A.1: Irreducible Concepts
This book represents but one step in the direction towards the unification
of physical science. Many previous steps have been taken by scientists and
many more will have to betaken before unification is complete. Perhaps
that never will be, but it is desirable at least to make the effort. Past success
itself justifies it, for already the discovery of ever wider generalizations in
physics and our increasing knowledge of the relation between apparently
isolated phenomena have greatly diminished the number of irreducibles
with which physicists have to work. Those that remain can be placed in
three classes, namely:
(a) Irreducible physical laws and principles,
(b) Irreducible physical constants,
(c) Irreducible constituents of the material universe.
The members of each class are today still too numerous to make complete
unification look very near, but not so numerous as to make it look hope-
less. In this first appendix I want to do no more than define the problem
as I see it. Detailed discussion would not be in place here. I shall
therefore not attempt to elaborate such suggestions as I am putting
forward but merely express the hope that they may encourage further
Some of those generalizations that are called laws or principles in
physics are tautologies: their truth is implied by their verbal formulation.
These form a big field of study and are of great importance in scientific
method, but again this is not the proper occasion for their detailed
discussion. It must suffice to note three facts about them.
(Here I am using the word 'tautology' to designate a statement about a
concept that is implicit in the definition of the concept. Such a statement
repeats in different words (or mathematical symbols) what is said when the
concept is defined. As technical terms in philosophy and logic have never
been standardized and are used with different meaning by different
people there must be some who would prefer a different word to 'tautology'.
But that need not affect what I have to say here.)
Firstly, far from being superfluous, tautologies serve a most useful
purpose in physics and are more abundant than is often supposed.
Mathematical statements belong to this category. The multiplication table,
for instance, is implicit in the definitions of number and of the process of
multiplication. In a similar way it can be seen that some of those scientific
laws that can be expressed in algebraic form are tautologies. They provide
a means of expressing in letter symbols what is implicit in the concepts
that are being handled.
Secondly, tautologies, like everything true by definition, are not only
sometimes true or only approximately true. When they are formulated
correctly they are always true and necessarily true. This characteristic helps
with the detection of tautologies. One need not look for them among those
looser statements that may be subject to exceptions or no more than first
approximations. Nor need one look for them among experimentally
Thirdly, and consequently, a tautology does not have to be verified
by experiment or observation. It can be proved with the help of pencil and
paper guided by logic. This holds even though experimental confirmation
of a tautology may also be possible. The multiplication table can be verified
by experiments with an abacus or by counting oranges, but its validity does
not depend on such procedure. This characteristic provides further help
with the detection of tautologies. Sometimes a law has been discovered by
experiment and its true nature is not immediately obvious. But it may
afterwards be satisfactorily 'explained' with the help of pencil and paper.
The explanation then proves that the law could have been discovered
without the experiment, that it is implicit in the definition of the
concepts that are embodied in its formulation, that, in short, it is a
One part of the work of unifying physical science is the detection of
A.3: Laws that are Implicit in More Comprehensive Ones
Another part of unification is concerned with those laws that are not
recognized as tautologies. Their number diminishes whenever it is
found that a law is implicit in more comprehensive ones. It is with
this part of the work of unification that the present book has been
largely concerned. If its reasoning is sound, several laws have been
taken out of the category of irreducible laws and placed in the category
of the deducible ones. Among them are the law that requires nebulae
to have spiral arms and the one that requires every large inert mass
to be the source of a gravitational field. I venture to suggest that the
method adopted here could, with advantage to the progress of physical
science, be recognized more generally and followed more systematically
than it has been in the past.
A.4: The Reduction of Physical Constants
The irreducible physical constants with which we are still left today
include the velocity of light, the quantum of action, and the electric
charges that occur respectively on the proton and the electron. What has
been said here in previous chapters suggests that the rate of origin of
matter and its half-life should be added to the list. But when all necessary
additions have been made, the list is today not a very long one.
The unification of physics requires that a relation be found between
the values in the list so that one would be able to say that these values were
inter-dependent. If it were so, one could infer one of the values from others.
Eddington attempted the task, though apparently without success. There
were some at the time who criticized him, oddly enough, not because they
had proof that he failed but because they thought he ought never to have
A.5: Establishing a Relationship between Different Constituents of the
The irreducible constituents of the material universe include an
inconveniently large number of so-called elementary particles, the proton,
the electron, the antiproton, the positron, many mesons, the neutron, the
neutrino. They also include radiation, energy, space and time. The list
would have been longer half-a-century ago. It would, for instance, have
included ninety-two distinct chemical atoms. Still earlier it would have
included all those substances that have been since reduced to their chemical
Previous reductions have sometimes occurred when it was found that
what had been thought of as elementary was really composite. It was so
with the reduction of chemical substances to chemical elements and of the
elements, in turn, to protons, neutrons and electrons.
Previous reductions also sometimes occurred when it was found that
one apparently irreducible constituent could be converted into another.
Both then appeared as different manifestations of the same single con-
stituent. Ice and water are a simple illustration. These two commodities
are reduced to one by substituting for both the single symbol H2O.
The process can be called unification by substitution. It was particularly
fruitful when the concept energy was introduced. This was found to be
a commodity that may occur in many different forms. One of them is called
work, another heat; in one form it is defined by the symbol ½mv2, in another
it is stored in the electrostatic field and takes the form ½ CV2. Einstein discovered that it may also occur in the form of inert mass.
The unifying power of this discovery was very great. For it showed
that mass, which we think of as 'quite real', and energy, which we tend to
think of as conceptual only, are both manifestations of one basic reality.
Yet another step towards the unification of physics has often been
the complete elimination from this science of some constituent of reality
that was previously included in it. Nature's horror of a vacuum is an
example. Feelings are today known to have no place in physics but to
belong to other disciplines.
Sensations are now in the same category. The sensation of a given
colour, the smell and taste of things, aesthetic and ethical values, every-
thing in short that depends on the experience of a particular observer, has
been excluded from a unified physics.
Constituents that have been postulated for the convenience of the
imagination have gone the same way. The reason for their elimination has
been that they depend on the observer in his capacity as thinker or image-maker. For this reason Newton's space has been eliminated from the list of
constituents. A space of which it can be said that no place differs in any
way from any other place is purely conceptual and plays no part in physics.
It has been replaced by Einstein's space, which is, as I have suggested
elsewhere in these pages, no more than a synonym for environment.
In the same way absolute time has had to go. The notion of undifferentiated time as a background for events has proved as meaningless in
physics as the notion of featureless space as a container of things.
Einstein's time, I am inclined to think, is no more than a synonym for
distance divided by velocity. Thereby the concept, time, has been stripped
of its subjective implications and quite a useful step taken towards unification.
A.6: Brief Summary
Above, five methods have been discussed in the briefest of words by
which the number of irreducibles that occur in physics may be caused to
decrease. For convenience let me condense even more and enumerate
them in the form of a list. The five methods are:
Reduction by finding that something is a tautology;
Reduction by finding that something is implicit in some other more
Reduction by finding that a variety of distinct things are all com-
posed of a small number of elementary components;
Reduction by finding that two or more distinct things can convert
into each other;
Reduction by finding that something does not belong to the
I should not like to venture a guess as to which of these methods is
likely to prove the most rewarding. They will probably all have been
used lavishly before complete unification has been achieved.
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