2.1: 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, which may
be energy of motion, of position, of chemical structure, 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 and articles
in scientific journals, 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 con-
servation 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; state-
ments 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
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 applied by scientists
already and more often than is always appreciated.
2.2: 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. For 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.
2.3: Use of the Principle of Minimum Assumption 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 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 recognized as a specific one, it was only because
it was not recognized as an assumption at all; it was thought of as a self-evident truth. Nevertheless, mathematicians had already shown that
other geometries were logically possible. But 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' 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
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 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, but 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
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 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 could be multiplied indefinitely. When a minimum assump-
tion 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.
2.4: The Concept of a Cosmic Statute Book
I have introduced the expression Cosmic Statute Book into Chapter I
and have discussed this concept in detail elsewhere. 1,2 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 charac-
terizes 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
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. Nor would any country 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 in-
ferred 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 is so
represented with afrequency that is determined by statistical considerations
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 ofwha)
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 has its advantages
sometimes. 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 do 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 explana-
tion 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.
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1R. 0. Kapp, Science versus Materialism, Methuen, 1940, Chapters XXII and XXV.
2R. 0. Kapp, Facts and Faith, Oxford University Press, 1955.