The Explanation of Laws

by Norman Campbell

Explanation in general is the expression of an assertion in a more acceptable and satisfactory form. Thus if somebody speaks to us in a language we do not understand, either a foreign language or the technical language of some study or craft with which we are not familiar, we may ask him to explain his statement. And we shall receive the explanation for which we ask if he merely alters the form of his statement, so as to express it in terms with which we are familiar. The statement in its new form is more acceptable and more satisfactory, because now it evokes a definite response in our minds which we describe by saying that we understand the statement. Again we sometimes ask a man to explain his conduct; when we make such a demand we are ignorant, or pretending to be ignorant, of the motives which inspired his action. We shall feel that he has offered a complete explanation if he can show that his motives are such as habitually inspire our own actions, or, in other words, that his motives are familiar to us.

But expressions, or the ideas contained in them, may be more acceptable and more satisfactory, on grounds other than their familiarity; and all explanations do not consist of a reduction of the less to the more familiar. Indeed it would seem that the explanations which, in the view of the man in the street, it is the business of science to offer do not involve familiar ideas at all. Thus we may expect our scientific acquaintances to explain to us why our water- pipes burst during a frost or why paint becomes dirty sooner in a room lit by gas than in one lit by electricity. We shall be told in reply that the bursting of the pipes is due to the expansion of water when it is converted into ice, and the blackening of the paint to the combination of the white pigment with sulphur present in coal-gas to form compounds that are dark, and not white, in color. Now in these instances the ideas involved in the explanation are probably less, and not more, familiar than those that they are used to explain. Many more people know that water- pipes burst during a frost than know that water (unlike most liquids) expands when it freezes; and many more know that their paint goes black than know that lead carbonate (one of the commonest white pigments) is converted by sulphur into black lead sulphide.

Why then do we regard our questions as answered? Why do we feel that, when we have received them, the matter is better understood, and our ideas on it clearer and more satisfactory? The reason is that the events and changes have been explained by being shown to be particular examples of a general law. Water always expands when it freezes, although it does not always burst household pipes; for it may not be contained in pipes or in any closed vessel. And lead carbonate always reacts with sulphur in the form present in coal-gas, even if it is not being used as a pigment. We feel that our experience is no longer peculiar and mysterious; it is only one instance of general and fundamental principles. It is one of the profoundest instincts of our intellectual nature to regard the more general principle as the more ultimately acceptable and satisfactory; it is this instinct which led men first to the studies that have developed into science. In fact, what was called... the “ordering” of experience by means of laws might equally well have been called the explanation of that experience. Laws explain our experience because they order it by referring particular instances to general principles; the explanation will be the more satisfactory the more general the principle, and the greater the number of particular instances that can be referred to it. Thus, we shall feel that the bursting of the pipes is explained more satisfactorily when it is pointed out that the expansion of water when converted into ice explains also other common experience, for instance that a layer of ice forms first on the top of a pond and not on the bottom.

Doubtless there are other kinds of explanation; but it is important for our purpose to notice that the explanations of common life often depend on these two principles—that ideas are more satisfactory when they are more familiar and also when they are more general; and that either of these principles may be made the basis of an explanation.

When it is asked what is the nature of the scientific explanation of laws—and it is the purpose of this chapter to answer that question—it is usually replied that it is of the second kind, and that laws are explained by being shown to be particular examples of more general laws. On this view the explanation of laws is merely an extension of the process involved in the formulation of laws; it is simply a progress from the less to the more general. At some stage, of course, the process must stop; ultimately laws so general will be reached that, for the time being at least, they cannot be included under any more general laws. If it were found possible to include all scientific laws as particular instances of one extremely general and universal law, then, according to this opinion, the purpose of science would be completely achieved.

I dissent altogether from this opinion; I think it leads to a neglect of the most important part of science and to a complete failure to understand its aims and development. I do not believe that laws can ever be explained by inclusion in more general laws; and I hold that, even if it were possible so to explain them, the explanation would not be that which science, developing the tendencies of common sense, demands. The first point is rather abstruse and will be dismissed briefly. It certainly seems at first sight that some laws can be expressed as particular instances of more general laws. Thus the law (stating one of the properties of hydrogen) that hydrogen expands when heated seems to be a particular instance of the more general law, that all gases expand when heated. I think this appearance is merely due to a failure to state the laws quite fully and accurately, and that if we were forced to state with the utmost precision what we mean to assert by a law, we should find that one of the laws was not a particular case of the other. However, I do not wish to press this contention, for it will probably be agreed that, even if we have here a reference of a less general, to a more general law, we have no explanation. To say that all gases expand when heated is not to explain why hydrogen expands when heated; it merely leads us to ask immediately why all gases expand. An explanation which leads immediately to another question of the same kind is no explanation at all.

What is a Theory?

How then does science explain laws? It explains them by means of “theories,” which are not laws, although closely related to laws. We will proceed at once to learn what a theory is, and how it explains laws. For this purpose an example is necessary, even though its use involves entering more into the details of science than is our usual practice. A great many laws are known, concerning the physical properties of all gases; air, coal-gas, hydrogen and other gases, differ in their chemical properties, but resemble each other in obeying these laws. Two of these laws state how the pressure, exerted by a given quantity of gas on its containing vessel, varies with the volume of the vessel, and with the temperature of the gas. Boyle's Law states that the pressure is inversely proportional to the volume, so that if the volume is halved, the pressure is doubled; Gay-Lussac’s states that, at a constant volume, the pressure increases proportionally to the temperature (if a certain scale of temperature is adopted, slightly different from that in common use). Other laws state the relation between the pressure of the gas and its power of conducting heat and so on. All these laws are “explained” by a doctrine known as the Dynamical Theory of Gases, which was proposed early in the last century and is accepted universally today. According to this theory, a gas consists of an immense number of very small particles, called molecules, flying about in all directions, colliding with each other and with the wall of the containing vessel; the speed of the flight of these molecules increases with the temperature; their impacts on the walls of the vessel tends to force the walls outwards and represent the pressure on them; and by their motion, heat is conveyed from one part of the gas to another in the manner called conduction.

When it is said that this theory explains the laws of gases, two things are meant. The first is that if we assume the theory to be true we can prove that the laws that are to be explained are true. The molecules are supposed to be similar to rigid particles, such as marbles or grains of sand; we know from the general laws of dynamics (the science which studies how bodies move under forces) what will be the effect on the motions of the particles of their collisions with each other and with the walls; and we know from the same laws how great will be the pressure exerted on the walls of the vessel by the impacts of a given number of particles of given mass moving with given speed. We can show that particles such as are imagined by the theory, moving with the speed attributed to them, would exert the pressure that the gas actually exerts, and that this pressure would vary with the volume of the vessel and with the temperature in the manner described in Boyle's and Gay-Lussac’s Laws. In other words, from the theory we can deduce the laws.

This is certainly one thing which we mean when we say that the theory explains the laws; if the laws could not be deduced from the theory, the theory would not explain the laws and the theory would not be true. But this cannot be all that we mean. For, if it were, clearly any other theory from which the laws could be deduced, would be equally an explanation and would be equally true. But there are an indefinite number of “theories” from which the laws could be deduced; it is a mere logical exercise to find one set of propositions from which another set will follow; and anyone could invent in a few hours twenty such theories. For instance, that the two propositions (1) that the pressure of a gas increases as the temperature increases (2) that it increases as the volume decreases, can be deduced from the single proposition that the pressure increases with increase of temperature and decrease of volume. But of course the single proposition does not explain the two others; it merely states them in other words. But that is just what logical deduction consists of; to deduce a conclusion from premises is simply to state the premises in different words, though the words are sometimes so different as to give quite a different impression. If all that we required of a theory was that laws could be deduced from it, there would be no difference between a theory which merely expressed the laws in different words without adding anything significant and a theory which, like the example we are considering, does undoubtedly add something significant.

It is clear then that when we say the theory explains the laws we mean something additional to this mere logical deduction; the deduction is necessary to the truth of the theory, but it is not sufficient. What else do we require? I think the best answer we can give is that, in order that a theory may explain, we require it—to explain! We require that it shall add to our ideas, and that the ideas which it adds shall be acceptable. The reader will probably feel that this is true of the explanation of the properties of gases offered by the dynamical theory. Even if he did not know (and he probably does not know apart from what I have just told him) that the laws can be deduced from the theory, he would feel that the mere introduction of moving particles and the suggestion that the properties of a gas can be represented as due to their motion would afford some explanation of those properties. They would afford some explanation, even if the laws could not be deduced correctly; they would then offer an explanation, although the explanation would not be true.

And this is, I believe, the reason why he would feel thus. Only those who have practiced experimental physics, know anything by actual experience about the laws of gases; they are not things which force themselves on our attention in common life, and even those who are most familiar with them never think of them out of working hours. On the other hand, the behavior of moving solid bodies is familiar to everyone; everyone knows roughly what will happen when such bodies collide with each other or with a solid wall, though they may not know the exact dynamical laws involved in such reactions. In all our common life we are continually encountering moving bodies, and noticing their reactions; indeed, if the reader thinks about it, he will realize that whenever we do anything which affects the external world, or whenever we are passively affected by it, a moving body is somehow involved in the transaction. Movement is just the most familiar thing in the world; it is through motion that everything and anything happens. And so by tracing a relation between the unfamiliar changes which gases undergo when their temperature or volume is altered, and the extremely familiar changes which accompany the motions and mutual reactions of solid bodies, we are rendering the former more intelligible; we are explaining them.

That is to say, the explanation of laws offered by theories (for this example has been offered as typical) is characteristically explanation of the first of the two kinds with which the chapter started. It is explanation by greater familiarity, essentially similar to that offered when a statement is translated from an unknown to a known language. This conclusion may be surprising, and indeed it is not that generally advanced. Before developing our view further, it will be well to examine the matter from another point of view.

Difference Between Theories and Laws

It was stated before that it has been usually held that the explanation of laws consists in showing that they are particular examples of more general laws. If this view were applied to the example under 'discussion, it might be urged that the dynamical theory explains the properties of gases because it shows that they are particular examples of the laws of dynamics; the properties of gases are explained because they are shown to be the consequences of the subjection of the molecules, of which the gases consist, to the general laws of all moving bodies. Here, it might be said, is the clearest possible instance of explanation by generalization, a simple extension of the process involved in the discovery of laws.

But, against this view, it must be pointed out that the most important feature of the theory is not that it states that molecules are subject to dynamical laws, but that which states that there are such things as molecules, and that gases are made up of them. It is that feature of the theory which makes it a real explanation. Now this part of the theory is not a particular instance of any more general law; indeed it is not a law or anything that could be an instance of a law. For it is not... part of the proper subject matter on which science builds its foundations. Molecules are not things which we can see or feel; they are not, like the ordinary material bodies to which the laws of dynamics are known to apply, objects discernible to direct perception. We only know that they exist by inference; what we actually observe are gases, varying in temperature and pressure; and it is only by these variations that we are led to suspect the existence of the molecules. We may apply once more our fundamental test of universal agreement which serves to distinguish the objects concerned in laws from any others. If somebody denied the existence of molecules, how could we prove him wrong? We cannot show him the molecules; we can only show him the gases and expound the theory; if he denied that the theory proved the existence of the molecules, we would be powerless. We cannot prove by his actions that he is perverse or deluded; for his actions will be affected only by the properties of gases, which are actually observed, and not by the theory introduced to explain them. Actually the dynamical theory of gases has been denied by men of science of high distinction. Usually the denial was based partly on the assertion that the laws of gases could be deduced accurately from the theory, but it has often been accompanied by the contention that, even if they could be deduced accurately, the theory was not true, and not worthy of acceptance. No denial of that case would be possible if the theory were indeed a law.

We conclude, therefore—and the conclusion is vital to the view of science presented here—that a theory is not a law, and consequently, that the explanation afforded by a theory cannot simply be the explanation by generalization which consists in the exhibition of one law as a particular instance of another. It does not follow that theories have nothing to do with laws, and that it is immaterial for the theory that the laws of dynamics are true, and of very great generality. We shall see presently that this feature is of great importance. But it does not involve that the theory is itself a law.

The Value of Theories

After this protest against a dangerous misunderstanding, let us return and develop further our view of theories, So far the truth of a theory has been based on two grounds; first, that the laws to be explained can be deduced from it; second, that is really explains in the sense that has been indicated. But actually there is in addition, a third test of the truth of a theory, which is of great importance; a true theory will not only explain adequately the laws that it was introduced to explain; it will also predict and explain in advance laws which were unknown before. All the chief theories in science (or at least in physics) have satisfied this test; they have all led directly to the discovery of new laws which were unsuspected before the theory was proposed.

It is easy to see how a theory may predict new laws. The theory, if it is worthy of consideration at all, will be such that the old laws can be deduced from it; so far as the theory is concerned, these others differ in no way from the known laws, and if the theory is to be true, these laws that are consequences of it must be true. As a matter of fact, it is very seldom that a theory, exactly in its original form, predicts any laws except those that it was proposed to explain; but a very small and extremely natural development of it may make it predict new laws. Thus, to take our example, in order to explain the laws (Boyle's and Gay-Lussac’s) to which the theory was originally applied, it is unnecessary to make any assumption about the size of the molecules; those laws can be deduced from the theory whatever that size (so long as it is below a certain limit) and the assumption was at first made for simplicity that the molecules were mathematical points without any size at all. But obviously it was more natural to assume that the molecules, though extremely small, have some size and once that assumption is made, laws are predicted which had not been discovered at the time and would never have been suspected apart from the theory. Thus, it is easy to see that, if the molecules have a definite size, the behavior of a gas, when the number of molecules contained in a given vessel is so great that the space actually occupied by the molecules is nearly the whole of the space in the vessel, will be very different from its behavior when there are so few molecules that practically all that space is unoccupied. This expectation, a direct result of the theory, is definitely confirmed by experiments which show a change in the laws of a gas when it is highly compressed, and all its molecules forced into a small volume.

This test of predicting new and true laws is always applied to any theory when it is proposed. The first thing we do when anyone proposes a theory to explain laws, is to try to deduce from the theory, or from some slight but very natural development of it, new laws, which were not taken into consideration in the formulation of the theory. If we can find such laws and prove by experiment that they are true, then we feel much more confidence in the theory; if they are not true, we know that the theory is not true; but we may still believe that a relatively slight modification will restore its value. It is in this way that most new laws are actually suggested... At the present time, in the more highly developed sciences, it is very unusual for a new law to be discovered or suggested simply by making experiments and observations and examining the results (although cases of this character occur from time to time); almost all advances in the formulation of new laws follow on the invention of theories to explain the old laws. Indeed it has been urged that the only use of theories is thus to suggest laws among which some will be found to be true. This opinion has been much favored by philosophers and mathematicians and has always been accompanied by the opinion that it is the end and object of science to discover laws. It has also been professed (especially at the end of the nineteenth century) by people who know something about science and actually practiced it; but I think that these people only professed the view because they were afraid what the philosophers might say if they denied it. At any rate, for myself, I cannot understand how anybody can find any interest in science, who thinks that its task is completed with the discovery of new laws.

For the explanation of laws, though it is formally quite a different process from the discovery of laws, is in its object merely an extension of that process. We seek to discover laws in order to make nature intelligible to us; we seek to explain them for exactly the same reason. The end at which we are aiming in one process as in the other is the reconciliation without intellectual desires of the perceptions forced on us by the external world of nature. What possible reason can be given for attaching immense importance to one stage in the process and denying all intrinsic importance to another? Surely so long as anything remains to be explained it is the business of science to continue to seek explanations.

Questions to Consider