The Online Library of Liberty

A project of Liberty Fund, Inc.

• The Selected Works of Gordon Tullock
• Introduction
• Preface and Acknowledgments
• The Organization of Inquiry
• Chapter I: The Social Organization of Science
• Chapter II: Why Inquire?
• Chapter III: The Subject and Methods of Inquiry
• Chapter IV: Data Collection
• Chapter V: The Problem of Induction
• Chapter VI: Verification and Dissemination
• Chapter VII: The Backwardness of the Social Sciences
• Chapter VIII: Practical Suggestions

CHAPTER V

THE PROBLEM OF INDUCTION

C. D. Broad has pointed out that inductive reasoning, which is the glory of science, is also the “scandal of philosophy.”1 Scientists go happily along engaging in what they call induction in spite of Hume’s destruction of the possibility of logical progression from the particular to the general. Philosophers, on the other hand, make attempts which have grown more and more desperate over the centuries to develop a logical basis for induction. This chapter, in a sense, is another effort in this direction, although it will take the course of trying to avoid Hume’s problem rather than solving it.2

Strictly speaking, this is a digression from the main purpose of this book. The rest of our study is devoted to the operations of a social organization of a certain type. At this point we will consider a process which takes place in the mind of an individual man, although it can eventually influence the social organization. What interests us as outsiders about a new theory is not the process which the discoverer followed in reaching his theory, but whether it is correct. The correctness or incorrectness is determined, as has been pointed out by Popper, by testing the idea. If it survives difficult tests, then we have confidence in it and would have such confidence even if the discoverer confided that the idea had been revealed to him by Brahma in a vision. Even if the discoverer were revealed as the permanent resident of an insane asylum whose management tolerantly permitted him to bombard the editors of various journals with crackpot contributions, we would still accept his theory if it survived adequate tests.3

But if the real basic problem is “Is it true?” not “How was it obtained?” the second question is still of enough importance to warrant discussion. This discussion, ever since Hume’s day, has turned on the problem of induction. In a sense, this represents an oversimplification, since a great many discoveries are the result of deductive rather than inductive reasoning. Efforts to explain both processes by the same theory are naturally handicapped. I shall first discuss the development of new theories and hypotheses by deductive reasoning and then turn to the cases which genuinely cannot be explained on deductive grounds.

The first method of producing a new hypothesis by deduction is simply to deduce a further consequence from an existing, accepted theory. Sometimes this process is initiated in order to test the theory. This is, in fact, the best way of “validating” any theory. More frequently the newly deduced hypothesis is developed for other purposes, and, in this case, it is not likely to be thought of as a hypothesis. Thus, the Michelson-Morley experiment was deduced from Newtonian physics, but the two experimenters were not trying to test the validity of the Newtonian system. On the basis of deductive reasoning based on a system in which they had complete confidence, they built an instrument to measure certain characteristics of light. The end result did not illuminate the problem which had been the object of the investigation, but instead cast grave doubt on the Newtonian mechanics.

The same type of deductive reasoning is very important in applied science. All sorts of improvements are developed as the result of deductive reasoning based on various received theories. Here, however, failure of an experiment is normally not considered as a disproof of the original theory, largely because of the modesty of the investigators. The applied scientists are perhaps more aware than the pure researchers of the likelihood of experimental failure due to outside factors. Partly, of course, this reflects real differences in the conditions faced by workers in these two fields. The applied scientist is trying to make something useful and must keep a careful eye on the likely costs of the ultimate product. Thus he cannot take such elaborate precautions to eliminate external influences as can the pure scientist. It really makes no great difference to the man trying to improve an automobile engine whether the failure of his experimental attempt to apply a deduced theory arises from the falsity of the original theory or from the existence of external influences which can be eliminated only through extremely costly modifications of the engine. Thus he is unlikely to undertake experiments to elucidate this problem and will normally simply assume the existence of external influences in his particular device. Sometimes, of course, he will decide that he has disproved the theory. He may also report his unsuccessful experiment in a form which attracts a pure scientist to test the original theory and thus at second hand provide for its validation (invalidation).

More important than this type of direct deduction, at least in numbers of hypotheses deduced, if not in the individual significance of the hypothesis, is the process of probable deduction. This is a system of perfectly ordinary deduction which proceeds from a probable premise to a probable conclusion.4 Thus:

If A probably B

A

Probably B

The use of this device in research can be readily illustrated. It was discovered that certain chemicals belonging to the family of nitrogen mustards were usable in a small way in the treatment of cancer. From general chemical experience and the principle of the uniformity of nature, it had been previously deduced that chemicals which are generally similar, but not identical, will have some similar effects on biological organisms and some different. It thus appeared probable that other members of the nitrogen mustard family would be better or worse than the originally tested chemicals, and that at least one member of the family would be markedly better. This probable deduction could then be converted into the hypothesis, “Chemical A will be useful in the cure of cancer” (or, “will not be useful”) and the hypothesis tested. This hypothesis is obviously the result of a probable deduction, with the final probable conclusion then converted into a certain statement solely for the purpose of rendering it testable. (Unfortunately, the hypothesis has so far been falsified.)

If none of the probable deductions survive tests, then we can turn to possible deductions which follow the same process, but with less likely results. A large part of cancer research has, in fact, followed this route. Slim and distant analogies have been searched out and testable hypotheses deduced from them. Although I have undertaken no statistical studies, I suspect that this type of probable/possible deductive reasoning is far and away the principal source of hypotheses for science; it is the normal route which research into any given problem follows. Data are carefully assembled, probable generalizations which seem to apply are searched out, and the data and the probable generalizations are combined to deduce probable conclusions. These probable conclusions are then shifted to testable form and tests are made. If the tested hypotheses are falsified, other less probable generalizations are applied to the same data and so on. Eventually, either the process produces a hypothesis which survives testing, or it plays out and the research project is abandoned.

Another, rather rare but highly important, use of deductive reasoning in the development of hypotheses involves the careful examination of an existing theory. Let us suppose an existing theory which from certain premises deduces properly a large number of physical phenomena. From premises A . . . E, we deduce the results of experiments 1. . . . N. An investigator deduces another experiment, N + 1, from the premises, and on testing, the experimental outcome contradicts the predicted outcome. Assuming there are no errors, this falsifies the original theory. The investigator then may consider carefully the experimental results, the original premises, and the logical chain connecting them. It may be possible, by straight deductive logic, to determine which of the original premises must be changed to explain the new results. It may even be possible to deduce what new premise must be substituted for the “falsified” old ones. This process played a great part in the development of the special theory of relativity. A careful examination of the logical basis of Newtonian physics led Einstein to realize that it was based in part on an implicit assumption of the invariance of time. He tried changing this premise (the particular change he used could not have been obtained by deduction) and got his theory.

It should be noted, however, that this process works only sometimes. Further, it produces only hypotheses. It is possible to produce an infinite number of theories fitting any given set of facts. Only if the theory “predicts” facts not involved in its development can it be regarded as tested and, therefore, as probably true. Thus, the new theory, although obtained by deduction from an old theory and new data, is not, by those facts, proved. It must survive further investigation before we can put much confidence in it.5

So far this discussion of possible deductive ways of obtaining hypotheses has failed to mention the principle of the uniformity of nature, although this will normally be involved in any deductive system of natural laws. Since there seems to be no method of deducing this principle, the fact that it is an indispensable step (although frequently unstated) in the logical chain from which any logical theory is deduced might seem to invalidate any deductive theory. Since the problem has been much discussed by the philosophers of science, I can confine myself to a few sentences. The question of whether the principle should be regarded as a testable hypothesis itself, or whether it is inherently untestable, and hence “metaphysical” in Karl Popper’s terminology, would appear still to be an open one. For our purposes, however, it is largely irrelevant. Everyone believes in the principle of the uniformity of nature regardless of the basis of that belief. All our deductive processes in connection with the real world, and, indeed, most of our day-to-day activities, are based upon this firm conviction. Thus, regardless of the philosophical question of the justification of this principle, people do use it to deduce hypotheses, and that is all we need to know for our present purposes.

Another possible problem in connection with the deduction of hypotheses concerns the validity of deductive reasoning itself. The best answer to this question was given by Morris Kline: “Who decides . . . which forms of deductive reasoning are valid? There is a simple answer to this question. Those people who agree on what is valid deductive reasoning band together and call the others insane.”6 This may not be philosophically satisfying, but it suffices for the student of society who need only know how people act.

Much of what I have been discussing so far is customarily called induction. My argument that deductive reasoning is really involved may seem hair-splitting, but it has a real objective. After we have taken away all of the methods for obtaining hypotheses listed above, a residue will remain. There are hypotheses which were not obtained by these methods. Of the remaining hypotheses, some will be found to have resulted accidentally through errors of one sort or another, but some, I contend, are the result of another mental process, for which I wish to reserve the term “induction.”

The usual definition of induction is the process of getting from the particular to the general. Thus an experiment has been repeated several times, and a given result has been obtained. The movement from this statement to the statement “This experiment will always give that result” is the normal example of induction. From my standpoint, this is a deductive operation in which the general principle of the uniformity of nature and the experimental data form the two premises of a syllogism from which the conclusion can be deduced. To take an example, we observe that the incidence of lung cancer correlates highly with the smoking of cigarettes. From this observation and a general principle that if two things are correlated they are probably causally related with the probability proportional to the strength of the correlation, we deduce a causal relation. From my standpoint, there is no induction except, possibly, in the original development of our general principle. Induction, in my usage, involves the discovery of general principles or patterns in terms of which deductive logic can explain factual data. The steps of explanation, on the other hand, are strictly deductive.

To make the point even clearer, let us consider a graphic example. Experimental evidence indicates that consistent sets of phenomena b, c, d, and f exist; b, c, and d are all deducible from theory Z. It is thought likely that a new theory, Y, will be eventually developed from which it will be possible to deduce Z and another hypothetical subtheory which permits deduction of f and some other as yet undiscovered set of phenomena, e. Obviously, we cannot deduce Y from Z and f because that would be going from the conclusion of an argument to its beginning. This is possible only in special cases where the conclusions and some of the premises are given, and they are so chosen as to permit only one set of additional premises to fit the situation. We cannot expect nature to be so accommodating in the normal case. In fact, we can feel no security that we have even properly guessed the general area in which the new higher-level theory will operate. It is perfectly possible that no theory Y exists and that the next step in the advance of knowledge will lead to the discovery of theory X, connecting Z with the as yet unknown set of phenomena a.

Getting to Y, then, involves a “logical jump.” It is not possible by use of deductive logic, but if we do not use deductive logic to obtain our theories, how do we obtain them? The simple reiteration of the term “inductive logic” obviously does little to solve this problem without some explanation of how “inductive logic” works. The rest of this chapter therefore will be a sort of theory of theories. It must, however, again be emphasized that no matter how we get our hypotheses, what counts is how well they stand up to tests, and this is a matter of deductive logic and experience rather than induction.

Let us consider a schoolboy trying to solve a problem set by his math teacher. Some such problems may be solved by simple routine carrying out of prescribed rules. Multiplying 138,975,017 by 2,386,945 is tedious, but it involves no “insight.” Most mathematical problems, however, require that the student make certain choices in his operations. The process to be followed in reaching the answer is not fully prescribed by the problem, but must be worked out by the student. X²−3x+2=0, for example, can be solved only by first deciding whether to use the formula or to try factoring. If factoring is tried, then it will be necessary to try several sets of factors in most cases. It might be possible to make a list of all procedures which can be followed in reaching the solution of the problem and then to apply them in turn. This would constitute a fixed procedure like that used in multiplying, and we could still use straight deductive reasoning in reaching our result. This requires two things, a finite number of possible procedures and a method of recognizing a correct answer when we reach it.

The problem, of course, concerns the requirement that the list of possible methods of solution be finite.7 In the examples normally given in textbooks, the list is finite because the student knows that all problems have been selected to be at a given level of difficulty and therefore knows that the more difficult procedures have been ruled out. In an algebra problem the student would know that if it was factorable, the factors would be small whole numbers and, thus, that they must be contained in quite a small list of possible solutions. This, however, is because of the kindness of the authors of texts; we cannot expect nature to be so co-operative. In practice, however, the better students do not simply exhaust the possible procedures in solving equations. They “see”8 the solution and proceed directly to it, albeit after a few false starts. Now it might be argued that what really happens is that all possible procedures are tested by the subconscious mind, and the “illumination” occurs when one works, but considerations of time taken and introspective examination of my own procedure in such cases convinces me that this is not so. Somehow a pattern of the whole problem appears in the mind and is then tested by working it out. The process of producing the pattern is the process of induction.9

Scientists, of course, are normally not presented with such neat problems as mathematics students are. Typically they have a mass of data, accumulated by the methods which we have already discussed, and they are searching for a pattern in that data which will permit its theoretical elucidation. The formulation of all possible patterns and the systematic testing of them one by one is the only method deductive logic can offer for this problem. A statistician can provide convenient and efficient tests for the hypothesis and possibly assist in the ordering of the hypothesis, but normally no more. In fact, it is usually impossible to specify a procedure which would, even if applied infinitely, produce all possible explanatory hypotheses. The infinity of possible hypotheses will thus be one of Cantor’s higher-order infinites.

Logically the problem would appear insoluble, yet experience indicates that such problems are solved daily. Before turning to my attempt to explain how it is done, let us consider one possible procedure for somewhat simplifying the search. Let us assume that we have a quantity of data on some subject which we suspect to be logically interrelated, but for which we do not know the interrelationships. Making a comprehensive list of all possible interrelations and then testing each one is clearly an infinite process. As a simplification, suppose we follow the following procedure: From the main mass of data, we select random or non-random subsamples of fairly small size. These small samples are then tested for a list of possible hypotheses more limited than the total number available for the larger sample. This would in part be necessary since some possible patterns for the whole body of data would not be testable in the smaller sample, but let us suppose that we test only the stronger patterns among those which could be tested in the smaller sample. If we get no results with our first set, we select larger subsamples and try out more hypotheses, including some that are more complicated than those included in the first run.

This process, of course, is also an infinite one, but it has one very great advantage. If any reasonably likely pattern is strongly present in the data, it will be detected toward the beginning of the process of search. This process organizes the hypotheses to be tested in order of simplicity and strength and thus is much more likely to find a simple, strong pattern in the data than would almost any other system. I shall later argue that the human mind does, in fact, do something very like this, although I shall also argue that the human mind has a special tool to use in the search which is not available in mathematics.

Turning now to induction proper, in order to explain my theory, I must begin with a discussion of a very common phenomenon, the recognition of another person. Suppose you are walking down the street when you recognize someone coming toward you. It may be a close friend or relation or a very slight acquaintance; it makes no difference. As he comes closer, suddenly you realize that you are mistaken; he is not the person you thought he was. Continuing to observe him as the two of you approach each other, you will normally be surprised at your initial recognition. At closer hand, he not only does not look like the man you thought him to be, he appears to have practically no characteristics in common with him. How, then, did it happen that you mistook him for an acquaintance?

The whole phenomenon of recognition of other people is mysterious. A man who has not seen a childhood acquaintance for years may instantly recognize him in spite of apparent great physical changes. Recognition of a person whom you have known as an adult, even after many years’ absence, is usually easier. The regularity with which our papers report the accidental recognition and capture of fugitives who have disguised themselves and lived peacefully for a number of years and are then noticed by someone who “knew them when” is evidence of the efficiency of the recognition process. At the same time, it points up the difficulty of explaining it. Of course recent research has indicated that almost all forms of perception are much more complicated than was formerly believed. The sense impressions transmitted to the brain are apparently subjected to quite elaborate manipulation.

It seems probable that what we recognize in others is a “pattern” of characteristics, not any given characteristic. What we see in the distant passer-by is a collection of attributes which fits the pattern held by our mind for some acquaintance. As he comes closer and the senses absorb more attributes, the additional attributes do not fit the pattern, and we therefore no longer recognize him. We know people, if this theory is true, not by individual characteristics, but as a pattern of attributes. The sense data are compared with memorized patterns of data and recognition or non-recognition follows. If we have less data, which would be true if the man were some distance off, these data are more likely to fit one of the patterns in the mind than if we have more data.

Similar patterns can be seen in much identification work. The art expert has no difficulty in telling us who painted a picture which he has never before seen. If he cannot tell us the painter, he can at least tell us a good deal about him. Normally he will know the nationality, probable date, and school. Now, he cannot really explain how he knows this. Art experts, it is true, write books “explaining,” but no one tries to learn how from the books, and the books do not even advise this. The prospective art expert simply studies a large number of attributed paintings. Eventually, a pattern for the work of given painters, schools, and nations will emerge in the budding expert’s brain. This pattern he will not be able to explain to others (although he may mistakenly think he can and write a book on the subject), but it will serve his purpose. He has learned this pattern by examining the paintings, not by reading what his predecessors have written (he may or may not have read a good deal in the literature). No one even pretends that the explanations of how this is done contained in the literature in themselves will teach the art. Further, careful examination of the work of various experts will reveal that their explanations of their abilities are inconsistent.

In each of these cases, then, I contend that the mind, in fact, perceives patterns in the sense data it receives from the sensory organs. The patterns must be directly perceived by the mind, since no “intellectual” explanation is available. No one can explain the pattern by means of which he recognizes a given person. Nevertheless, we have no difficulty recognizing people; the type of skill possessed by the art experts is also widely available. The efficient mechanic normally has it in a great degree. What seems to be clear is that there is some pattern to the data and that the mind perceives this pattern directly, without having it explained by someone else. The pattern must exist, at least approximately, since we can and do use it to recognize things, and we must have some method of detecting it, since it is manifest that we do. The mathematical argument at the beginning of this chapter proves that we cannot obtain this recognition of the pattern by successively testing all possible patterns, because that would take an infinitely long time. The only remaining possibility is that the human mind has some direct method of discerning a pattern in received data, if a pattern is there.

How this process works, I cannot imagine. All of our formal knowledge of the reasoning powers of the human mind turns on the use of deductive reasoning and thus can give us little assistance in talking about another, non-deductive power of the mind. To demonstrate the existence of another power of the mind is not to explain it, but we also do not really understand how we deduce. We have good step-by-step descriptions of the process, and it is possible to deduce general rules from given assumptions which seem reasonable, but we have no explanation as to why the human mind reasons this way. Just as there are non-Euclidian geometries, there may well be non-Aristotelian logics. Thus pattern detection, as an attribute of the human mind, is really no more mysterious than deductive reasoning. It is simply a somewhat newer idea, and hence a little harder to accept.

If the human mind is capable of directly detecting patterns in data, then this would explain “induction,” which could then be taken as simply the perception of a pattern in the data. This pattern, in some cases, would be a logical pattern, i.e., a pattern of deduction, and, in some cases, like the recognition of another human being, not. In any event, the perceived pattern might be incorrect.10 As we obtained more data, we might realize that this new data did not fit the pattern achieved on the basis of the less refined information. It is also quite possible that the pattern-perceiving process is less dependable than the deductive reasoning system. If this is so, then our tendency to put greater weight on deductive reasoning would be justified. That we do put at least some reliance on directly perceived patterns in the absence of contrary evidence is obvious to anyone who takes the trouble to examine carefully his own thought patterns.

The part played by pattern detection in science can best be understood if we start by examining some examples of pure “pattern” theories. The Greeks, particularly the Pythagoreans, put great emphasis on discovering patterns in nature. Although we now tend to consider this work largely number magic, it cannot be doubted that it had a great part to play in the early development of science. The general importance of such patterns may be illustrated by the gloomy speculations of physicists immediately after the discoveries of Li and Wang which destroyed one axis of symmetry around which the nuclear particles had previously been tentatively grouped.11

Better examples of pure pattern thinking, however, can be drawn from the history of chemistry in the nineteenth century. Early in the century it was noticed that the atomic weights of the various elements approximated a series of whole-number multiples of the weight of the lightest of them, hydrogen. This was a pattern and gave a logical way of listing the elements, and it could also be considered a hypothesis, or more exactly, a part of a hypothesis concerning the atoms. Insofar as it was hypothetical, it could be tested by two methods. There were gaps in the series, and it could be guessed that new elements would be discovered to fill these gaps; and this gradually happened. The other test, however, was much more precise. If the weights of the atoms were really simple multiples of the weight of hydrogen, then the irregularities shown in the existing data would be progressively reduced as the methods of measuring improved. Great progress in such measurements was, in fact, made. In 1912 the Nobel Prize for physics went to a scientist who had made extraordinarily precise determinations of the weights. There was, however, not the slightest tendency for the more precise measurements to approach simple multiples of the weight of hydrogen. Chlorine, in fact, persisted in approaching with greater and greater accuracy a figure about half way between thirty-five and thirty-six times the weight of hydrogen.

All of this made no difference to the chemists. The pattern was still there, even if it was rather fuzzy, and they continued to think it significant. Eventually, of course, it was discovered that elements, in the state of nature, are composed of mixtures of various isotopes and that the weights are consequently little more than coincidences. The pattern is still there, however, and is still considered important by chemists.

A more elaborate development of the pattern of the elements was developed by Mendeleev, who noticed that certain chemical characteristics seemed to recur at regular intervals if the elements were considered in order of their weight. This regular pattern was expressed in the form of the periodic table, which ordered all elements on two axes. Again, the system was both a pattern and a hypothesis, but the hypothesis was rather quickly disposed of. It was hypothesized that newly discovered elements would fill the blank spots on the table. Some of them did, but the rare earths resolutely refused to fit in, and the majority of elements discovered since Mendeleev’s time are rare earths. This has never bothered the chemists very much. They simply print up the periodic table in a form which gives the rare earths special status and go on with it. Eventually, in this case, atomic physics produced explanations for the regularity and the irregularity of the chemical characteristics of the elements, but even the nuclear physicists have standard periodic tables in their offices because they find the pattern important in itself.

The pure pattern type, although important to my theory of theories, is a rather rare type of theory in the present age. Most modern theories are logical and deductive. Certain premises to certain conclusions is the normal form of a modern theory. This is, from my standpoint, simply a particular type of pattern. The ordering which the mind perceives in nature is a logical ordering, not some type of symmetry as in the periodic table of elements or in the organic ordering which is involved in recognizing the pattern of an individual. The logical pattern, however, has a very important special characteristic. It is frequently testable. It is possible to deduce from such a theory at least one testable hypothesis, and the theories which survive such tests are much more reliable than those that do not. Even among such logical theories, however, there are some which cannot be tested. The theory of evolution has so far been untestable.

The testable characteristic of logical patterns, in general, accounts for their predominance among present scientific theories. We prefer to depend on theories which are subject to deduction and experimental testing rather than on those which are “verified” solely by our perception of a pattern. In the case of the former there is further evidence in addition to the existence of the pattern, and we therefore feel more confident. A good many theories, as was demonstrated earlier, are the product of deduction, rather than induction, in their original form. These constitute logical patterns, but the logical pattern is the result of deductive operations, not a previously discovered pattern.

We are now in a position to describe the process of forming a hypothesis by induction. In practice, of course, a great many hypotheses are obtained by deduction. Further, the hypotheses are tested on being discovered, and the information obtained from the tests of unsuccessful hypotheses is then available for the formation of new theories. The interrelation between data collection, hypothesis formation (both deductive and inductive), and testing is complex. Each hypothesis is formed on the basis of available information, much of which may have been obtained in the testing of previous hypotheses, and the remainder of which presumably was obtained as a result of a hypothesis on the desirability of collecting some sort of data. The hypothesis is then checked, which normally involves further data collection, and either accepted or rejected. If rejected, the data collected (and the failure of the hypothesis is part of the obtained data) will then be used in attempts to form further hypotheses.

We can, however, analytically confine ourselves to the history of one hypothesis. This will involve considering the collection of data preceding the formulation of the hypothesis in relation only to that hypothesis and ignoring the other hypotheses which were actually involved. We can, also, for purposes of study, consider only inductive hypotheses, since the deductive hypotheses raise rather different issues. As a final simplification, we shall temporarily ignore the process of checking on the hypothesis once formed and stop our analysis at the point when we have a hypothesis. We have, then, a man whose information on some problem is increasing as a result of data accumulation. He will continue to increase his data and search for a hypothesis until he either formulates a theory or grows tired of the problem. Even after he has tired of the problem, his mind may continue to work, and he may produce a hypothesis long after he stops specific data accumulation.

The process of “induction,” in my opinion, consists of examining the data for patterns, using the mind’s ability to perceive such patterns directly.12 The human mind is limited, however, in the number of facts which it can hold and even more limited in the number which it can hold at the front of the mind for the purposes of such a search. Analogously, we can assume that a scientist who has ten thousand “bits” of knowledge about a given problem can at any one time consider a group of one hundred of them while relegating the rest to the back of his mind. He then searches this group for a pattern, and if he fails to find one, he selects another group of one hundred “bits” (which may or may not include members of the first group). This description sounds rather mechanical, but I think it is what actually happens. The mind considers a selected group of facts. If no pattern is perceived, a new group is considered. Apparently, the process can be carried on by the subconscious mind while the conscious mind is otherwise engaged, since discoveries sometimes “come” to people who are consciously thinking of something else.

Earlier in this chapter, I pointed out that considering small samples of data selected out of a larger mass was a method of ordering the potential patterns in the mass so that the stronger patterns would be perceived first. This procedure of the human mind, therefore, is a good one for finding the strongest patterns, but almost insures that less conspicuous patterns will be overlooked. This is unfortunate, but patterns which are so feeble that they can be detected only by considering more data than the mind can hold at one time cannot be discovered except by accident. Only if such a pattern was deducible from other theories or if it was one of a finite universe of possible patterns for a given area would it be possible for the human race to become aware of it.

The mind, in considering a small part of all the data available to it, does not follow a truly random process. Since the theory sought is not known at the beginning of the procedure, and, indeed, since it is not even known that there is a theory, the initial selection, however, may as well be called random. Normally the procedure from this point forward, however, is to consider the given group of “bits” of information. If no hypothesis appears, a few more bits of information will be brought into consideration, and this shifts a few others out. This process of gradual shift in the information under consideration continues until the problem is solved or abandoned, or until it is decided to undertake a radical change in the approach to the problem. In the latter case, the investigator tries a new starting point using a radically different selection of information and then goes through the same process of gradual change.

As time goes on, two processes somewhat improve the “span” of information which the investigator can consider at one time. In the first place, as he becomes more bound up with the problem, the amount of extrinsic information kept in the forefront of the mind declines. He devotes more of his mind to the problem and less to other matters, and this permits him to keep somewhat more knowledge under consideration at a time. Thus a pattern too faint to detect at the beginning of the study may now become visible.

More importantly, the investigator begins to group the information in clusters and then to think of the clusters as “bits.” In the ideal case, these clusters of information themselves are genuine theories, which have been well tested. The simplification of the whole mass of data and the increase in the amount which can be brought under active consideration at any given moment by the development of such theories are of the very greatest importance. If a whole range of data, comprising thousands of “bits,” can be compressed into a theory which itself can be treated as one or a few bits, then the real capacity of the mind is vastly increased. The greater the generality of the theory, the more useful it is in promoting thought by this process.

Unfortunately, most of the “clusters” which the mind will construct out of individual “bits” of information, if it considers a given problem for a long time, are not highly validated theories, but rather vague associations. The clusters, even if much less than genuine theories, are still of great value in permitting the mind to carry on active consideration of a larger range of information than would be possible without this labor-saving device. This is the reason why an expert in a given line is so greatly superior to others in solving completely new problems in his specialty. Familiarity and the ability to carry a good many more bits of information in the forefront of his mind (in the form of “clusters”) give him the ability to reach almost immediate solutions to problems which are completely beyond the capacity of less well-trained minds.

This phenomenon, at the same time, explains why sometimes an outsider, or a man just learning a new field, will discover things which have escaped all the experts. The expert thinks in terms of the “clusters” of information which he has developed over the period of his experience. It may happen, however, that these clusters will be inappropriate for a new problem. In this case they actually handicap the search for a new solution, and a fresh approach will more probably be successful. A new mind will, of necessity, have a fresh approach and is unlikely to develop exactly the same clustering of ideas as the older experts have. Thus the procedure, often used by administrators attempting to expedite the solution of some problem, of bringing in a new man with a “fresh” viewpoint or of “going back to first principles” is rational. If the problem resists solution by the regular experts, either bringing in people who do not have the same mental clusters or trying to rearrange the clusters in the heads of the existing experts is a rational step.

It should not, however, be forgotten that the reason that the problem resists solution may be that it is insolvable, either in the present state of knowledge or permanently. Probably no problem has attracted as much intellectual effort as the trisection of an angle with ruler and compass. The eventual outcome was a proof that it was impossible, but only after almost two millennia of efforts for a solution. Any new problem we try to solve may be just as insolvable. We cannot tell an insolvable problem from a solvable one by examining the problem (unless, as in the trisection problem, we happen to have a proof of insolvability), nor does the fact that we have so far failed to solve a problem prove it insolvable. We may simply not yet have tried the right method. Problems may be divided into three classes, a small group that we have solved, another small group that we can prove to be insolvable, and the vast majority of problems about which all we know is that we cannot now solve them.

Any effort to solve a given problem, therefore, may be simply wasted. We cannot be certain that the problem is solvable at all. Even more, we cannot be certain that it is solvable with present knowledge and techniques. Many problems which bothered the Greeks were insolvable with their equipment but are very simple to our scientists. Similarly, we must expect that many problems which are not really insolvable are insolvable in the present state of knowledge. We may hope for the solution to a given problem and direct resources into the area, but we cannot really plan on its solution. A tendency to ignore this fact has characterized much recent writing about science. Doubtless we must make some anticipations of future developments in science as in other fields if we wish to invest our resources wisely, but it should not be forgotten that they are guesses. Somehow the fact that these guesses are guesses about science seems to carry the implication that they themselves are scientific. In fact, the one area of human activity in which we have a good logical proof that it is not possible to foresee developments is science. We cannot know today what we will discover tomorrow.

This principle has, however, certain encouraging aspects. If we cannot plan today on what we will discover tomorrow, we also cannot say what we will not discover. The process of induction which we have been discussing in this chapter is essentially unplannable. Discoveries by this method may occur in the most surprising ways. A man investigating one problem may suddenly see a pattern completely outside his field. It may, in fact, be argued that most important developments have occurred through this process. Certainly, it is common and important. In our earlier explanation of the process of investigation leading to the perception of a pattern, we spoke of the accumulation of data and the successive consideration of samples of that data. We said this process continued until a hypothesis was achieved, but we did not specify that the hypothesis would concern the original problem. In practice, it frequently does, but in a surprising number of cases it does not. The scientist, if he is a pure scientist, is motivated both by his particular curiosity about this specific problem and by a more general curiosity. The general curiosity is likely to remain with him while he investigates the specific problem, and the “induction” may well concern some problem other than the one he is specifically investigating. The same line of reasoning may be applied to practical investigations. The man who started out to find a substitute for ivory in billiard balls and made a whole vast series of important discoveries in the field of plastics en route is legendary, but not even slightly improbable.

Here we have another reason for not trying too hard to plan science. The solution of problem A may most easily be reached through investigation aimed at a solution of the apparently unrelated problem B. If this is so, then resources put into attempts to solve A would, from the standpoint of their own objective, be simply misdirected. Obviously, since we have no way of telling which non-A line of research is the most likely one, we will concentrate on direct approaches if we wish to solve A. Nevertheless, we should not consider our “plan” as more than a rather poor guess, relied on only because we have no better guesses. When we begin a given line of research, we cannot know whether the problem is solvable, solvable with our present knowledge, or best approached by the methods we have chosen. On the other side of the coin, it is quite possible that our investigation will lead to the solution of another problem which is more important.

As a final reason for not trying too hard to plan research, we must return to our discussion of the methods used by the mind in searching for a pattern. It is my theory, as will be recalled, that successive samples of data are examined for patterns by the mind. Now, planning of research must involve deciding on areas to be given priority, and priority will normally involve the assignment of more personnel. The number of samples scanned will, by this method, be increased, but not proportionally. There will be at least some duplication between individuals. The same samples will be scanned by each of two scientists working on the same problem at least occasionally. Further, the amount of such overlapping will increase exponentially with the number of men in the field. The marginal return on increased personnel is thus a declining function of their number, and it is therefore wise to keep scientific investigators dispersed in their interests. Concentrating them in one or a few fields will only marginally increase the rate of discoveries in that field, but will greatly reduce the rate of discoveries in the fields from which they have been drawn.

Thus, I finish my digression on the operation of the mind of the individual investigator and will return to the social organization of science. The possibility that I am completely wrong here is high enough that I feel an apology for including it is in order. I take comfort, however, in Hume’s observation that “the errors of philosophy are only ridiculous and its extravagances do not influence our lives.”13

[1. ]C. D. Broad, The Philosophy of Francis Bacon (Cambridge: Cambridge University Press, 1926), p. 67.

[2. ]Jerrold J. Katz, The Problem of Induction and Its Solution (Chicago: University of Chicago Press, 1962), is the best recent discussion of the problem. It contains a strict proof of the impossibility of logically justifying “inductive” reasoning.

[3. ]It would probably be hard for such a person to get his theory tested. Michael Polanyi has emphasized the role of an “orthodoxy” in the development of science, and our lunatic would probably find it most difficult to get orthodox scientists to pay attention to his idea. Once it had been tested, however, it would stand or fall in terms of the test, not of its origin.

[4. ]J. O. Wisdom, Foundations of Inference in Natural Science (London: Methuen, 1952), pp. 132 et seq.

[5. ]G. Polya, Mathematics and Plausible Reasoning (2 vols; Princeton: Princeton University Press, 1954), discusses the solving of scientific problems from a different viewpoint. He is interested in the procedures and attitudes of mind which are most likely to lead a scientist to success in research. This leads to a quite different approach, but his rules seem eminently sensible. “Flashing the subconscious,” however, is left out of his analysis. See also Polya’s How to Solve It (Princeton: Princeton University Press, 1945).

[6. ]Mathematics and the Physical World (New York: Crowell, 1959), p. 16.

[7. ]This is the basic problem in the work of Herbert A. Simon in attempting to program machines to “think.” His programs do appear to produce original solutions to problems, but normally out of a finite set. For an introduction to his work, see A. Newell, J. C. Shaw, and H. A. Simon, Self-Organizing Systems (New York: Pergamon Press, 1960).

[8. ]“Illumination” in the vocabulary of Poincaré.

[9. ]N. R. Hanson, Patterns of Discovery (Cambridge: Cambridge University Press, 1958), emphasizes the importance of “seeing” patterns. If I understand Professor Hanson, however, he either thinks of these patterns as creations of the mind or believes that there are a number of equally valid patterns and that the mind selects one or more of them. This differs considerably from my position.

[10. ]Dr. James Buchanan has pointed out to me that the pattern-perceiving process may well be the explanation for the phenomena of religious conversion. It seems likely that the human being is not only equipped to perceive patterns; he is strongly motivated to find them. The “ordering” of his universe is necessary for the mental equilibrium of the individual. Thus the sudden “perception” of a religious pattern which orders a large part of reality might well be as impressive an experience as the descriptions of “conversion.” This would also explain both the extreme reluctance of most religious believers to accept simple disproofs of their religion, and their relative willingness to switch to another. See also Charles Joseph Singer, A Short History of Scientific Ideas to 1900, p. 239.

[11. ]For an extreme example of the domination of pattern thinking before Li and Wang’s work, see Murray Gell-Mann and E. P. Rosenbaum, “Elementary Particles,” Scientific American, 197, No. 34 (July, 1957), 72. In spite of the blow dealt by Li and Wang, Gell-Mann, this time with Geoffrey Chew and Arthur H. Rosenfeld, has another article, in the February, 1964, issue of the same magazine, which is once again dominated by two elaborate charts showing a strong pattern (“Strongly Interacting Particles,” 74–93).

[12. ]Although I have no idea of the process by which the human mind finds “patterns,” it is possible that we may be about to learn. Computers that can be trained to recognize patterns now exist. They have been developed to the point where at least one company, Bendix, is advertising its product in full pages of the Scientific American, 213, No. 1 (July, 1965), 12. Although I have no very clear idea of exactly how these machines make their decisions, their operations do not appear very much like human pattern perception. They are, however, a first step toward understanding the phenomenon.

[13. ]A Treatise on Human Nature, Part IV, Section VII.