The randomness refers to the process of generation and

can only be detected in a long sequence of such numbers. A

pseudo-random-number generator is a device which produces

a sequence of numbers (usually the digits 0, 1, 2, . . . , 9 or

the binary digits 0 and 1), which for all practical purposes

are indistinguishable from a purely random sequence. And

what would a purely random sequence look like? It would

look like a sequence from a purely random generator! We

might begin to specify some of the required properties. To

begin with, the number occurring at any point in the sequence

should not depend on anything that has gone before. If it did,

it would be, in part, predictable. More generally, nothing we

could learn from part of the series should contain any informa-

tion for predicting any other part. This all depends on asking

about the mechanism which generated the series. According

to this approach one does not ask whether a particular string

is random. Any string whatsoever could have been generated

by a random process. A sequence of 20 sixes generated by

rolling a six-sided die has probability (1/6)20 which, although

an exceedingly small number, is exactly the same probability

as any other sequence, however random it might look. Instead

one asks whether the generating mechanism was such that all

possible digits had the same probability of occurring inde-

pendently of all other outcomes. From this one can deduce

various properties which a series should have. For example,

in the binary case, each digit should appear roughly equally

often; each time a 1 occurs it should be followed equally often

by a 1 or 0, and so on. By making these comparisons one

can never categorically rule out any series as non-random but

one can get closer to saying whether it is likely to have been

generated by a random generator.

Chaos out of order 63

The second approach is to ask how close the actual series

is to randomness. This is based on a clever idea of Gregory

Chaitin1 (see his chapter 2 in Gregerson, 2003a) though, like so

many other good ideas, it had been anticipated some years ear-

lier. To avoid confusion it would be more accurate to describe

the method as measuring the amount of pattern rather than the

degree of randomness. A highly non-random series exhibits a

large degree of pattern and conversely. For example a sequence

of 100 ones is highly patterned and it can be described very

concisely as 100 ones. The sequence 01010101 . . . is almost

equally simple and could be described as 01 repeated 50 times,

say. Chaitin™s idea was that the simpler the pattern, the fewer

the words which were needed to describe it. At the other

extreme, randomness occurs when there is no detectable pat-

tern at all and then there is no shorter way of describing it

than to write the sequence out in full. Actually one has to be a

little more precise than this. Descriptions have to be in terms

of the length of the computer program (on an ideal computer)

necessary to reproduce the sequence. Nevertheless the idea

is essentially very simple. Randomness is measured by the

degree to which one can describe the series more economi-

cally than by using the series itself. We shall meet this second

way of measuring randomness again in chapter 7.

c h ao s

In ordinary usage the word chaos carries a much wider con-

notation than described in this section and this tends to spill

over into theological discussions. I am talking here about what

Gregory J. Chaitin, born in 1947, is a mathematician and computer scientist

1

who apparently had the basic idea mentioned here while still at school.

There are many accounts of this way of measuring randomness but the

chapter referred to here (entitled ˜Randomness and mathematical proof™)

has the merit of being written by Chaitin himself.

God, Chance and Purpose

64

is more accurately described as mathematical chaos.2 In fact it

is, at least, debatable whether anything exists in the real world

corresponding exactly to mathematical chaos.

Science has been particularly successful where it can predict

change over a long period. Tide tables are published well

in advance because tides depend on accurate knowledge of

the relative positions of the earth, moon and sun and how

they change with respect to one another. Similarly the great

successes of engineering are the result of good knowledge of

materials and how they respond to the strains and stresses that

are put upon them. The study of dynamical systems has also

depended on being able to determine how the rates of change

depend on the current state of the system.

Mathematical chaos had its origins in the discovery that

there are physical processes which do not appear to behave

in a regular and predictable way. This is especially true of

the weather. Forecasters attempt to model weather systems.

They measure such things as air and sea temperatures at many

points on the earth™s surface and use models to predict how

they will change and then construct their forecasts. Yet despite

the growing complexity of their models and vastly increased

computing power available it has proved impossible to look

more than a few days ahead. Edward Lorenz3 was one of

the ¬rst to study the situation mathematically and to discover

chaotic behaviour in an apparently deterministic model. He

wrote down a set of differential equations which, he thought,

There are many books on chaos aimed at different audiences. A popular

2

account is given in Chaos: Making a New Science (1987) by James Gleick. An

excellent, somewhat more technical account “ but still not making undue

mathematical demands “ is Explaining Chaos (1998) by Peter Smith.

Edward Lorenz (1917“) was a meteorologist working at the Massachusetts

3

Institute of Technology. It was in simulating weather patterns that he dis-

covered, accidentally apparently, that the development of a process could

depend heavily on the initial conditions.

Chaos out of order 65

provided a reasonable approximation to real weather systems.

As he projected them into the future it became apparent that

the ˜futures™ which they generated depended critically on the

precise starting assumptions. Thus was born chaos theory.

A characteristic feature of chaotic processes is this extreme

sensitivity to initial conditions. It is this fact which makes long-

term forecasting almost impossible and which justi¬es the

introduction of the description chaotic. Two systems, which are

extremely close together at one time, can subsequently pursue

very different trajectories. This near lack of predictability is

the source of the term chaotic.

Weather is not the only phenomenon which exhibits this

sort of behaviour. Heart beats and the turbulent ¬‚ow of li-

quids are two other examples which have attracted attention. It

is not, perhaps, immediately obvious whether chaotic pro-

cesses such as I have described bear any relationship to the

pseudo-random-number generators discussed above. They

do in the sense that one can choose simple indicators charting

the path which a chaotic process takes which are, essentially,

sequences of digits. In this form, their properties can be com-

pared with sequences of pseudo-random numbers.

This similarity to random processes has certain attractions

for theologians. For example, for those who are uneasy with

any true randomness and hanker after the certainties of deter-

minism, it holds out the prospect of a fully controlled universe

with a fully sovereign God. For if the unpredictability of some

dynamic processes displays the lack of predictability of chaos

then, maybe, all the uncertainties of life are rooted in mathe-

matical chaos. This, as I note elsewhere (for example in chapter

12), poses problems of another kind for theology, but at least

it avoids the most ¬‚agrant introduction of pure randomness.

A second attraction of chaos theory for theologians is that

it offers the prospect of space for God™s action in the world

God, Chance and Purpose

66

which does not violate the lawfulness which is so integral to

science. For if such different paths are followed by processes

starting from positions which differ in¬nitesimally, then per-

haps God can engineer changes of direction at the human

level by such minuscule adjustments that they are not humanly

detectable. This view of things creates enough space for God

to act without the possibility of detection by humans.4 Like

so many seemingly simple recipes, this one raises enormous

problems which we shall come to later. Note here, however,

that it makes the major assumption that the world can, in fact,

be controlled in this way and secondly, that it implies things

about the nature of God from which the orthodox might shrink

if they were made fully explicit.

For the moment I simply add chaos theory to the list of

deterministic processes which contribute to the unpredictabil-

ity of the world.

The idea that chaos theory provides space for God to act without violating

4

the lawfulness of nature is particularly associated with the name of John

Polkinghorne. He clari¬es his views in his contribution to Russell et al.

(1995) (pp. 147“56, see especially the section on Chaos theory, p. 153).

In the same volume, p. 348, Nancey Murphy makes a similar point but it

should be emphasised that the views of both authors are more subtle than a

super¬cial reading might suggest. We return to the subject of God™s action

in the world in chapters 8 and 9.

c h a pt e r 5

What is probability?

Theologically important arguments sometimes depend on the correct-

ness of probability calculations. In order to evaluate these arguments

it is necessary to know something about elementary probability the-

ory, and in particular, about mistakes which are often made. In this

brief, non-technical, chapter I introduce some basic ideas of measuring

probability and then identify several common fallacies. These involve

the importance of the assumption of independence, conditionality and

the so-called prosecutor™s fallacy.

t wo k i n d s o f p ro ba b i l i t y

Chance is the general-purpose word we use when we can see

no causal explanation for what we observe. Hitherto I have

not needed to quantify our uncertainties but we shall shortly

move into territory where everything depends on exactly how

large these uncertainties are. I have already spoken of prob-

ability in unspeci¬c terms but now I must take the next step.

Probability is a numerical measure of uncertainty and numer-

ical probabilities have been used extensively in theological

discussions. As we shall see shortly, there are events, such as

the appearance of life on this planet, for which the way in

which probabilities are calculated is absolutely crucial. Cer-

tain events are claimed to have extremely small probabilities

and on their validity hang important consequences for what

we believe about life on earth or, even, the existence of God.

67

God, Chance and Purpose

68

Many of the arguments deployed are often little more than

rather wild subjective opinions, so it is vital to be as precise as

possible about probability statements. There is no need to go

into technicalities but it is important to be aware of some of the

major pitfalls and to be able to recognise them. The purpose of

this short chapter is to set out the more important things that

the reader needs to know before we come to such contentious

issues as Intelligent Design, where the correct calculation of

the probabilities is crucial.

Probabilitytheorymightseemtobeoneofthemoreesoteric

corners of the ¬eld which can be passed over without losing the

general drift of the argument. But much can turn on whether

probabilities are correctly calculated and interpreted. This

is just as true in ¬elds outside theology, so I begin with an

example of considerable public interest from forensic science

to illustrate this point. The outcomes of trials and subsequent

imprisonment have depended on probabilities. Sally Clark, a

solicitor and mother of two young children, was imprisoned

for the murder of her two children in 1999.1 The prosecution

argued that the probability of two cot deaths in one family was

so small that it could be ignored in favour of the alternative

explanation of murder. The probability calculation played a

major part in the conviction of Sally Clark and her subsequent

release after two appeals. DNA matching is another ¬eld where

This case raised a number of important issues about the presentation of

1

scienti¬c evidence in courts of law. In this particular case the expert witness

was a distinguished paediatrician and his evidence was crucial in the court™s

decision to convict Sally Clark of murder. It was subsequently recognised

that he was not competent to give expert testimony on statistical matters.

The Royal Statistical Society issued a press release pointing out that a seri-

ous mistake had been made and urging that steps should be taken to use

appropriate expertise in future. In the United Kingdom the Crown Prose-

cution Service has issued guidelines for expert witnesses in such matters as

DNA pro¬ling.

What is probability? 69

it is becoming increasingly common for probabilities to be

used in criminal cases. Probabilities of one in several millions

are not uncommonly quoted in court as supporting evidence.

A probability is a number purporting to measure uncer-

tainty. It is commonly expressed as a number in the range 0“1,

with the upper end of 1 denoting certainty and the lower end of

0 corresponding to impossibility. A probability of 0.5, which

is halfway, means ˜as likely as not™. Anything larger than 0.5 is

then more likely than not. The betting fraternity do not often

use probabilities in this form but use odds instead. These are

an equivalent way of expressing uncertainty and are, perhaps,

more widely understood. Odds of 3 to 1 in favour of some

outcome is saying the same thing as a probability of 0.75 and

˜evens™ is equivalent to a probability of 0.5. Odds tend to be

favoured when probabilities are very large or small. In Sally

Clark™s trial the estimated odds of a cot death were 1 in 8543, a

¬gure whose import is, perhaps, more readily grasped than the

corresponding probability of 0.00012. Odds are, however, less

convenient when we come to make probability calculations or

to be explicit about the assumptions on which the probability

is calculated.

But where do the numbers in which such probabilities are

expressed come from? For our purposes we can identify two

sorts of probability.2 First there are those based on frequencies.

These are used by actuaries when constructing life tables for

This twofold division of probability is, of course, somewhat oversimpli¬ed.

2

Elementary textbooks often begin with ˜equally likely cases™ and do not

get much beyond problems concerning coins, dice and cards, where the

possible outcomes are, self-evidently, equally likely. Traces of this idea

appear in what we shall meet later where complicated biological entities are

mistakenly treated as what we shall call combinatorial objects (see chapter

7, p. 110). What are here called degrees of belief can be further subdivided

into objective and subjective (or personal) probabilities.

God, Chance and Purpose

70

insurance purposes. If, for example, 60 per cent of forty-year-

old men survive to their seventieth birthday we may say that

the probability that a randomly selected forty-year-old man

reaching that age is 0.6. Armed with such probabilities one

can go on to estimate the probabilities of certain other events

concerning forty-year-old men. The probability of a baby

suffering cot death, quoted above, would have been estimated

by reference to a large number of babies at risk and then

observing how many succumbed. Frequency probabilities are

probabilities of events.

The second kind of probability is more subjective and is

often referred to as a degree of belief. Betting odds are an

example. Bookmakers arrive at such odds by re¬‚ecting on all

the factors of which they are aware and making a subjective

judgement. Although they may be very experienced and have

copious information at their disposal, in the end the number at

which they arrive is subjective. There is no absolute standard

to which their judgement can be referred to decide whether it

is right or wrong. Degrees of belief, like relative frequencies,

can also refer to events as, for example, if we consider the

possibility that there will be a change of government at the

next general election. They can also refer to propositions, such

as when we consider the probability that God exists. There are

sophisticated ways in which degrees of belief can be elicited,

by offering hypothetical bets, for example, and there are also

crude subjective judgements arrived at with little thought. For

the moment I have lumped all of these together in the second

category of degree of belief.

m u lt i p ly i n g p ro ba b i l i t i e s

Probability theory is not so much concerned with estimating

individual probabilities as with combining them to determine

What is probability? 71

probabilities of complicated events. For present purposes, by

far the most important of these relate to conjunctions. If some-

thing occurs with probability 0.5 then what is the probability

that it will happen ten times in a row if it is repeated? If the

chance of a baby dying from cot death is 0.00012, then what is

the probability of two babies in the same family dying from

the same cause? The answer to this question was given in Sally

Clark™s trial as 0.00012 — 0.00012 = 0.000000144 or about one

in 69 million. What is the justi¬cation for multiplying prob-

abilities in this way? The answer is that it is only valid if the

events in question are independent; that is, if what happens in

one case is entirely unin¬‚uenced by the other. When tossing

coins this is a plausible assumption because it is dif¬cult to

see how the two tosses could be linked physically in any way.

But in the case of cot death it is far from obvious that two

deaths in the same family would be independent. Common

in¬‚uences, genetical and environmental, are not unusual in

families and it would be surprising if events happening to sib-

lings were independent. If events are not independent then

the second probability in products such as the above must be

modi¬ed to take account of what happened to the ¬rst baby.

This could substantially alter the answer. The same consid-

erations apply if there are more than two events. Multiplying

several very small numbers produces a product which can be

very small indeed. It is on such calculations that many of the

more startling conclusions in the science“religion debate rest.

The validity of the implicit assumption of independence is

seldom commented on or even noticed!

In a section entitled ˜Universes galore: the problem of dupli-

cate beings™, Paul Davies (2006, pp. 201“3) discusses calcula-

tions purporting to show that in a large enough universe, or

collection of universes, all sorts of unlikely things are bound to

happen sooner or later. Thus one would certainly ¬nd another

God, Chance and Purpose

72

planet, exactly like the earth, on which lives an individual

exactly like oneself in every detail. This would pose obvious

problems about personal identity. Davies begins with coin

tossing showing that, if you toss a coin suf¬ciently often, such

extremely unlikely sequences as a thousand heads in a row

will certainly occur. He then reports some calculations by the

cosmologist Max Tegmark about how far, for example, one

would expect to travel through the universe before ¬nding an

exact copy of oneself. The distance is enormous (about 1029

metres), but we are told that ˜Weird though these conclusions

may seem, they follow incontrovertibly from the logic of sim-

ple statistics and probability theory.™ Such conclusions may be

weird but they certainly do not follow from ˜the logic of simple

statistics and probability theory™. In addition to assumptions

about the uniformity of the laws of physics and suchlike, we

may be sure that, lurking somewhere, there will be assump-

tions of independence such as those used in the introduction

about coin tossing; these do not come as part of the probability

package. They have to be imported as part of what is assumed

to be ˜given™. Probability theory must not be saddled with

responsibility for every weird conclusion drawn from its use.

The weirdness almost always lies in the assumptions, not the

logic. If one has a suf¬ciently lively imagination it is possi-

ble to make all sorts of assumptions for whose truth one has

no evidence whatsoever. Conclusions are no better than the

assumptions on which they are based. Science ¬ction thrives

on improbable scenarios, and though readers may enjoy the

thrill of being transported into realms of near-fantasy on the

wings of elementary statistics, they should be on their guard

against being taken for a ride by calculations based on a grain

of fact and a mountain of guesswork.

The probability of life emerging on earth or the remarkable

coincidence of the parameter values on which the evolution of

What is probability? 73

an inhabitable world depends are other examples we shall meet

later. Only rarely, with such arti¬cial events as coin tossing,

is one likely to be con¬dent in assuming independence.3

c on d i t i ona l p ro ba b i l i t i e s

One of the most common traps for the unwary is in fail-

ing to notice that all probabilities are conditional probabilities.

This means that the numerical value of a probability depends

on what we already know about the event or proposition in

question. The reason that this rather obvious fact is so often

overlooked is that the conditioning circumstances are often

˜understood™. For example, we talk about the probability that

a coin falls heads without feeling the need to specify all the

circumstances of the toss because they do not, as far as we

can see, affect the outcome. The probability arguments used

in physics are often of this kind and the notation used makes

no mention of such irrelevant factors but many of the falla-

cious probability arguments we shall meet in the next chapter

fail by overlooking this elementary fact. If a doctor is try-

ing to diagnose the cause of a patient™s symptoms, then it is

obvious that the probability of a particular diagnosis depends

on how much the doctor knows about the patient. Tests will

be carried out to clarify the position and as the results come

in so will the probability of the possible diagnoses change.

In order to make this explicit, and to keep track of what is

going on, it is necessary to introduce a notation which will

incorporate all the relevant information. All that the reader

will have to cope with is a new notation which is no more

The late William Kruskal, a Chicago statistician, also had a serious interest

3

in theology and he clearly recognised the pitfalls of ignoring independence.

His paper on ˜Miracles and statistics™ was subtitled ˜The casual assumption

of independence™ (Kruskal 1988).

God, Chance and Purpose

74

than a convenient shorthand. Probabilities will be written as

follows:

P(A g i v e n B).

The P stands for probability and the A and the B within the

brackets denote the two things we need to know to spec-

ify a probability. A is the thing (event or proposition) in

whose probability we are interested. The B speci¬es the con-

ditions under which the probability is to be calculated (what

is assumed to be known). Thus, for example, we might be

interested in the probability that someone who is now forty

years of age would still be alive at seventy. This obviously

depends on a variety of things about the person. So B might,

in this case, be male or female and the numerical value of the

probability will depend on which is chosen. Again, we might

be interested in the effect of smoking on the probability. Thus

we would want to calculate,

P(40-year-old alive at 70 g i v e n that they are male and a smoker).

The conditioning event in this case is a compound event

depending on two things about the subject. In general, the

conditioning event may be as complicated as is necessary. We

would not ordinarily include in B anything which we knew

to have no effect on the probability. Thus although we could

write

P(head g i v e n six heads in a row already)

we would not usually do so because, in independent tosses, it

is generally recognised that the probability is unaffected by

what has happened prior to that point. The important point to

notice and to bear in mind, especially when we come to con-

sider particular probabilities in the next and later chapters, is

that every probability is calculated under speci¬c assumptions

What is probability? 75