As for the nature of such a transition to maturity, it deserves a far fuller discussion than it has received in this book, especially from those concerned with the development of the modern social sciences. For that purpose it may help to point out that such a transition need not—and I now think should not—be associated with the prior acquisition of a paradigm.
The members of every scientific community, including the schools of the “pre-paradigm” period, share the sorts of basic elements that I have collectively called “a paradigm.”
What changes with the transition to maturity is not the presence of a paradigm but rather its nature. Only after that change does normal puzzle-solving research become possible. I shall now discuss the various features of advanced science that I earlier associated with the acquisition of a paradigm as the consequences of acquiring a paradigm of that kind: a paradigm that identifies what the challenging puzzles are, provides clues for their solution, and guarantees success to truly capable specialists. Only those who have taken heart from the assurance that their own field—or school—possesses paradigms are likely to feel that something important is sacrificed by the change.
The second problem, more important at least for historians of science, concerns the implicit one-to-one identification made in this book between the subject matter of science and the scientific community. In other words, I have repeatedly behaved as though, for example, “physical optics,” “electricity,” “heat,” and the like must name the subject matter of research. The only alternative my argument seems to allow is that all these subjects belong to the physics group. Yet, as my historian colleagues have often pointed out, an identification of that sort will not usually withstand examination. Before the middle of the nineteenth century, for example, there was no physics group; it was formed by the merger of two previously separate groups, mathematics and natural philosophy (physique experimental). What is today the subject matter of a single broad scientific community was in the past divided in various ways among multiple groups. Narrower subjects, such as the theory of heat or of matter, existed for many years without becoming the special domain of any single scientific community. But normal science and revolution are both activities grounded in scientific communities; one must first uncover the changing social structure. In the first case, normal science, a paradigm governs not the subject matter of a science but a group of practitioners. Any consideration of paradigm-directed or paradigm-shattering research must begin by locating the group, or groups, responsible.
When the analysis of scientific development is approached in that way, several difficulties that have been crucial points of concern may be removed. Several critics, for example, have pointed to the theory of matter and have regarded me as having greatly exaggerated the uniformity of scientists’ allegiance to a single paradigm. They note that, until relatively recently, theories of matter were a subject that produced continuing disagreement and debate. I agree with that description, but I do not think it is a counterexample. Theories of matter, at least until around 1920, were not the special domain or subject matter of any scientific community. Rather, they were tools for a number of groups of specialists. Members of different scientific communities sometimes chose different tools and criticized the choices of other groups. More important, the theory of matter is not the sort of topic on which even the members of a single group must necessarily agree. The need for agreement depends on what work the scientific community performs. Chemistry in the first half of the nineteenth century provides one example of this point. Although several of the community’s basic tools—the laws of definite proportion, multiple proportion, and gaseous reaction—were accepted as universal properties as a result of Dalton’s atomic theory, chemists after that event could carry out their research on the basis of these tools while at times disagreeing sharply about the existence of atoms.
I believe that other difficulties and misunderstandings can be resolved in the same way. Partly because of the examples I chose, and partly because of related ambiguities, some readers have concluded that I am primarily, or even exclusively, concerned with major revolutions associated with Copernicus, Newton, Darwin, or Einstein. But a clearer description of the structure of scientific communities will help reinforce the quite different impression I have tried to bring out. I take a revolution to be a special kind of change that involves a certain reconstruction of group commitments. But it need not be a change on a large scale, and it may not appear revolutionary to anyone outside a single group composed of, say, twenty-five or fewer people. Precisely because changes of this kind, scarcely recognized or discussed in the literature of the philosophy of science, occur regularly on such a small scale, revolutionary change, as opposed to cumulative change, must be understood.
Finally, one revision is closely related to the preceding ones and may do much to encourage that understanding. Several critics have expressed doubt as to whether a crisis—that is, a common awareness that something has gone wrong—so invariably precedes revolution as the first edition seemed to imply.
But nothing important in my argument depends on crisis being an absolute prerequisite for revolution. Crisis is merely the usual prelude: in other words, it provides a self-correcting mechanism that ensures that the rigidity of normal science will not continue forever unchallenged. Revolutions may be induced in other ways, though in my view such cases seem rare. In addition, I would point out here something that was left obscure above by the absence of an adequate discussion of the structure of scientific communities. A crisis need not be produced by the research of the very scientific community that experiences it and sometimes passes through a revolution as a result. A new instrument, such as the electron microscope, or Maxwell’s laws may arise in one specialty, and their assimilation may produce a crisis in another.
2. Paradigms as the Constellation of Group Commitments
Let us now return to paradigms and discuss what they might actually be.
My original text leaves no question more obscure or more important. One sympathetic reader, who shares my conviction that “paradigm” names the central philosophical elements of this book, prepared a partial analytical index and concluded that the term paradigm is used in at least twenty-two different ways. 7) I now think that most of these differences are due to a lack of formal consistency—that is, Newton’s laws are sometimes paradigms, sometimes parts of paradigms, and sometimes used paradigmatically—and they can probably be eliminated relatively easily. But once that editorial work is done, two quite different uses of the term remain, and these require separation. The broader meaning will be the subject of this subsection. The other meaning will be considered in the next subsection.
Once a particular professional community has been distinguished by the techniques just discussed, a useful question can be raised: what do its members share that accounts for the fullness of their professional communication and the relative unanimity of their professional judgments? To this question, the first edition of this book offered as its answer a paradigm, or a set of paradigms. But for this usage, unlike the one to be discussed below, the term is not appropriate. Scientists themselves would say that they share a theory or a set of theories, and I would be pleased if that term could ultimately be reclaimed for this use. But as it is currently used in the philosophy of science, “theory” implies a structure far more restricted in character and scope than what is required here. Until the term can be freed from its present implications, adopting another will avoid confusion. For the present purpose, I propose to use “disciplinary matrix”: “disciplinary” because it concerns the common possession of the practitioners of a particular discipline; “matrix” because it is composed of various kinds of orderly elements, each of which requires a high degree of specification. Most or all of the objects of group commitment that the first edition made into paradigms, parts of paradigms, or things paradigmatic constitute components of the disciplinary matrix, and as such their elements form a whole and function collectively. But they will no longer be discussed as though they were all of a piece. I shall not attempt an exhaustive list here, but noting the principal types of components of a disciplinary matrix will both clarify the character of my present approach and prepare the ground for my next point.
One important type of component I shall call “symbolic generalizations.” By this I have in mind expressions, deployed by group members without question or dissent, that can be put into a logical form such as (x)(y)(z) φ(x, y, z). They are the formal, or readily formalizable, components of the disciplinary matrix.
Sometimes they are already found in symbolic form. Examples are f=ma or I=V/R. Others are ordinarily expressed in words, such as “elements combine in fixed proportions by weight” or “action equals reaction.” If such expressions were not generally accepted, group members would have no point of attachment for the powerful techniques of logical and mathematical manipulation in their puzzle-solving activity. Although examples from taxonomy suggest that normal science can proceed with only a few such expressions, the power of a science seems, quite generally, to increase with the number of symbolic generalizations available to its practitioners.
These generalizations look like laws of nature, but their function for the members of the group is often not only that. Sometimes it is. For example, the Joule-Lenz law is H = RI². When that law was discovered, the members of the scientific community already knew what H, R, and I each stood for, and these generalizations merely told them something they had not yet known about the behavior of heat, current, and resistance. But as the discussion earlier in this book suggests, symbolic generalizations commonly display a second function at the same time, a function usually sharply separated in analyses by philosophers of science. Like f=ma or I=V/R, symbolic generalizations function in part as laws, but in part they also function as definitions of some of the symbols they deploy. Moreover, the balance between their inseparable legislative and definitional force changes over time. In another context these points will be analyzed again in detail, because the nature of a commitment to a law is altogether different from that of a pledge to a definition. Laws can often be corrected piecemeal, but definitions, as tautologies, cannot. For example, part of what acceptance of Ohm’s law required was the redefinition of both “current” and “resistance.” If these terms had continued to mean what they had meant before, Ohm’s law could not have been correct. This is why Ohm’s law met with such fierce opposition, whereas, for example, the Joule-Lenz law did not.8) This situation is probably typical. These days I suspect that every revolution involves, among other things, the abandonment of generalizations whose force had previously been in some part tautological. Did Einstein prove that simultaneity is relative, or did he alter the concept of simultaneity itself? Were those who felt a paradox in the phrase “relativity of simultaneity” simply wrong?
Next we must consider a second sort of component of the disciplinary matrix, one about which I spoke at length in the first edition under such headings as “metaphysical paradigms” or “the metaphysical parts of paradigms.” What I have in mind here are shared commitments to such beliefs as the following: heat is the kinetic energy of the constituent parts of bodies. If I were rewriting this book now, I would describe such commitments as beliefs in particular models. And I would extend the category of models to include a considerably heuristic variety as well. An electric circuit may be regarded as a steady-state hydrodynamic system. The molecules of a gas move as if tiny elastic billiard balls were moving at random. Although the strength of group commitment varies along the spectrum from heuristic to ontological models, with nontrivial consequences, all models have similar functions. Among other things, they provide the group with preferred or permissible analogies and metaphors. By doing so, they alter what will be accepted as an explanation; in other words, they help determine the roster of unsolved puzzles and evaluate the importance of each. It should be noted, however, that members of a scientific community need not share even heuristic models—though they usually do. I have already pointed out that during the first half of the nineteenth century, membership in the community of chemists did not require belief in atoms.
As a third sort of element in the disciplinary matrix, I shall here describe values.
Usually they are accepted more widely across different scientific communities than either symbolic generalizations or models, and such values contribute greatly to giving natural scientists, as a whole, a sense of community. They are always in operation, but the time when their importance becomes especially conspicuous is when the members of a particular scientific community must identify a crisis, or later, when they must choose between incompatible ways of practicing their field. Perhaps the most deeply held values concern prediction. Predictions should be accurate.
Quantitative inference is preferable to qualitative inference. Whatever the permissible margin of error, prediction should be consistently satisfied within a given field. There are many others. But there are also values used in evaluating entire theories. First and foremost, they must make puzzle-formulation and puzzle-solution possible. As far as possible, they should be simple, coherent, plausible, and not in conflict with other theories then being deployed. (I now consider it a weakness of the first edition that I paid so little attention to values such as internal and external consistency when considering the sources of crisis and the factors in theory choice.) In addition, there are other kinds of values—for example, that science should be socially useful, or need not be—but I believe the preceding account suggests what I have in mind.
One characteristic of shared values, however, requires special mention. To a far greater degree than any other kind of component in the disciplinary matrix, values can be shared by people who differ in their application of them. Judgments of accuracy are relatively—though of course not entirely—stable from one age to another, and they do not differ greatly from one member of a particular group to another.
But judgments of simplicity, consistency, plausibility, and the like often vary greatly from individual to individual.
What for Einstein was an intolerable contradiction in the old quantum theory, one that made the pursuit of normal science impossible, was for Bohr and others merely a difficulty that could be expected to resolve itself by normal means. More important still, in cases where values must be applied, different values alone often led to different choices. One theory may be more accurate than another, but less consistent or less plausible. Here again the old quantum theory provides an example. In short, although values are widely shared by scientists, and although commitment to them is profound and constitutive of science, the application of values may be considerably affected by the characteristics of individual personality and experience that distinguish the members of a group.
To many readers of the earlier parts of this book, this characteristic in the operation of shared values has appeared to be a major weakness in my position. Because I insist that what scientists share is not sufficient to command uniform agreement on such matters as the choice between competing theories or the distinction between an ordinary anomalous phenomenon and one that provokes a crisis, I have often been accused of glorifying subjectivity and even irrationality.9) But such a reaction ignores two characteristics displayed by value judgments in any field.
First, even though the members of a group do not all apply values in the same way, shared values can be important determinants of group behavior.〔If this were not so, there would be no distinctive philosophical problems concerning value theory or aesthetics.〕People did not all paint alike during periods when representation was a primary value, but the pattern of development in the plastic arts underwent extreme change when that value was abandoned.10) Imagine what would happen in science if consistency ceased to be a primary value. Second, individual variability in the application of shared values may perform a function essential to science. The matters to which values must be applied are also, without exception, the very points at which risks must be taken. Most anomalies are resolved by normal means. Most proposals for new theories turn out to be wrong. If every member of a scientific community responded every time to an anomaly as a source of crisis, or willingly accepted every new theory advanced by a colleague, science would come to a halt. On the other hand, if no one responded to anomalies or to high-risk new theories, revolutions would occur rarely or not at all. In matters such as these, for a scientific community to rely on shared values rather than on shared rules governing individual choices may be a way of distributing risk and ensuring the long-term success of its research activity.
I now want to turn to a fourth sort of element in the disciplinary matrix, not the only remaining sort, but only the last sort to be discussed here. For this element, the term “paradigm” would be exactly appropriate, both linguistically and lexically. This is the component of a group’s shared commitments that originally led me to choose that word. But since the term seems to have taken on a life of its own, I shall here replace it with the word “exemplars.” What I mean by this term is the concrete problem-solutions that students encounter from the beginning of their scientific education, whether in the laboratory, on examinations, or at the ends of chapters in science textbooks. To these shared examples, however, must be added at least some of the technical problem-solutions found in periodicals. These are the ones scientists encounter in their research life after completing their education, and they also show scientists by example how their work is to be carried out. More than any other component of the disciplinary matrix, differences among sets of exemplars provide the scientific community with the fine structure of science. For example, all physicists begin by learning the same examples: problems such as the inclined plane, the conical pendulum, and Kepler’s orbits.
Instruments such as the vernier, the calorimeter, and the Wheatstone bridge are also included. But as scientists’ training advances, the symbolic generalizations they share are increasingly explained by different exemplars. Although physicists in solid-state theory and field theory both share the Schrödinger equation, only its more elementary applications are common to both groups.
3. Paradigms as Shared Examples The paradigm as shared example is now the central element of what I regard as the most novel and least understood part of this book. Accordingly, exemplars require more attention than any other kind of component in the disciplinary matrix. Philosophies of science have not ordinarily discussed the problems a student encounters in the laboratory or in science textbooks, because such things are considered to provide only practice in applying what the student already knows. Unless the student first learns a theory and some rules for applying it, it is thought, he cannot solve problems at all.
Scientific knowledge is embedded in theories and rules. Problems are provided so that one may become skilled in applying them. But I have been arguing that it is wrong to localize the epistemic implications of science in that way. After students have solved many problems, perhaps by solving still more they are merely cultivating facility. But at the beginning, and for some time thereafter, solving problems is learning coherent matters about nature. If such standard examples did not exist, the laws and theories students had already learned would have very little empirical content.
To indicate what I mean, let us now return briefly to the generalization of symbol use. As one widely shared exemplar, consider Newton’s second law of motion, generally expressed as f=ma. A sociologist or linguist who discovered that members of a given specialist group uttered and accepted an equivalent expression without question would not, without further consideration, learn much about what that expression or each term in that formula meant, or how the scientists of that community related the formula to nature. In fact, the mere fact that they acknowledge it without question and use it as a key for introducing logical and mathematical operations does not in itself mean that they are in complete agreement on issues such as meaning and application. Of course they will agree to a considerable extent, and if they do not, that fact will soon appear from their subsequent conversation. But this question may arise.
At what point, and by what means, did they come to do so? Faced with a given experimental situation, how did they come to know how to pick out the relevant force, mass, and acceleration?
In fact, although this aspect of such situations has very rarely been noticed, or not noticed at all, what students must learn is far more complicated than this. It is not like the case in which logical and mathematical operations are applied directly to f=ma. On examination, that relation proves to be a law-sketch, or law-schema. As the student or the scientist engaged in research moves from one problem situation to the next, the symbolic general formula to which these operations are applied changes. In the case of free fall, f=ma becomes mg=md2s/dt2. For a simple pendulum, the relation is transformed into mgsinΦ=-mld2Φ/dt2. For a pair of interacting harmonic oscillators, it is written as two equations, the first of which is m1d2s1/dt2+k1s1=k2(s2-s1+d). And in a case as complicated as a gyroscope, it takes yet another form, so that it becomes difficult even to recognize its resemblance to the formula f=ma. Nevertheless, as a student learns to identify force, mass, and acceleration in various physical situations he has not previously encountered, he also comes to grasp how to devise the appropriate modifications of f=ma that relate those expressions to one another; usually, such transformed expressions take forms whose literal equivalents he has never experienced before. How did the student come to know how to do this?