JAN C. GEERTSEMA

Department of Statistics 
Potchefstroom University for Christian Higher Education
 Potchefstroom 2520, South Africa

However, in recent years, influential new directions of thought appeared in the Philosophy of Science. These stand in many ways in opposition to the traditional view. it is notable that one of the originators of these new directions, T.S. Kuhn, comes from Physics - one of the more exact sciences. Kuhn (1962) emphasizes science as activity in context ("within a paradigm," in his terminology). Using examples from the history of science, Kuhn attempts to show that in practice science does not function as prescribed by the traditional view. In fact, an important role is ascribed to the scientific community with its commitment to its own criteria and its own pattern of educating new scientists. The result is that the human aspect of science comes to the fore and that it becomes clear that science cannot be seen in isolation-it must be seen in context.

Given this new climate of thought, there is more room for the development of a Christian analysis of assumptions in scientific thought and practice and a Christian approach to the way science is used and taught. As far as Statistics is concerned, a further incentive exists, namely an intense current discussion of the foundations of Statistics on an international scale in various symposia (see Godambe and Sprott 1971, Harper and Hooker 1976) and many journal articles.

A Christian Contextual View of Science

For the viewpoint of science in context, I am indebted to the Christian philosopher Stoker (1976). He views science in connection with a coherently ranging series of evermore encompassing contexts. Examples are the social context encompassed by the context of mankind, again encompassed by the context of the cosmos and finally encompassed by the context of God and His relation to the cosmos and human beings. Intersecting these are other contexts, of which the historical is of particular significance. In scientific work, scientists cannot evade the predispositional contextual views concerning the matter being explored. These contextual views are views to which scientists are committed, telling them what are valid problems, relevant criteria and appropriate methods. In this way their research, thoughts and observations are directed by the contextual views, but the views cannot be proved by logic and experiment. For Christians doing science, their contextual views are in part based on biblical teaching about man and nature, and conditioned by biblical instruction as to bow they should see their tasks. as scientists.

The contextual view of science developed by Stoker is in opposition to the standard view mentioned above. In the standard view, science is seen in isolation from its contexts. But this is not acceptable to Christians, since Christians see mankind as a unity that has connections with the narrow scientific material as well as the various contexts in which they are situated. Furthermore, they all belong to God's creation and cannot therefore be separated. The contextual view of science is a broad view which examines the narrow scientific material as well as the contextual views as part of science.

This view of science demands of Christian practitioners of science that they not only work analytically, but in a contextual way as well. We have a responsibility to examine the predispositions. This is especially important, since in most fields of science there are reigning paradigms and these could be in opposition to our Christian views without our even being aware of them.

In what follows, special attention is given to Statistics and various contexts which should be taken into account. This should make clear that, even in an exact ;cience such as Statistics, a Christian view is meaningful.

The Historical Context

In a discussion of the historical context of Statistics, emphasis will be on the influence of philosophical currents of thought and their role in the development of the subject. This should lead to a better understanding of the factors influencing present day Statistics and ,f their historical roots. As Statistics is concerned with he formation of knowledge from numerical data, there ; a close relationship with the philosophical problem of induction, which concerns itself with generalizing from complete information. Attempts to overcome this problem and to obtain practical statistical procedures for doing so, show two main and opposing lines of fought in history.

The dissension can be traced back to the publication 1764 of a paper by the English clergyman, Thomas eyes. The paper was published posthumously, being submitted for publication by his friend, Richard Price, ho added his own notes to it. The paper contained a preliminary form of what is now known as Bayes' Theorem, a source of controversy through the centuries. The theorem contains a method of induction and shows how, given the results of an experiment, the :probability of a hypothesis may be calculated. The problem here is the requirement that prior probabilities should be assigned to the various hypotheses even before the experiment is performed. The question is how this should be done. Some have argued that the assignment should be carried out on the basis of a principle of thought, but others have totally disagreed about the possibility of doing this in a scientifically acceptable way.

'The circumstances under which Bayes' paper was published contribute toward the controversy. In a preface to the paper, Price remarks that Bayes had first derived his theorem under a certain assumption about the prior probabilities, but then did not believe that all readers would accept the assumption. Instead of the assumption, Bayes then described an auxiliary experiment concerning balls running on a table and coming to rest at random points. His mathematics could then be seen as a model for the experiment and the assignment of prior probabilities became more acceptable because of the correspondence with an experiment. if one adds to this the fact that Bayes apparently was reluctant to publish his paper, one seems to be able to conclude that he was doubtful about the possibility of assigning prior probabilities in an acceptable way, as required by his theorem (e.g., see Fisher 1956:10).

It is remarkable though, that the Frenchman Laplace formulated Bayes' Theorem more generally at a somewhat later stage and used it freely in his statistical work. Laplace apparently had no doubts about the assigning of prior probabilities, and his approach was very influential for several decades. He is considered to be the real originator of what nowadays is called Bayesian statistics.

Since Bayes' Theorem plays a central role in the present discussion, it might be fruitful to illustrate the theorem, its use and the difficulties surrounding it in a simple (and artificial) example. Suppose (as is often done in probability theory) that we have urns containing colored balls, We have two urns, labeled I and 2 respectively. Urn I contains one red and one green ball and urn 2 contains two red balls and one green ball. The composition of the urns is known to the investigator. An experiment consists of selecting an urn and afterwards a ball from the selected urn. It is not known which urn was selected. Suppose the ball that was drawn is found to be green and that the drawing of the ball was done in a random way (i.e., each ball in the selected urn had equal probability of being chosen). If we denote by H, and H2 selection of urns I and 2 respectively, a question is how to decide between H, and H2 on the basis of the observed evidence (the green ball). In other words, how do we decide whether urn 1 or urn 2 was selected, given the information that a randomly selected ball was green? If we denote by G the event to obtain a green ball, by P the probability and by P(G I HI) the conditional probability of G given that H, has occurred, et cetera, Bayes' Theorem can be stated by the following formula in the present case:

P(H_1|G) = P(G|H_I) P(H_I) /P(G|(H₁) P(H₁) + P(G|H₂) P(H₂)                  

and similarly for P(H₂|G). Thus the formula gives a way of calculating the so-called posterior probability of H, given the evidence of a green ball. The calculation requires knowledge of two kinds of probability. First, probabilities like P(G|H_I) which do not give rise to any particular difficulties because of the random choice of a ball. Thus P(G |H,) = 1/2, P(G | H₂) = 1./3. The second  kind of probability is the  probabilities of P(H₁) and P(H₂).

These are called prior probabilities (before evidence is obtained) and are controversial. In the simple example which we consider, it could be arranged beforehand that the choice of an urn be random, so that P(H_i) = P(H₂) = 1/2. However, if nothing is known about the way the urn was selected, the prior probabilities of the hypotheses H₁. and H₂ are unknown. Some have advocated the use of equal probabilities in such a situation as a principle of thought. Others have advocated the assignment of prior probabilities on the basis of introspection, and then making explicit one's beliefs concerning H₁ and H₂.  Still others claim that it is impossible and unacceptable to assign prior probabilities if it is not known how the urn was selected. But if one is willing to assume equal probabilities for H_I and H₂, the above formula gives:

P(H₁|G) = (1/2 x 1/2)/(1/2 x 1/2 + 1/3 x 1/2) = 3/5

and similarly P(H₂|G) = 2/5.

Since H₁ has the higher posterior probability, Bayesian statistics prescribe that H₁, should be chosen on the basis of the evidence. Note that the decision might be wrong, but it has been reached in a systematic way.

Having illustrated Bayes' Theorem, we are now in a better position to discuss the two lines of thought mentioned above. Boldrini (1972:137-138) makes the following statement concerning the differing philosophical backgrounds of Bayes and Laplace:

One can see here the different mental outlook of the two: the Englishman who derives his thinking from the tradition of Locke and Hume, and the Frenchman who aligned himself with the development of Cartesian rationalism.

It seems then that Bayes was part of the empiricist tradition of the England of his time and would rather use an auxiliary experiment than a principle of thought in connection with the question of prior probabilities, if indeed be was willing to make his views about this public. On the other hand, Laplace, in the more rationalist tradition of the European continent, seems not to have had any doubts about the validity of an approach with prior probabilities. Thus we observe the beginning of an empiricist versus a rationalist line of thought in Statistics which would continue to exert influence over the centuries and up to the present day.

On the continent, Lalplace's standing ensured that his approach was well established for most of the past century, but in England it was much criticized by people such as Boole and Venn. By the beginning of this century, however, one of the great minds in modern science got involved in Statistics. He was the Englishman R. A. Fisher, who also was one of the leaders in the field of Genetics. Fisher was vehemently opposed to Bayesian statistics and he proceeded to lay the foundations of an alternative statistical theory. This theory was taken further by people such as J. Neyman and E.S. Pearson, and became a comprehensive and established theory; so much so, that is is nowadays known as the classical theory (Barnett 1973).

The classical theory is empirical in nature. The probability concept used is an empirical concept: probability is to be thought of as an observed ratio of the number of occurrences of an event in an unlimited series of repetitions of an experiment. Thus, prior probabilities as used in Bayesian statistics make no sense. Also, only situations which are repeatable may be considered in a statistical analysis, and all statements concerning properties of procedures have to be interpreted in terms of "the long run."

But the Bayesian approach was never completely dead. Some years ago it experienced a remarkable revival under the leadership of figures such as H. Jeffreys, L.J. Savage, B. de Finetti and D.V. Lindley. This revival was accompanied by many new developments, recognized contemporary issues and claimed to be the approach to statistics. Bayesian statistics is more intellectualistic in character. It is often justified as the inevitable consequence of certain self-evident requirements for rational behavior. These requirements are called axioms and are mathematical formulations of concepts such as consistency and coherence. The Bayesian approach then follows by logical deduction from the axioms. In contrast with classical statistics, the probability concept here is subjective. It reflects the beliefs of the statistician in a unique given situation.

It may be mentioned that attempts have been made to compare the revival of Bayesianism with a scientific revolution, in the sense of Kuhn (Lindley 1980), with Bayesianism emerging as the victorious paradigm. Perhaps this conclusion is a bit premature, but one does have the impression that the foundations of Statistics are subject to areas of major disagreement. (A more comprehensive discussion of the foundational controversy in Statistics is contained in Geertsema 1983.)

Thus, if Statistics is viewed in a historical context, it becomes clear that the development of the subject is connected with broader philosophical issues. The old struggle between empiricism and rationalism in the rise of modern science becomes visible in Statistics also. In this arena, Christians should understand the issues in view of their faith. For instance, Hooykaas (1972) in an insightful analysis of the role of the Christian faith in the development of science from Greek antiquity up to the seventeenth century, points out that Greek science was strongly rationalist, mainly because of the Greek view of the world as a rational being, the product of a rational creation. The rationalist outlook retarded the progress of science, but as the influence of a biblical world view came to be felt more and more, an empirical approach gradually appeared in science. This was due to the biblical view of God, the Creator, who created according to His own will, not bound by human rationality or other prescriptions. Thus the biblical view of the world is that of a creation which can only be understood by man through observation. Thus an empirical approach is called for, but one should point out that the role of the mind is not hereby denied. In fact, observation as well as thought is needed in the formation of scientific knowledge. Although a full analysis cannot be given here, it appears that a Christian point of view concerning the two main approaches in Statistics would be a balanced one, appreciating both and steering away from one-sidedness.

A second example of the influence of philosophical currents of thought in the history of Statistics is to be found in the ideology of the Eugenics movement during the second half of the 19th century and the first part of this century in Britain. Mackenzie (1981) gives a penetrating analysis of these developments and argues that the social circumstances and patterns of thought not only influenced the motivation of the leading figures, but also the content of their statistical work. Notable amongst these leaders were Gaiton, Karl Pearson and R. A. Fisher, all of whom were dedicated to the ideal of improving the human race; an ideal that they pursued with almost religious fervor, motivated by a scientific naturalism that rejected the supernatural and saw scientists as best equipped to lead society. Their ideals led them into genetic research and this, in turn, required statistical tools which they strove to develop. Many of their tools, though cleaned and sharpened, still form the basis of modern statistical theory. Thus, the philosophical influences from the time of the formation of these tools still linger in the background, and need to be understood from a Christian perspective for a proper insight in their meaning. Such a discussion will not be attempted here.

The Scientific Context

Statistics is connected to all empirical sciences and acts as a mathematical auxiliary to them. This is because all sciences make use of data to some extent, and Statistics is just that ancillary science which has the task of handling empirical data. This unique characteristic of Statistics implies a closely assumed relationship between Statistics and Philosophy of Science. Kempthorne (1976) describes Statistics as applied Philosophy of Science. This is because Statistics is concerned with

questions such as: What is a random sample? Do the data support a given model? What are useful ways of analyzing data? How should a probability be judged? He notes, however, that in practice very little interplay between Statistics and the Philosophy of Science materializes-a deplorable situation. As Statistics may be viewed in the context of the empirical sciences as a whole, interpretations of these also have a bearing on Statistics. This is more so since there is the very special relationship between Statistics and Philosophy of Science. Many of these views are controversial, however, and call for a Christian reply.

The real reason why a Christian statistician
avoids harmful effects on people taking part in experiments
 is to be found in the commandment to love one's fellowman.

As an interesting example, let us consider the attempt by Kemptborne (1976:286-288) to view the Philosophy of Knowledge, and consequently Statistics, from an existentialist perspective. In his view, the existentialist phrase "life is absurd" means that life is not perfectly predictable. He probably thinks that this is important for science and has a connection with Statistics where the study of random phenomena is central. A closely related existentialist phrase, "existence precedes essence," implies to him that a complete rational explanation in science is impossible. Of course these phrases belong to an atheistic philosophy and cannot be accepted by a Christian. Sartre (1948) explains that life is absurd because there is no God who gives man an essence by planning him before he comes into existence. Man makes his own essence by developing according to his own will. This is entirely opposed to the Christian faith in God who creates man and has a calling for him.

As another example of philosophical views of science which have implications for Statistics, one may mention the contribution of the pragmatist philosopher C.S. Peirce at the end of the nineteenth century. Kempthorne and Folks (1971:507-508) are of the opinion:

The general philosophy of pragmatism as put forward by Peirce @ms to lie at the root of statistical practice.

They also conclude that for Peirce knowledge was public and not personal, which is in opposition to the Bayesian view that the opinion of the individual is what is important. Also his ideas about the nature of scientific inference provide insight and should be studied with statistical application in mind. Apparently Peirce attempted the formation of a philosophy which would encompass both science and Christianity. He was deeply upset by the religious controversies which followed Darwin's views. (See Murphey 1968:531). Thus, Peirce's ideas warrant special attention from the Christian community.

The Social Context

Statistics also functions within society. This leads to various ethical problems to which biblical norms of love, justice, truth, honesty and authority should apply. Of course, the statistical profession is well aware of the ethical problems and has taken a firm stand for professional integrity. The most recent statement in this direction is the International Statistical Institute Declaration on Professional Ethics (1986). This declaration calls on the statistician to guard against misinterpretations or misuse of statistical material, to make an impartial assessment in a statistical study, not to accept contractual conditions that are contingent upon a particular outcome from a proposed statistical inquiry, to respect confidentiality requirements but allow colleagues to assess their methods, to avoid undue intrusion into the privacy of people, to protect experimental human subjects against potentially harmful effects, et cetera. It is clear, however, that a Christian statistician should view the ethical issues from a deeper dimension, namely from the biblical norms mentioned above. For instance, the real reason why a Christian statistician avoids harmful effects on people taking part in experiments is to be found in the commandment to love one's fellowman. One sees the immediate relevance of the Christian faith in statistical applications in society.

An interesting instance where ethical questions arise is in political (and other) opinion polls, which rest on statistical sampling theory and are often quite controversial. It is sometimes contended that the results of opinion polls exert influence on voters. People like to be on the winning side and are then persuaded to vote for the candidate who is shown to be the winner by the polls. Candidates with a poor showing in the polls have a hard time getting campaign funds, because nobody wants to support a loser. It is even claimed that the results of polls are manipulated by the pollsters, and that there are sometimes deviations from correct statistical sampling techniques in order to save money.

Concerning polls which are not aimed at election results, but other important public issues, there is even more criticism. It is contended that the "public opinion" does not consist of the opinion of a number of equally important people who independently cast their votes. Also, experience has taught us that large segments of the public are very uninformed. There is also reason to suspect that individuals, rather than admit that they are uninformed, blindly choose one of the alternatives presented to them in a poll. In defense, the pollster, George Gallup, takes the point of view that opinion polls are almost indispensable in a democracy, and that the final stage in the development of a democracy will arrive when the will of the people is known at all times. He is quoted as saying: "The task of the leader is to decide how best to achieve the goals set by the people," and "This job almost makes you an evangelist for democracy."

The pollsters contend that there is no proof that polls exert an influence on the voters. They cite the British election of 1970 as an example to the contrary (the Conservatives won, even though the polls consistently pr dicted a victory for Labor). They also point out that the procedures that are used are not secret in any way-they have been carefully described in scientific journals. Much has been learned from the mistakes of the past and methods are continually refined to take into account, for instance, voters who do not show up to vote or the ignorance of members of the public. Also in defense of polls, it can be argued that a potential loser could gain by knowing the weakness of his support, so that he can work harder.

A Christian point of view is very relevant in this controversy. Opinion polls can be of great help to a government that is committed to the biblical norm that those who are in a position of authority have the duty to serve those over whom they have authority. In order to serve well it is important that they know the wishes and opinions of the nation and take these into account. The reason for this is not that government should be a government according to the will of the people government should be according to the principles of the Word of God. Thus opinion polls can help a government to serve well. In this connection one is reminded of what Winston Churchill said during the Second World War:

Nothing is more dangerous in wartime than to live in the temperamental atmosphere of a Gallup poll, always feeling one's pulse and taking one's temperature.... There is only one duty, only one safe course, and that is to try to be right and not to fear to do or say what you believe to be right.

Furthermore, truth and honesty require that opinion polls should be of a high standard. Methods with an accepted statistical basis should be used and no compromise should be allowed due to a shortage of money and time, even though requirements such as randomness in sampling may be expensive to achieve. Questions concerning the improper influence of political polls concern injustice which should be eliminated. This could

be done, for instance, by the requirement that the polling be done by independent organizations and not by political parties. These independent organizations should do their work on a high statistical standard and should not compromise on the requirements set by statistical theory.

The Religious Context

There is a connection between the Statistics of a statistician and the god be worships. Christian statisticians should be aware of this and not be lured into uncritically accepting certain predispositions in the work of their colleagues. An example is to be found in the views of Karl Pearson, whose name has already been mentioned in connection with the Eugenics movement. He made many important contributions to Statistics, but was also very interested in philosophical matters. His son, E.S. Pearson, wrote two lengthy articles (1936 and 1937) shortly after the death of his father and gave many interesting details of his life and his views. From these articles we also get some glimpses of his religious views. For instance, towards the end of his life, Karl Pearson wrote (E.S. Pearson 1936:196):

1 can only say that till [sic.] this day I think Spinoza the sole philosopher who provides a conception of the Deity in the least compatible with scientific knowledge.

He also was the author of four lengthy articles on Spinoza. Without pursuing this much further, let us note that a characteristic of Spinoza's philosophy is the identification of God and nature. It is remarkable that Einstein, a contemporary of Pearson, is quoted to have said:

I believe in Spinoza's God, who reveals himself in the orderly harmony of all that exists, not in the God who concerns himself with fates and actions of human beings. (Golden 1979:66)

Another interesting point is that Pearson's search for his own confession of faith is visible in his early writings. E.S. Pearson mentions that his father was driven by a "Moral force" in his scientific work and views. Such thinking was typical of the Victorian era, which was characterized by the penetration of scientific thought right through the traditions of ortbodox Christianity. In spite of this, his father felt the compelling need for his own confession of faith; this can be found in his life ideals and in conjunction with his view of science. In K. Pearson's own words, this confession of faith leads to the phenomenon that men:

serve science from love as men in great religious epochs have served the Church. (E.S. Pearson 1936:194)

He had as ideal the search for truth, and believed that scientific knowledge would bring salvation to man. rhus one can almost say that Pearson's god was science, and one sees that his "religious" enthusiasm was intimately connected with his scientific and statistical work.

Another more recent example is a statement by Kish (1978), as president of the American Statistical Association, on the role of chance in human life. He sees chance as a phenomenon that occurs everywhere, and argues that statisticians have a special duty to prepare people for the effects of chance. One almost gets the feeling that he sees Chance as the god that must be served because it reigns supreme in the world. He states:

The tragic biblical Job might have been happier and wiser if he knew that his plagues were due to chance. The triumphs or the problems of your children may be due to chance, not only to your behavior-despite what Freud may say; a statistical view may protect parents against false pride or against guilt and despair. But we are not mere helpless puppets of chance and we can improve our chances-for example, by quitting smoking, with regular exercises, and by losing weight. Recognition of the interplay of chance with discernable causes may yet lead us to a better way of life and to a better moral philosophy. Somebody may even start a new religion of Statisticology!

A last example shows that questions concerning truth and science have a profound religious significance. In his book, Scientific Truth and Statistical Method, Boldrini (1972) (member of the Pontifical Scientific Academy) discusses the role of Statistics in the search for scientific truth and is guided by the belief that truth is ultimately found only in Jesus Christ. In the preface of this book he states:

What can one say by way of introduction to this book? The answer is to be found in a trial which took place 2,000 years ago, when some immortal words were spoken. 'Quid est veritas?' the perplexed Pontius Pilate asked himself after an interrogation which had left him full of anguish. Jesus, the accused, had already given an answer when he stated with authority, 'Ego sum Veritas'. In that tragic moment the answer reached but few hearts, but it set out on its way down the centuries.

This book is entirely concerned with the development of Pilate's question and ends by accepting, on the very last page, the answer of Jesus. Indeed, the opinion was once firmly held that scientific truth was something essential and predetermined, a hidden principle of the physical world and a difficult objective for studious minds to achieve stage by stage, through conjecture and experiment. That opinion has now been shown to be mistaken. By scientific truth is still meant, of course, a certainty but a subjective one, transitory, a special relation between man and world, adapting itself to the progress of knowledge and to changes of interpretation and of human requirements.

Conclusion

We have given examples of various contexts within which Statistics may be viewed. Clearly, there is room for a Christian point of view. In some of the more encompassing contexts, such as the religious and social contexts, biblical perspectives could be used directly. However, in those which are less encompassing, such as the scientific context, philosophical questions have to I be answered in a philosophical manner, and here the importance of a Christian Philosophy becomes clear.

It has been pointed out that study of these questions is the responsibility of a Christian statistician in order to form a single integrated world and life view. But the view of Statistics sketched here also has implications for t the teaching of Statistics by the Christian. Students should not only be taught "the facts" which modern textbooks present. They should also know that there are different presumptions as to what constitutes a "fact," as well as different interpretations and uses of them. Students should therefore be helped to realize that belief, and thus their own belief, is connected to the subject which they are studying.

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