A heritage of thought about the process of scientific learning comes to us from such classical writers as Aristotle, Galen, Grossteste, William of Occam, and Bacon who have emphasized aspects of good science and have warned of pitfalls.

Iteration Between Theory and Practice

One important idea is that science is a means whereby learning is achieved, not by mere theoretical speculation on the one hand, nor by the undirected accumulation of practical facts on the other, but rather by a motivated iteration between theory and practice such as is illustrated in Figure 1(a).

Matters of fact can lead to a tentative theory. Deductions from this tentative theory may be found to be discrepant with certain known or specially acquired facts. These discrepancies can then induce a modified, or in some cases a different, theory. Deductions made from the modified theory now may or may not be in conflict with act, and so on. In reality this main iteration is accompanied by many simultaneous subiterations.

Flexibility

In this view efficient scientific iteration evidently requires unhampered feedback. The iterative scheme is shown as a feedback loop in Figure 1(b) In any feedback loop it is, of course, the error signal-for example, the discrepancy between what tentative theory suggests should be so and what practice says is so-that can produce learning. The good scientist must have the flexibility and courage to seek out, recognize, and exploit such errors-especially his own. In particular, using Bacon's analogy, he must not be like Pygmalion and fall in love with his model.

Parsimony

Since all models are wrong the scientist cannot obtain a “correct” one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameteriza- tion is often the mark of mediocrity.

Worrying Selectively

Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.

Role of Mathematics in Science

Pure mathematics is concerned with propositions like “given that A is true, does B necessarily follow?” Since the statement is a conditional one, it has nothing whatso- ever to do with the truth of A nor of the consequences B in relation to real life. The pure mathematician, acting in that capacity, need not, and perhaps should not, have any contact with practical matters at all. In applying mathematics to subjects such as physics or statistics we make tentative assumptions about the real world which we know are false but which we believe may be useful nonetheless. The physicist knows that particles have mass and yet certain results, approximating what really happens, may be derived from the assumption that they do not. Equally, the statistician knows, for example, that in nature there never was a normal distribution, there never was a straight line, yet with normal and linear assumptions, known to be false, he can often erive results which match, to a useful approximation, those found in the real world. It follows that, although rigorous derivation of logical consequences is of great importance to statistics, such derivations are necessarily encapsulated in the knowledge that premise, and hence consequence, do not describe natural truth. It follows that we cannot know that any statistical technique we develop is useful unless we use it. Major advances in science and in the science of statistics in particular, usually occur, therefore, as the result of the theory-practice iteration. The researcher hoping to break new ground in the theory of experimental design should involve himself in the design of actual experiments. The investigator who hopes to revolutionize decision theory should observe and take part in the making of important decisions. An ap- propriately chosen environment can suggest to such an investigator new theories or models worthy to be enter- tained. Mathematics artfully employed‘ can then enable him to derive the logical consequences of his tentative hypotheses and his strategically selected environment will allow him to compare these consequences with practical reality. In this way he can begin an iteration that can eventually achieve is goal. An alternative is to redefine such words as experimental design and decision so that mathematical solutions which do not necessarily have any relevance to reality may be declared optimal.    

The Perils of Open Loop:cookbookery and mathematistry

The progress in science is a result of feedback etween theory and practice. Feedback requires a closed loop. By contrast, when for any reason he loop is open, progress stops. Such stagnation can occur with the normally iterative) cycle stuck either in the practice mode or in the theory mode. The maladies which result may be called cookbookery and mathematistry. The symptoms of he former are a tendency to force all problems into the molds of one or two routine echniques, insufficient thought being given to the real objectives of the investigation r to the relevance of the assumptions implied by the imposed methods. Concerning the latter, Fisher’s apparently bivalent attitude towards mathematicians has often been remarked and has been the cause of perplexity and annoyance. He himself was an artist in the use of mathematics and emphasized the importance of mathematical training for statisticians-the more mathematics known the greater the potential to be a good statistician. Why then did he sometimes seem to refer so slightingly to mathematicians? The answer is that his real target was “mathematistry.” It is to make the distinction that the word is introduced here. Mathematistry is characterized by development of theory for theory’s sake, which since it seldom touches down with practice, has a tendency to redefine the problem rather than solve it. Typically, there has once been a statistical problem with scientific relevance but this has long since been lost sight of. Fisher felt strongly about this last point, particularly when he himself had produced the originally useful idea. He felt the development of distribution-free tests misused ideas initiated in Chapter III of his book Design of Experiments [III, p. 481]. Another annoyance was the generalization to what he felt was absurdity of his applications of group theory and combinatorial mathematics to experimental design. The penalty for scientific irrelevance is, of course, that the statistician’s work is ignored by the scientific community. But this does not come to the notice of a statistician who has no contact with that community. It is sometimes alleged that there is no actual harm in mathematistry. A group of people can be kept quite happy, playing with a problem that may once have had relevance and proposing solutiom never to be exposed to the dangerous test of usefulness. They enjoy reading papers to each other at meetings and they are usually quite inoffensive. But we must surely regret that valuable talents are wasted at a period in history when they could be put to good use. Furthermore, there is unhappy evidence that mathematistry is not harmless. In such areas as sociology, sychology, education, and even, engineering, investigators who are not themselves statisticians sometimes take mathematistry seriously. Overawed by what they do not understand, they mistakenly distrust their own common sense and adopt inappropriate procedures devised by mathematicians with no scientific experience. An even more serious consequence of mathematistry concerns the training of statisticians. We have recently been passing through a period where nothing very much was expected of the statistician. A great deal of research money was available and one had the curious situation where the highest objective of the teacher of statistics was to produce a student who would be another teacher of statistics. It was thus possible for successive generations of teachers to be produced with no practical knowledge of the subject whatever. Although statistics departments in universities are now common place there continues to be a severe shortage of statisticians competent to deal with real problems. But such are needed. •