The Information Content of Laws | On Beyond Darwin - Chapter 5


The Information Content of Laws

In a letter to a scientific colleague Newton spoke about what he considered to be the best method in natural philosophy—science:

For the best and safest method of philosophizing seems to be, first diligently to investigate the properties of things and establish them by experiment, and then to seek hypotheses to explain them. [1]

Fact finding comes first, then comes the search for “hypotheses” which will explain the facts. Newton found that the same set of experimental facts could be explained by quite different hypotheses. It was impossible to resolve an argument about which hypothesis was right provided all the hypotheses did was fit the known facts. In the heat of one argument he swore off making hypotheses—in Latin, of course, by “hypotheses non fingo” which means “I do not make hypotheses”. In fact, he went right on making hypotheses but avoided arguments. This remark of Newton’s is sometimes interpreted as meaning that we should not try to explain laws by devising some underlying mechanism but be content that the facts are explained when the law has been formulated that fits the facts. Newton’s law of gravitation is usually taken as the example—we tend to accept the law as the final explanation of gravity without asking further questions as to why this particular law holds. In a way, Newton is pressing for empiricism here, telling us to skip the theorizing because all it does is lead to arguments. Newton himself made many hypotheses. To have a model of what is going on—a theory, if you will—can be useful. Often, a theory—law or hypothesis—not only fits facts that have been found by observation and experiment but also indicates—or predicts—that other facts, not known at the time, should be true. The theory contains then some part that is not strictly empirical—it contains assumptions about what might be the case. Often, we can make a choice between two alternative theories that fit the known facts by looking for consequences of the theories that differ from each other. An experiment is then performed to resolve the argument about which theory is better. Bridgman describes the process:

If we can show that any of the indirect consequences are opposed to experiment then the assumption is false. But, the concept of true is not applicable. I think we would want to invent a new concept to cover the situation: probably the word “possible” has enough of the required connotations to meet our needs… Doubtless a great many alternative theories will be possible and we shall have to choose between them on grounds of simplicity or convenience of calculation or perhaps on purely aesthetic considerations. [2]

I have already indicated that I am accepting as scientifically true any law, or theory that fits the facts. False to me applies to one that does not fit the facts. Several different theories can be true as far as I am concerned. Bridgman wants to call theories that fit the facts “possible theories,” thus avoiding the problem of calling two alternative theories true. It is just a matter of words here. He indicates that if we have two equally good—possible—theories, in the sense that they explain the facts, we choose one or the other on some arbitrary grounds such as “simplicity or convenience of calculation.” The choice in favor of the simple, or the beautiful law was made originally because scientists, like Newton, believed that it accorded better with the mind of the Creator. It was thus, more likely to be “really” true. Mach also said the simpler explanation should be chosen but for an entirely different reason. This is how Feuer summarizes Mach:

Natural laws were economical summations of experience, labor-saving devices, that enable the labor of other men to be substituted for one’s own…Occam’s Razor, the so-called principle of simplicity in scientific method, was no longer to be founded on a metaphysical belief in the simplicity of the universe; rather it was a principle of economy expressive of man’s biological aim to do things with the least expenditure of energy. [3]

Or in Mach’s own words:

Science itself, therefore, must be regarded as a minimal problem consisting of the completest possible presentment of facts with the least possible expenditure of thought. [4]

It is a pragmatic point of view. Again Mach:

Those ideas that hold good throughout the widest domains of research and that supplement the greatest amount of experience are the most scientific. [5]

This reason of Mach’s for choosing a simpler theory over a more complicated one, because the simpler theory is more scientific, has little force when the simpler theory emerges some time after the more complicated one and does not predict anything different from the original theory. Louis de Broglie makes this clear:

Indeed, we may assert it as a sociological law of the scientific community that no new theory, whatever its appeal of elegance or simplicity, will generally supersede the old unless it leads to new experimental discoveries of fact; such resultant discoveries may, with the help of auxiliary hypotheses, be rendered consistent with older doctrines, but would not have been foretold by the latter’s adherents. [6]

A theory becomes established and will not easily be deposed by a new simpler theory which fits the same facts, even though, from Mach’s standpoint—and mine—it is better.

The ability of a theory to predict facts that can be later verified is perhaps overemphasized. Certainly, if a theory does predict facts beyond the known facts that it was designed to fit, we should check to see whether or not the additional facts can be verified. If they cannot, we would have to alter the theory or get a new one. The alteration of a theory is quite possible—we could for instance, impose additional restrictions so the theory would not predict the nonexistent facts. Schrödinger, the scientist who invented the wave mechanical model of the atom, is quoted by Feuer as saying:

The following process is recurrent in physical science. A certain amount of special knowledge, empirically accumulated and asserted, is tentatively cast into a comprehensive theoretical aspect. The theory, after having been gradually corrected by further experiments… tends to acquire an unforeseen general validity. But, strangely enough… the knowledge which its proportions are supposed to convey turns out to be more and more tautological.

Schrödinger is indicating that the theory which emerges after the processes of fact fitting and correction are ended begins to seem more than just a “possible’ explanation—in Bridgman’s terminology. It becomes stamped with an authenticity that makes it practically inevitable, if not obvious, or true by definition. It becomes established, and any change from the established view is taken only if the established view is proved false.

Louis de Broglie hints that the choice between equivalent theories is often made on the basis of personal philosophical beliefs in “hidden harmonies” in the universe—the order of things:

A few examples don’t suffice to prove that there are always an infinite number of possible theories for explaining the same experimental facts, and it seems certain to us that even, when there are a great number of logically equivalent theories, the physicist has the right to believe that one of them conforms more to underlying physical reality, and is more capable of generalization, more apt to reveal the hidden harmonies. [7]

I will be writing in a later chapter about the philosophical beliefs of a number of modern scientists which slanted their science towards certain types of theories and made those theories, to them, seem inevitable. Tzara, a Dadaist philosophically, wrote:

Science throws me off as soon as it pretends to be a philosophical system; for it loses its useful character… I detest that pat objectivity and harmony with which science finds all in order. [8]

The Dadaists hated the idea of design and logic, and featured chance in all their thinking and art. I must admit that science throws me off sometimes with concepts such as the “wave-particle duality of matter and radiation,” as if duality were of the essence of nature.

Sometimes after a theory has been established for a while there are alternative approaches which contain much the same information, in that they explain the same facts that are accepted by the scientific community. For example, the energy point of view in mechanics—which is the study of motion of interacting objects—was formulated a considerable length of time after Newton first devised his laws of motion. There is absolutely no additional information provided by the introduction of the concept of energy, but often greater insight into a physical situation can be gained by using it. It is a labor saving device in some computations. So we have two alternative viewpoints on motion, the force-acceleration viewpoint and the energy viewpoint. Sometimes the new viewpoint leads to the discovery of new information. In studying the electromagnetic interaction, the concept of the electromagnetic field used by Maxwell helped him to add a piece to the theory that was not based on any experimental evidence. Maxwell’s enormous success has meant that the field concept has dominated thinking about electromagnetism, and the electric charge, as the source of the field, has been somewhat neglected. Scientists frequently introduce concepts, like energy or electromagnetic field, which are defined in terms of other, more fundamental, concepts. These others are more fundamental only because they are closer to our own sense experiences. Newton introduced the concepts of force and mass which he defined in terms of the prior and more fundamental concepts of position, velocity, and acceleration. Einstein explains this all rather well:

We shall call “primary concepts” such concepts as are directly and intuitively connected with typical complexes of sense experiences. All other notions are—from the physical point of view—possessed of meaning, only in so far as they are connected, by theorems, with the primary notions. These theorems are partially definitions of the concepts (and of the statements derived logically from them) and partially theorems not derivable from the definitions, which express at least indirect relations between the “primary concepts,” and in this way between sense experiences. Theorems of the latter kind are “statements about reality” or laws of nature, i.e., theorems which have to show their usefulness when applied to sense experiences comprehended by primary concepts. The questions as to which of the theorems shall be considered as definitions and which are natural laws will depend largely upon the chosen representation. It really becomes absolutely necessary to make this differentiation only when one examines the degree to which the whole system of concepts considered is not empty from the physical point of view. [9]

Einstein indicates that concepts like energy and electromagnetic fields must be connected with primary concepts through definitions. As well as the definitions, there will be other—usually mathematical—statements that can be made. These are often called laws of nature, for example, the law of conservation of energy. But in fact there is always confusion about what is a definition and what is a law of nature. We tend to think that a definition does not contain any information about the physical world in it but that a law of nature does. Einstein points out that it is “the whole system of concepts” that must be thought of as containing information about the world. When I speak of the information content of laws , I mean the same thing as what Einstein calls “the degree to which the whole system of concepts considered is not empty from the physical point of view.” Here is another opinion, this time from Richard Feynman:

Although it is interesting and worth while to study the physical laws simply because they help us to understand and to use nature, one ought to stop every once in a while and think, “What do they really mean?” The meaning of any statement is a subject that has interested and troubled philosophers from time immemorial, and the meaning of physical laws is even more interesting, because it is generally believed that these laws represent some kind of real knowledge. [10]

Feynman uses the phrase “some kind of real knowledge” where I speak of “information content.”

In the next chapter, I will examine Newton’s laws of motion which are one of the expressions of the information about the motion of objects as they interact with each other. In performing this analysis, I must consider the whole system, both laws and definitions, because, as Einstein indicates, the boundary between them is very fuzzy. Then there is the confusion about information content when there are alternative formulations. Because there are alternative systems for mechanics, I would have to say that one is as true as another. That brings up another related problem. If the whole of mechanics can be explained without any reference to the concept energy, is there anything real about energy? If you could explain mechanics without the concept of force is there anything “real” about force? And what about explaining electromagnetism without the concept of the electromagnetic field? Then there is the other side of this. Newton said that he would make no hypothesis about the law of gravity. Was he saying that he would not try to explain why the force of gravitation produced by an object depended on its mass and fell off inversely as the square of the distance from the object? Why should it depend on the mass, a concept he introduced to express his laws of motion? And why should it vary as the inverse square of the distance? Explain that Sir Isaac!—Answer: “Hypotheses non fingo.”—No, his real answer is that it is the will of the Creator who designed the universe by laying down a set of universal laws from which everything follows. End of explanation. When you have this attitude perhaps you shut doors that should remain open. Newton hated controversy, but science must have differences of opinion, or it cannot survive. Kuhn points this out:

The resolution of revolutions is the selection by conflict within the scientific community of the fittest way to practise future science… And the entire process [development of science] may have occurred, as we now suppose biological evolution did, without benefit of a set goal, a permanent fixed scientific truth, of which each stage in the development of scientific knowledge is a better exemplar. [11]

Kuhn, like me, doubts that there is an absolute scientific truth to which science is approaching closer and closer.

So that is my program—in the next chapter I will look at Newtonian mechanics and see what I can read into it as far as information content is concerned. Then, in the following chapter, I will examine electromagnetic theory. I will be acting somewhat as a critic in that I will be tearing the laws—and definitions—apart hunting for meaning—content. Bridgman describes this kind of activity as follows:

The material for the physicist as critic is the body of physical theory just as the material for the physicist as theorist is the body of experimental knowledge. [12]

Remember that my aim is twofold—first, to show that the idea of general laws is an illusion and second, to offer some explanation as to why any specific facts are simple, if they are.

One of the dangers of this sort of exploration is that, in trying to present a coherent world view, I must plug some of the holes with suggestions that I perhaps have not thought out carefully enough. Actually, most of the suggestions have already been made by other people and I am merely selecting ones that fit in with my scheme.

These suggestions have not had widespread interest shown in them because they departed from the orthodox view and were in a sense contradictory to that view, even though they had features that were appealing. At the end of each of the following chapters I will try to list those unorthodox ideas that I am including in my world view, both those of others and my own. It must be remembered that the point of this investigation is to see whether it is possible to construct a world view based on the premise that there are no general laws. Each particular suggestion must not be taken as being the only one that might serve.

I have a somewhat similar aim to Mach who in the preface to his book on the Science of Mechanics said:

The present volume is not a treatise upon the application of the principles of mechanics. Its aim is to clear up ideas, expose the real significance of the matter and get rid of metaphysical obscurities. [13]

By “significance of the matter,” I mean the “information content.” By getting rid of “metaphysical obscurities,” I mean that my aim is to show that there is no evidence in physics for or against an underlying pattern of things.

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