Towards a Unified Cosmology

by Reginald O. Kapp

2. The Principle of Minimum Assumption


Contents List:

Occam's Razor
What is a Minimum Assumption?
Use in Physics
Paul Dirac, 1902-84
The Concept of a Cosmic Statute Book

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See also:

A Cosmic Statute Book?
Is There a Cosmic Statute Book?

Occam's Razor

Any search for greater unification of physics must have a reasoned beginning; it must be conducted methodically; the unifying principle sought must conform to the requirements of scientific method. So let the starting point of this inquiry be one of the commonplaces of scientific method: the rule of economy of hypotheses, sometimes called 'Occam's razor'. This says that when more than one explanation of an observation is available, one must provisionally choose the one that involves the least number of assumptions. The rule is so well known and so generally accepted that there is no need to illustrate it by examples. Most of us never doubt that it is a good rule, although we may differ as to how rigidly it should be applied. To assess its value, we must first consider its nature and then the place that it occupies in scientific research. We shall find that in the history of physics, the importance of this rule is very great indeed.

The rule of economy of hypotheses is one of the canons of scientific method. It tells the scientist what to do when more than one explanatory hypothesis is available; but it does not guarantee that the recommended choice will be justified by events. This is evident from use of the word 'provisionally'. By advising that the minimum assumption be made provisionally, the rule allows for the possibility that another hypothesis, one that does not meet the criterion of minimum assumption, may have to replace it some day.

The great principles of physics are in a different category. They are not mere rules of procedure but statements about the very nature of the physical world. They are so well established that the word 'provisional' is omitted from their formulation. The principle of conservation of energy is an example. It is concerned with the energy in a given system and may be energy of motion, of position, of chemical structure, or of mass. The principle asserts that, provided the system be self-contained, the total quantity of energy in it is constant. Many conclusions can be inferred from this great principle. One of them is that a perpetual motion machine is impossible. So the principle of conservation of energy suffices to refute a person who claims to have invented such a machine. It would be highly exaggerated caution to tell him that his idea has to be rejected provisionally; one rejects it outright. One does not say that it is unlikely that the machine will work; one says that the principle of conservation of energy proves the inventor's idea to be wrong.

It is not so with a statement that violates the rule of economy of hypotheses. Almost every week, letters appear in the daily press, articles in scientific journals, and papers are presented to learned societies in which hypotheses are put forward that involve more, sometimes much more, than a minimum assumption. One may deplore such hypotheses but it is not customary to use Occam's razor to prove them wrong.

Here is a significant difference between a principle and a rule of procedure. One can refute a statement that violates the principle of conservation of energy, but one can do no more than deprecate a statement that violates the rule of economy of hypotheses.

In making this distinction, do we give a sufficient status to the rule of economy of hypotheses? It depends, I am venturing to suggest, on the discipline with which one is concerned. In history, in biology, in the social sciences, the rule can be no more than a useful guide to procedure; statements that conform to it can be accepted only provisionally; they may eventually have to be replaced by statements that violate the rule. But I wish to make the bold claim here that, in physics, the rule of economy of hypotheses can be so expressed and defined that it acquires a status far higher than the one usually accorded to it; I wish to raise it from a mere rule of procedure to one of the great universal principles to which the whole of the physical world conforms. At this level, it would be worded as follows: In physics, the minimum assumption always constitutes the true generalization. It needs a name so I propose to call it the Principle of Minimum Assumption.

This claim is, itself, an hypothesis and has to be justified. I propose to do so in subsequent chapters by showing that a unified cosmology is achieved by the consistent, uncompromising and methodical application of the Principle of Minimum Assumption to theories about the past and future duration of matter. But before I do this, I shall have to discuss the nature and meaning of the principle and show that it is already applied by scientists more often than is always appreciated.

What is a Minimum Assumption?

One sometimes hears the remark 'that is a big assumption' just as one hears 'that is a big lie'. The implication of such colloquial habits of speech is that quantitative distinctions can be made between different assumptions and between different lies; that one could arrange a collection of assumptions or lies in a row, with the biggest at one end and the smallest at the other; that the magnitude of assumptions and lies could be expressed in units, like those of temperature, hardness, and other measurable quantities.

It may be so for all I know. But I am not concerned here with the grading of assumptions according to size. I am concerned instead with the search for a criterion by which an assumption that is defined as a minimum one can be clearly distinguished from assumptions that cannot be so defined. I do not think that the criterion is hard to find. I think it is whether the assumption is specific or not; so I shall define a minimum assumption as one that is completely unspecific. What this means can best be explained with the help of some examples. The first of them will be deliberately chosen to be extreme to the point of absurdity.

A young man who has had a predominantly humanistic education has been fascinated by a popular book on astronomy and has become enthusiastic about what he calls the beauties of science. He has read in his book that most stars do not have planets but that a very small proportion of them do and that these form solar systems like our own. The total number of stars is great, he learns, and so even the tiny fraction that have solar systems amounts to millions of stars.

He has also read in earlier books that planets are sublime bodies, constrained by their noble natures to move in orbits of geometrical perfection. In doing so, he has gathered, they produce a lovely harmony known as the music of the spheres. He reaches the not unnatural opinion that, where perfection is displayed in terms of geometry and music, it must also be displayed in terms of number: so he arrives at the conclusion that every one of those millions of distant solar systems must have a pleasing number of planets. He can well believe that this may be the mystical number seven, or the virile number nine, or occasionally perhaps the round number ten. But he feels sure that no solar system can be cursed with the unlucky number thirteen.

An astronomer friend reproves him for his unscientific outlook. To believe that the number thirteen is precluded is, he says, a big assumption and an unjustified one. He tells the young man that some solar systems do have thirteen planets. The young man remains puzzled. Why, he asks, is it a big and forbidden assumption to believe that no solar system has thirteen planets and a small and permitted assumption to believe that some solar systems do have thirteen planets?

The question is not a silly one. It must not be dismissed with a shrug and a smile. It is basic to scientific method and it behoves us to find a clear and direct answer to it.

The answer is not that solar systems with thirteen planets have been observed. The astronomer admits that his belief in the existence of such solar systems is an hypothesis. The resolving power of our best telescopes is not sufficient to reveal any solar system but our own. So far as observation goes, we have no proof that there is any other solar system at all.

Nor is the answer that there is a law of physics by which some solar systems are required to have thirteen planets. The reason for the astronomer's belief is, on the contrary, the very absence of any known law to require that solar systems shall have a specific number of planets. Here the minimum assumption is the unspecific one, i.e., that any number of planets can occur. This is therefore the assumption that, in conformity with the demands of scientific method, is made by our astronomer. He would not feel justified in making any other.

From the above little story, one may learn the operative word by which to recognize a minimum assumption. It is 'any'. It need not apply only to numbers, but can also apply to quantities, properties, relationships, configurations, to any feature that one likes to mention. When there are only two alternative possibilities, the grammatical substitute for 'any' is 'either' as, for instance, when there is a choice between the positive and the negative sign. So a minimum assumption can be recognized by the use in its formulation of the words 'any' or 'either'.

In practice, it is not difficult to distinguish between a minimum assumption, as just defined, and one that is not a minimum one. But the question remains whether, when one has recognized a minimum assumption, one is always justified in making it. In the example of the number of planets in a solar system, one cannot be sure whether the minimum assumption is the true generalization or not. It is impossible to prove by observation that solar systems may have any number of planets. But there are many occasions in the history of physics when the minimum assumption has proved to be the true generalization; and I can recall no occasion in the history of physics when it has not been so. Let this be demonstrated with the help of some real examples.

Use in Physics

Minimum assumptions have not received as much attention as one might have expected in the philosophy of science. The distinction between specific and unspecific assumptions is not a textbook subject and so but little has hitherto been done to clarify thought about it. But the history of science shows that specific assumptions have, in the past, been made time and time again; that they have been treated as generalizations about the nature of the physical world; and that they have eventually had to be replaced by unspecific ones. Whenever this has happened, new light has been shed on a wide range of subjects; the unifying and explanatory power of the unspecific assumption has been demonstrated.

Thus it was assumed at one time that a specific law, applicable only to planets, constrained these to move in elliptical orbits. But Newton replaced the hypothesis that the planetary orbits were the consequence of a specific law by the hypothesis that they were the consequence of the circumstances in which the planets found themselves. The assumption that certain bodies are required by their natures to move in specific ways was replaced by the assumption that any body may move in any way, its actual path being determined by the forces exerted on it. This proved to be the true generalization.

Similarly, it was assumed at one time that nature has specific likes and dislikes; for instance, that she abhors a vacuum. This could be translated into contemporary language as the specific assumption that a law of physics prevents the density of matter from falling below a specific value. But it is now known that the true generalization about the density of matter is unspecific. The laws of physics permit any density, ranging from the high concentration that occurs in the white dwarf stars to the extreme tenuousness of extragalactic space.

A further illustration may be taken from more recent history: the geometry of space. Until some fifty years ago, it was assumed that this was required by the laws of physics to be of the kind known as Euclidean. If the assumption was not recognised as a specific one, it was only because it was not recognised as an assumption at all; it was thought of as a self-evident truth. Although mathematicians had already shown that other geometries were logically possible, very few persons believed that they were also physically possible.

Einstein, however, was prepared to take the same view about the geometry of space that the astronomer in our little story took about the number of planets in solar systems. Knowing of no law to preclude non-Euclidean geometries, he made the minimum assumption namely, that they can occur. The unifying and explanatory power of this assumption has proved to be enormous.

Yet another, rather simple, example is provided by the periodic table of the elements. The basic feature by which the chemical properties of an element are determined is the number of unit charges on the nuclei of its atoms. For the elements to be found in nature, the maximum number of such charges is ninety-two, a stability limit. This limit provided a logical reason why no nucleus could be observed in nature with more than ninety-two unit charges, for a greater number would be inconsistent with the definition of a stable (or at least relatively stable) particle. But there was no logical reason why a smaller number should not occur. The minimum assumption is that a stable nucleus may carry any number of unit charges up to ninety-two.

There was a time, not so very long ago, when observation had nearly, but not quite, justified this assumption. Nearly all the numbers had been observed, but there were a few gaps. One of these was seventy-two. Another was zero.

Those who subscribed to the doctrine that a scientist must never, never believe what he cannot observe would have been precluded from believing that nuclei with seventy-two unit charges could occur. But few, if any, scientists carried their faith in empiricism thus far. Their faith in the Principle of Minimum Assumption was the stronger at least about the number seventy-two. So an element corresponding to this number was predicted purely on the basis of this principle. Its properties were also inferred and predicted. The event justified faith in the principle, for the time came when an element with seventy-two unit charges was observed and found to have the predicted properties. It received the name 'Hafnium'.

Had faith in the Principle of Minimum Assumption been a little stronger and the word 'any' been taken a little more literally, scientists would also have predicted nuclei, or at least particles, with zero charge. For they would have noticed that there was neither a law nor a logical reason why such particles should not occur. Having satisfied themselves about their logical possibility, they would next have worked out what properties would follow logically from the definition of a particle with zero charge. They would have decided that it could not attract any satellite electrons; that it would pass through any atom without being deflected by the charges on the nucleus of the atom; that therefore no vessel could contain it; that it would not enter into any chemical reaction. They would, in short, have predicted the neutron together with its properties. Subsequent observation of neutrons would then have served to justify the choice of a minimum assumption. Actually, the sequence of events was reversed. The neutron was observed first and its occurrence and properties were explained afterwards. But no other hypothesis was needed for the explanation than that any particle may carry any number of unit charges, including zero. The neutron provides one of many illustrations of the great explanatory power of the Principle of Minimum Assumption. Here we have one of the great missed opportunities in science.

Paul Dirac, 1902-84

Let me quote one further illustration of the power of this principle. It is the discovery of the positron by Dirac. The scientific work that led him to predict it is recondite and need not be described in detail. A few salient facts, deliberately presented in an over-simplified form, will suffice to point a moral.

One of Dirac's equations had two solutions as happens when one solves a quadratic equation. One of the terms in both these solutions represented energy; but it occurred with the positive sign in one of them and with the negative sign in the other. The positive sign caused no difficulty. It represents energy as we know it. But the negative sign could mean only that the solution applied to a system that contained negative energy. It was difficult to give meaning to negative energy; but this was not the only objection to the second solution. Just as, at one time, no one had observed particles that had seventy-two or zero charges, so no one had observed a state of negative energy.

Had Dirac been a slave to the doctrine that what is not observable has no place in reality, he would have had to assume that a specific law of physics prohibits a state of negative energy. But instead of assuming this, he allowed himself to be guided by the Principle of Minimum Assumption. He saw that it would involve a specific assumption to deny the possibility of negative energy and a minimum assumption to accept that possibility. His reasoning was strictly analogous to that of the astronomer who finds it more consistent with scientific method to postulate rather than to deny that some solar systems have thirteen planets.

Dirac's next step was to seek possible reasons why negative energy had never been observed. He rejected the facile answer that there was no such thing. Having satisfied himself that negative energy was logically possible, he was convinced that it was also physically possible. The answer that Dirac did find need not concern us; what is relevant is that in seeking it, he reached the conclusion that evidence for negative energy would be provided by a particle with the mass of an electron and positive unit charge. This conclusion did not involve any additional hypothesis or assumption; it was a logical inference from work based on no other hypothesis than that the minimum assumption is always the true generalization.

That the predicted particle had no more been observed than a state of negative energy did not shake Dirac's faith in the principle a faith that was justified when, some years later, the particle was actually observed. It is called the 'positron'.

Until this happened, there were doubts about Dirac's prediction and it is important to appreciate their nature. It was thought that the reasoning might have been faulty; that some essential fact might have been over-looked; that the mathematics might have contained an undetected error. The eventual discovery of the positron served to allay doubts of this kind. But they were all doubts as to whether a state of negative energy was really logically possible. Few doubted that, if it was, it was also physically possible and would occur occasionally.

Examples could be multiplied indefinitely where, in physics, the Principle of Minimum Assumption has led to new and valuable discoveries, when it has served to predict, to explain, to unify. When a minimum assumption has been used as a basis of the subsequent reasoning, a number of valuable conclusions have followed from it by a process of logical inference and without the need for any additional hypotheses. But I cannot recall a single instance in physics where a specific assumption has, after scrutiny, been maintained as a true generalization. For such examples, one has to turn to history, to biology, to the social sciences to the study, in other words, of systems that come under the influence of life.

The Concept of a Cosmic Statute Book

I have introduced the expression Cosmic Statute Book into Chapter 1 and have discussed this concept in detail elsewhere. [See Science Versus Materialism and Facts and Faith. Ed.] It must suffice to mention here only one or two points connected with it.

Laws are always concerned with generalizations. The laws that govern the formation of companies, finance acts, the rule of the road, apply to all companies, to all tax-payers, to all road users. This holds equally for what are called the laws of physics. But there the resemblance ends.

Laws imposed by authority can be, and often are, recorded in statute books, so I shall say that they are of the statute book kind. What characterises them is that they demand a specific choice between alternatives, all of which are logically possible. They require, for instance, that traffic shall keep to a certain side of the road and prohibit passage on the opposite side.

In the formulation of such laws, the words 'any' and 'either' must not occur. If they did, the law would be meaningless. Further, such laws do not include what is the logical consequence of other accepted principles. If they did, they would be redundant. Thus no country would enact a law to say that people may drive either on the right or on the left. Such a law would enforce nothing and prohibit nothing, and no country would enact a law to require that two twos shall be four. It would be so whether the law were enacted or not.

The question now arises whether there are laws of physics that prohibit anything that is logically possible. We believe that it is logically possible for a solar system to have thirteen planets, and we know that it is logically possible for a particle to have seventy-two and zero unit charges, for space to have a variety of different geometries, for energy to occur in the negative state. If, nevertheless, there was a law of physics that prevented any of these possibilities from being realized, it would be of the statute book kind. Such a law would have no place in a unified physics. It could not be inferred from any known principle. It could not be explained. It could be discovered only by observation.

According to the Principle of Minimum Assumption there are no such laws in physics, and I have shown above with the help of a few examples that physicists often act on the belief that it is so. Their belief can be expressed by rewording the Principle of Minimum Assumption as follows: In physics, a generalization that is logically possible is also physically possible. It can therefore be represented by an actual example; and it is so represented with a frequency that is determined by statistical considerations only.

Yet another formulation of the same principle is as follows: For the physicist, there is no such thing as a Cosmic Statute Book.

This negative formulation brings the principle into the category of what Sir Edmund Whittaker called 'postulates of impotence'. It tells us what one cannot do. It says that one cannot base a true generalization in physics on a specific assumption. Therewith it has a faint resemblance to a formulation of the principle of conservation of energy that says: 'One cannot make a perpetual motion machine.' This negative wording sometimes has its advantages. In mechanics, one tests conclusions for their conformity to the principle of conservation of energy and one may reject them by saying: 'That is equivalent to inventing perpetual motion.' If we got into the habit of testing our conclusions for their conformity to the Principle of Minimum Assumption, we should similarly find ourselves saying sometimes: 'That is equivalent to an entry in a Cosmic Statute Book.'

However, the Principle of Minimum Assumption is far from having reached universal acceptance. People have not become very articulate about it. It is by no means applied with the uncompromising consistency that it needs. Those who apply it do so instinctively rather than deliberately, and many of them would oppose my plea for elevating the rule of economy of hypotheses to the status of a great principle of physics. Nevertheless, this is exactly what I shall do in the ensuing pages. On the more superficial view, this book might be regarded as an ingenious explanation of certain cosmic phenomena. But what I have to say was not found as a result of a search for explanations. It was found as a result of exploring some implications of the Principle of Minimum Assumption, and I am hoping that a more profound view will be taken of what I have to say than that it constitutes some ad hoc explanations. I should like it to be regarded as an example of the way in which the Principle of Minimum Assumption can be justified by its unifying and explanatory power.