A Note on the Methodology of Physics

IN the second of these lectures I raised the question whether the laws of physics make for cosmic order and I reached the conclusion that they do not. Therewith I introduced the only question about the laws of physics that is relevant to the dispute between monists and dualists. For while it is relevant to show that these laws do not serve the same kind of purpose that is served by the laws that are inscribed in statute books it is not relevant to discuss what purpose they do serve. Logic, which requires that only those facts be admitted to the argument that are strictly relevant and that all others be austerely banned, should therefore forbid any addition to what I have said already about the laws of physics. But a negative statement, even though it suffice for the immediate purpose, tends to provoke an all too clamorous demand for its positive counterpart. When those who are conscientiously trying to understand the nature of reality in its entirety are told what the laws of physics do not do it is only natural for them to ponder what these laws do do; and to ponder thus is to lose receptivity. Good logic is sometimes the enemy of good exposition.

It is the more likely to be so in the present instance because it has come to be taken for granted that a law, no matter of what kind, cannot conceivably serve any other purpose than that of providing both proof and guarantee of order. Just as there can be no order without law, some will be inclined to reason, so there can be no law without order. To say, as I have done, that the material universe is without any order must seem to them to ignore the very existence of the laws of physics. They may well then be left with the conviction that there must be a flaw somewhere in the reasoning that has been presented in these lectures, even if they have not been able to detect it. Anyone who is thus disturbed by the negative statements that I have made about the laws of physics may find those statements more acceptable if they are balanced by positive ones. So I propose in this Appendix to say a few words about the purpose that the laws of physics do serve.

An example will convey this purpose the more clearly the simpler it is and the less recondite scientific knowledge it presupposes. So I shall choose one from a subject with which many non-physicists are familiar, namely, map reading. But I should like to emphasize that it is more than an analogy; it provides a true illustration of the principle that is revealed in many of the most valuable laws of physics.

Mountains are irregular formations. They may be of any height, of any shape. Their approaches may have any gradient. Valleys, saddles, passes, escarpments, precipices, gentle slopes, jagged peaks may occur at any place. It would be difficult indeed to formulate any generalization about the shape of mountains. The evidence refutes any suggestion that there may be a law demanding that mountains shall be shaped thus and not otherwise.

For this reason a person set down without a map in unknown hilly country would not be able to say what the far sides of the mountains were like. But if he had a map and the map showed contour lines he could say many things about regions that were hidden from his view. He could, for instance, say what the gradient of every slope was. For he would have learnt of a rule that can quite accurately be called a law of contour lines. This states that the gradient of any slope is inversely proportional to the distance between the lines. The closer they are to each other the steeper is the slope.

This is but one of several laws of contour lines that help the map reader. Others enable him to predict where in his wanderings he will find a pass, where there is a spur and where a valley, what peaks can be seen by a person who climbs to a certain spot, and much else about the topography of the region. These laws enable a builder to estimate how much rock will have to be removed in order to provide a level foundation for a house that is to be perched on the hillside; they provide an engineer with means for choosing the best site for the dam that is to form part of a hydro-electric scheme; they help a general to dispose his troops and his guns to the best advantage.

The laws of contour lines are in striking contrast to the material reality that they help one to study. As has just been pointed out the subject about which information is sought by the map reader, a mountainous region, is one without order; no general laws can be formulated about it. And yet the laws of contour lines are general ones. There are no exceptions to them. Nothing that might happen could ever change them. They are not mere first approximations to the truth; they are completely true. Statements, in short, that have great universality, statements that have the right to be called unalterable laws, serve here for the study of a material system that does not conform to any laws at all.

The same can be said about Ohm's law, which makes it seem strange that it should be quoted more often than any other in support of belief in a Cosmic Statute Book. Ohm's law serves for the study of electrical circuits as the law of contour lines serves for the study of mountains; and no general statements can be made about electrical circuits, any more than they can about mountains. Electrical circuits may have any degree of complexity. They may comprise only a few elements or they may comprise a great many. They may consist of one single path for the current or of several parallel paths. Each path may contain any number of diverse components, to which such names as resistance, inductance, capacitance are given. Each of these components may have any numerical value, which may be measured in ohms, henries, or farads. These values may change in a variety of ways with a variety of circumstances. The resistance, for instance, may change with temperature, with voltage, perhaps with rate of change of voltage. There is, in other words, no law to require that electrical circuits shall be thus and not otherwise.

Yet it is in the study of such circuits that Ohm's law proves so useful. It has some resemblance to the law according to which the gradient of a given hillside is inversely proportional to the distance between contour lines. Ohm's law says that, for a part of an electrical circuit for which all constants except the resistance may be ignored, the voltage gradient is proportional to the current. To be quite accurate the words should be added 'provided the material of which the circuit is made is such that the resistance is independent of the voltage and that all circumstances that would cause the resistance to change are excluded'.

The wording is not as simple as that for the law of contour lines and Ohm's law is correspondingly more difficult to demonstrate. The choice of circuit has to be made rather carefully; for in many circuits the effects of other circuit constants, such as inductance and capacitance, cannot by any means be ignored. They often predominate to such an extent that the voltage gradient is very far from being proportional to the current. Circuits are in common use in which there is not even a remote resemblance to proportionality. But such circuits are not treated as exceptions, to which Ohm's law does not apply. Indeed Ohm's law is one of those that are used when the performance of such circuits is being calculated; circuits in which there is no direct proportionality between voltage gradient and current are as compatible with the law as any others. When a circuit appears to refute Ohm's law the real explanation is that the law has been carelessly and inaccurately formulated. When it is correctly formulated, with all necessary qualifications, it is just as general a statement as the law of contour lines. Like this law it could not be changed by anything that might happen. It is not a first approximation to the truth; it is completely true. Here again a statement of great universality and one that deserves to be called an unalterable law serves for the study of material systems that do not conform to any law at all.

The reason why both the law of contour lines and Ohm's law are absolutely universal and absolutely unalterable is that both are true by definition, the one by definition of a contour line, the other by definition of a resistance. If one knows what a contour line is one can deduce by simple mathematical reasoning that the gradient of a hillside is inversely proportional to the distance between contour lines drawn on a map. If, similarly, one knows what in electrical science is meant by the word 'resistance', one can deduce Ohm's law. In this sense both laws are tautologies. They are true for the same reason that it is true to say that five oranges taken seven times amount to thirty-five oranges. Another way of expressing this is to say that the law of contour lines, Ohm's law, and the multiplication table are all conceptually, and not empirically, true.

A law that is conceptually true can, of course, nevertheless be verified by experiment. One can prove the multiplication table by counting oranges. One can measure a horizontal and the corresponding vertical distance on a hillside, compute therefrom the gradient and compare it with the gradient that is predicted by the law of contour lines. That would provide experimental verification of the law. But people who have to multiply numbers and people who have to read maps know that such experiments are unnecessary. The laws in question are obvious enough to be understood without them. It can be agreed that the multiplication table and the law of contour lines could be formulated by a person who had never seen an orange or a mountain.

The experimental verification of Ohm's law is also possible, though not as easy as that of contour lines. The experimenter must devise a circuit for which all the constants except the resistance may be ignored. He must carefully control the conditions of the experiment and he must select a material for which the resistance is independent of the voltage. When he has thus reproduced a copy of the ideal circuit that is postulated in the formulation of the law his instruments do, of course, give readings that are in conformity with the law. But that is only because the truth of the law was assumed first and the experiment then planned to demonstrate it. What has really been proved by this verification of the law is that the experimenter was successful in constructing the required circuit and in controlling the experimental conditions.

I think that students are often misled about this. To suggest to them that Ohm's law is empirically and not conceptually true is to mask its real use and nature. For one thing it leads the students to the belief that the law is a statement about circuits and prevents them from understanding that it is only a statement about one particular element of a circuit, namely, resistance. There is a risk that this may leave the students with the notion that Ohm's law does not hold when the circuit being studied has non-linear constants or when it differs in any other way from the simple, ideal one employed for the demonstration. That makes it more difficult for the students to understand why Ohm's law can be correctly applied for calculating the performance of circuits when the voltage gradient is far from being proportional to the current. The pretence that Ohm's law is empirically true may, moreover, suggest the unfortunate notion that it is one of those laws that are no more than a first approximation to the facts. Should this happen the students would miss the universal and absolute character of Ohm's law. If students are not often thus misled it is only because their powers of abstract reasoning are proof against the regrettable implications of an experimental demonstration.

One can abstract from a collection of oranges a property called its number. One can abstract from among all the properties of a mountain one called the height of a certain spot above sea-level. One can imagine a line connecting all spots on the surface of the mountain that have this property in common. One can then represent this imagined line by a contour line drawn on a map of. the mountain. The contour line is a symbolical representation of a physical reality that has been abstracted from many other physical realities. In the same way one can abstract from among all the properties of an electrical circuit one called its resistance. One can represent this by the letter symbol R and make sundry statements that follow from its definition, such for instance as Ohm's law. One then has a statement about one abstracted property of electrical circuits. In other words the multiplication table is not a generalization about numbers of oranges or numbers of anything else in particular. It is a generalization about the abstract concept number. Similarly the law of contour lines is a generalization about the abstract concept contour lines and Ohm's law a generalization about the abstract concept resistance. It does not tell of any feature that all electrical circuits have in common.

To make the nature of those laws clear of which the truth is conceptual and not empirical let them be compared with laws that belong to a different category and are of the statute book kind. Suppose, for instance, that a biologist has discovered a law of growth for lobsters. It will not be a law about the abstract concept growth. It will tell of something that all lobsters have in common. It will be really and truly a generalization about lobsters. It will not be true by definition only; its basis will be in the world of concrete realities. It could not be discovered by a person who had never seen a lobster, but would depend for its verification on many measurements made on many growing lobsters. It would be found by the observations made that it was not absolutely true but only a good, working approximation to the truth. There would, moreover, be some exceptions to the law.

In consequence the law of growth for lobsters would yield a kind of information that was basically different from the information yielded by laws that are tautologies. It would enable one to predict the rate of growth of any lobster, even of a lobster that had never been seen. Laws of the statute book kind, which abound in the biological sciences, always serve that valuable purpose. In the third lecture I pointed out how many things such laws enable one to predict about an oak-tree on which no one has ever made any measurements, which has not even begun to grow yet.

Laws that are tautologies give no help at all with that kind of prediction. The information that they help one to obtain is always implicit in other information that has been obtained already. The multiplication table only helps one to predict that counting will reveal thirty-five oranges after one has counted seven heaps each containing five oranges. The law of contour lines only helps one to predict the gradient of a hillside after the mountain has been surveyed and the height of every spot above sea-level has been measured. One can only use Ohm's law to predict the performance of an electrical circuit after one has obtained numerical information about the circuit constants.

The help that these laws do give is in telling the scientist how to deduce the required information from the information that is presented to him. The multiplication table is a very simple illustration. The law of contour lines is not quite so simple. It shows how to make proper use of three numbers that provide explicit information; the scale of the map, the distance between the contour lines that are drawn on it, and the figures that are shown on the contour lines and stand for heights above sea-level. From these explicit numbers a map reader can, with the help of the law and some fairly easy arithmetic, arrive at an implicit one, the gradient.

Some of the applications of Ohm's law are much more difficult. This law often has to be combined with other laws of physics, which are also tautologies. The explicit data with which the scientist is presented is often very abundant. The processes of mathematical reasoning by which the required implicit information is made available is often very recondite. Without a wise choice of those tautologies that are called laws, the task of forming the right deductions would be impossible. The place of the laws of physics, we are led to conclude, is not in the world of physical reality but in the world of scientific methodology.

Much more remains, of course, to be said about the methodology of physics. This note only touches the fringe of a big and important subject. But in an appendix to a discourse with a different theme it would not be appropriate to go deeply into this subject. The immediate task has been only to help those who seek to reconcile two observations that may seem to contradict each other. One is the evidence that the inorganic world, the physicist's field of study, lacks order, the other the immense usefulness of those laws of physics of which Ohm's law is a typical example.

The reconciliation shows that one may, after all, be able to agree with those who say that there can be no order without law and no law without order. The laws on statute books prove that there is order in men's conduct. The laws of biology prove that there is order throughout the organic world. Though the laws of physics do not prove that there is order in the inorganic world, they do prove that there is order in the method adopted by scientists.

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