The physicist, in his study of natural phenomena, has two methods of
making progress: (1) the method of experiment and observation, and (2)
the method of mathematical reasoning. The former is just the collection
of selected data; the latter enables one to infer results about
experiments that have not been performed. There is no logical reason
why the second method should be possible at all, but one has found in
practice that it does work and meets with reasonable success. This must
be ascribed to some mathematical quality in Nature, a quality which the
casual observer of Nature would not suspect, but which nevertheless
plays an important role in Nature’s scheme.

One might describe the mathematical quality in Nature by saying that the
universe is so constituted that mathematics is a useful took in its
description. However, recent advances in physical science show that
this statement of the case is too trivial. The connection between
mathematics and the description of the universe goes far deeper than
this, and one can get an appreciation of it only from a thorough
examination of the various facts that make it up. The main aim of my
talk to you will be to give you such an appreciation. I propose to deal
with how the physicist’s views on this subject have been gradually
modified by the succession of recent developments in physics, and then I
would like to make a little speculation about the future.

Let us take as our starting-point that scheme of physical science which
was generally accepted in the last century – the mechanistic scheme.
This considers the whole universe to be a dynamical system (of course an
extremely complicated dynamical system), subject to laws of motion which
are essentially of the Newtonian type. The role of mathematics in this
scheme is to represent the laws of motion by equations, and to obtain
solutions of the equations referring to observed conditions.

The dominating idea in this application of mathematics to physics is
that the equations representing the laws of motion should be of a simple
form. The whole success of the scheme is due to the fact that equations
of simple form do seem to work. The physicist is thus provided with a
principle of simplicity, which he can use as an instrument of research.
If he obtains, from some rough experiments, data which fit in roughly
with certain simple equations, he infers that if he performed the
experiments more accurately he would obtain data fitting in more
accurately with the equations. The method is much restricted, however,
since the principle of simplicity applies only to fundamental laws of
motion, not to natural phenomena in general. For example, rough
experiments about the relation between the pressure and volume of a gas
at a fixed temperature give results fitting in with a law of inverse
proportionality, but it would be wrong to infer that more accurate
experiments would confirm this law with greater accuracy, as one is here
dealing with a phenomenon which is not connected in any very direct way
with the fundamental laws of motion.

The discovery of the theory of relativity made it necessary to modify
the principle of simplicity. Presumably one of the fundamental laws of
motion is the law of gravitation which, according to Newton, is
represented by a very simple equation, but, according to Einstein,
needs the development of an elaborate technique before its equation can
even be written down. It is true that, from the standpoint of higher
mathematics, one can give reasons in favour of the view that Einstein’s
law of gravitation is actually simpler than Newton’s, but this involves
assigning a rather subtle meaning to simplicity, which largely spoils
the practical value of the principle of simplicity as an instrument of
research into the foundations of physics.

What makes the theory of relativity so acceptable to physicists in spite
of its going against the principle of simplicity is its great
mathematical beauty. This is a quality which cannot be defined, any
more than beauty in art can be defined, but which people who study
mathematics usually have no difficulty in appreciating. The theory of
relativity introduced mathematical beauty to an unprecedented extent
into the description of Nature. The restricted theory changed our ideas
of space and time in a way that may be summarised by stating that the
group of transformations to which the space-time continuum is subject
must be changed from the Galilean group to the Lorentz group. The
latter group is a much more beautiful thing than the former – in fact,
the former would be called mathematically a degenerate special case of
the latter. The general theory of relativity involved another step of a
rather similar character, although the increase in beauty this time is
usually considered to be not quite so great as with the restricted
theory, which results in the general theory being not quite so firmly
believed in as the restricted theory.

We now see that we have to change the principle of simplicity into a
principle of mathematical beauty. The research worker, in his efforts
to express the fundamental laws of Nature in mathematical form, should
strive mainly for mathematical beauty. He should still take simplicity
into consideration in a subordinate way to beauty. (For example
Einstein, in choosing a law of gravitation, took the simplest one
compatible with his space-time continuum, and was successful.). It
often happens that the requirements of simplicity and of beauty are the
same, but where they clash the latter must take precedence.

Let us pass on to the second revolution in physical thought of the
present century – the quantum theory. This is a theory of atomic
phenomena based on a mechanics of an essentially different type from
Newton’s. The difference may be expressed concisely, but in a rather
abstract way, by saying that dynamical variables in quantum mechanics
are subject to an algebra in which the commutative axiom of
multiplication does not hold. Apart from this, there is an extremely
close formal analogy between quantum mechanics and the old mechanics.
In fact, it is remarkable how adaptable the old mechanics is to the
generalization of non-commutative algebra. All the elegant features of
the old mechanics can be carried over to the new mechanics, where they
reappear with an enhanced beauty.

Quantum mechanics requires the introduction into physical theory of a
vast new domain of pure mathematics – the whole domain connected with
non-commutative multiplication. This, coming on top of the introduction
of new geometries by the theory of relativity, indicates a trend which
we may expect to continue. We may expect that in the future further big
domains of pure mathematics will have to be brought in to deal with the
advances in fundamental physics.

Pure mathematics and physics are becoming ever more closely connected,
though their methods remain different. One may describe the situation
by saying that the mathematician plays a game in which he himself
invents the rules while the physicist plays a game in which the rules
are provided by Nature, but as time goes on it becomes increasingly
evident that the rules which the mathematician finds interesting are the
same as those which Nature has chosen. It is difficult to predict what
the result of all this will be. Possibly, the two subjects will
ultimately unify, every branch of pure mathematics then having its
physical application, its importance in physics being proportional to
its interest in mathematics. At present we are, of course, very far
from this stage, even with regard to some of the most elementary
questions. For example, only four-dimensional space is of importance in
physics, while spaces with other numbers of dimensions are of about
equal interest in mathematics.

It may well be, however, that this discrepancy is due to the
incompleteness of present-day knowledge, and that future developments
will show four-dimensional space to be of far greater mathematical
interest than all the others.

The trend of mathematics and physics towards unification provides the
physicist with a powerful new method of research into the foundations of
his subject, a method which has not yet been applied successfully, but
which I feel confident will prove its value in the future. The method
is to begin by choosing that branch of mathematics which one thinks will
form the basis of the new theory. One should be influenced very much in
this choice by considerations of mathematical beauty. It would probably
be a good thing also to give a preference to those branches of
mathematics that have an interesting group of transformations underlying
them, since transformations play an important role in modern physical
theory, both relativity and quantum theory seeming to show that
transformations are of more fundamental importance than equations.
Having decided on the branch of mathematics, one should proceed to
develop it along suitable lines, at the same time looking for that way
in which it appears to lend itself naturally to physical interpretation.

This method was used by Jordan in an attempt to get an improved quantum
theory on the basis of an algebra with non-associative multiplication.
The attempt was not successful, as one would rather expect, if one
considers that non-associative algebra is not a specially beautiful
branch of mathematics, and is not connected with an interesting
transformation theory. I would suggest, as a more hopeful-looking idea
for getting an improved quantum theory, that one take as basis the
theory of functions of a complex variable. This branch of mathematics
is of exceptional beauty, and further, the group of transformations in
the complex plane, is the same as the Lorentz group governing the
space-time of restricted relativity. One is thus led to suspect the
existence of some deep-lying connection between the theory of functions
of a complex variable and the space-time of restricted relativity, the
working out of which will be a difficult task for the future.

Let us now discuss the extent of the mathematical quality in Nature.
According to the mechanistic scheme of physics or to its relativistic
modification, one needs for the complete description of the universe not
merely a complete system of equations of motion, but also a complete set
of initial conditions, and it is only to the former of these that
mathematical theories apply. The latter are considered to be not
amenable to theoretical treatment and to be determinable only from
observation.

The enormous complexity of the universe is ascribed to an enormous
complexity in the initial conditions, which removes them beyond the
range of mathematical discussion.

I find this position very unsatisfactory philosophically, as it goes
against all ideas of the unity of Nature. Anyhow, if it is only to a
part of the description of the universe that mathematical theory
applies, this part ought certainly to be sharply distinguished from the
remainder. But in fact there does not seem to be any natural place in
which to draw the line. Are such things as the properties of the
elementary particles of physics, their masses and the numerical
coefficients occurring in their laws of force, subject to mathematical
theory? According to the narrow mechanistic view, they should be
counted as initial conditions and outside mathematical theory. However,
since the elementary particles all belong to one or other of a number of
definite types, the members of one type being all exactly similar, they
must be governed by mathematical law to some extent, and most physicists
now consider it to be quite a large extent. For example, Eddington has
been building up a theory to account for the masses. But even if one
supposed all the properties of the elementary particles to be
determinable by theory, one would still not know where to draw the line,
as one would be faced by the next question – Are the relative abundances
of the various chemical elements determinable by theory? One would pass
gradually from atomic to astronomic questions.

This unsatisfactory situation gets changed for the worse by the new
quantum mechanics. In spite of the great analogy between quantum
mechanics and the older mechanics with regard to their mathematical
formalisms, they differ drastically with regard to the nature of their
physical consequences. According to the older mechanics, the result of
any observation is determinate and can be calculated theoretically from
given initial conditions; but with quantum mechanics there is usually an
indeterminacy in the result of an observation, connected with the
possibility of occurrence of a quantum jump, and the most that can be
calculated theoretically is the probability of any particular result
being obtained. The question, which particular result will be obtained
in some particular case, lies outside the theory. This must not be
attributed to an incompleteness of the theory, but is essential for the
application of a formalism of the kind used by quantum mechanics.

Thus according to quantum mechanics we need, for a complete description
of the universe, not only the laws of motion and the initial conditions,
but also information about which quantum jump occurs in each case when a
quantum jump does occur. The latter information must be included,
together with the initial conditions, in that part of the description of
the universe outside mathematical theory.

The increase thus arising in the non-mathematical part of the
description of the universe provides a philosophical objection to
quantum mechanics, and is, I believe, the underlying reason why some
physicists still find it difficult to accept this mechanics. Quantum
mechanics should not be abandoned, however, firstly, because of its very
widespread and detailed agreement with experiment, and secondly, because
the indeterminacy it introduces into the results of observations is of a
kind which is philosophically satisfying, being readily ascribable to an
inescapable crudeness in the means of observation available for
small-scale experiments. The objection does show, all the same, that
the foundations of physics are still far from their final form.

We come now to the third great development of physical science of the
present century – the new cosmology. This will probably turn out to be
philosophically even more revolutionary than relativity or the quantum
theory, although at present one can hardly realize its full
implications. The starting-point is the observed red-shift in the
spectra of distance heavenly bodies, indicating that they are receding
from us with velocities proportional to their distances.* The
velocities of the more distant ones are so enormous that it is evident
we have here a fact of the utmost importance, not a temporary or local
condition, but something fundamental for our picture of the universe.

With this kind of cosmological picture one is led to suppose that there
was a beginning of time, and that it is meaningless to inquire into what
happened before then. One can get a rough idea of the geometrical
relationships this involves by imagining the present to be the surface
of a sphere, going into the past to be going in towards the centre of
the sphere, and going into the future to be going outwards. There is
then no limit to how far one may go into the future, but there is a
limit to how far one can go into the past, corresponding to when one has
reached the centre of the sphere. The beginning of time provides a
natural origin from which to measure the time of any event. The result
is usually called the epoch of that event. Thus the present epoch is 2
x 109 years.

Let us now return to dynamical questions. With the new cosmology the
universe must have been started off in some very simple way. What,
then, becomes of the initial conditions required by dynamical theory?
Plainly there cannot be any, or they must be trivial. We are left in a
situation which would be untenable with the old mechanics. If the
universe were simply the motion which follows from a given scheme of
equations of motion with trivial initial conditions, it could not
contain the complexity we observe. Quantum mechanics provides an escape
from the difficulty. It enables us to ascribe the complexity to the
quantum jumps, lying outside the scheme of equations of motion. The
quantum jumps now form the uncalculable part of natural phenomena, to
replace the initial conditions of the old mechanistic view.

One further point in connection with the new cosmology is worthy of
note. At the beginning of time the laws of Nature were probably very
different from what they are now. Thus we should consider the laws of
Nature as continually changing with the epoch, instead of as holding
uniformly throughout space-time. This idea was first put forward by
Milne, who worked it out on the assumptions that the universe at a given
epoch is roughly everywhere uniform and spherically symmetrical. I find
these assumptions not very satisfying, because the local departures from
uniformity are so great and are of such essential importance for our
world of life that it seems unlikely there should be a principle of
uniformity overlying them. Further, as we already have the laws of
Nature depending on the epoch, we should expect them also to depend on
position in space, in order to preserve the beautiful idea of the theory
of relativity there is fundamental similarity between space and time.
This goes more drastically against Milne’s assumptions than a mere lack
of uniformity in the distribution of matter.

We have followed through the main course of the development of the
relation between mathematics and physics up to the present time, and
have reached a stage where it becomes interesting to indulge in
speculations about the future. There has always been an unsatisfactory
feature in the relation, namely, the limitation in the extent to which
mathematical theory applies to a description of the physical universe.
The part to which it does not apply has suffered an increase with the
arrival of quantum mechanics and a decrease with the arrival of the new
cosmology, but has always remained.

This feature is so unsatisfactory that I think it safe to predict it
will disappear in the future, in spite of the startling changes in our
ordinary ideas to which we should then be led. It would mean the
existence of a scheme in which the whole of the description of the
universe has its mathematical counterpart, and we must suppose that a
person with a complete knowledge of mathematics could deduce, not only
astronomical data, but also all the historical events that take place in
the world, even the most trivial ones. Of course, it must be beyond
human power actually to make these deductions, since life as we know it
would be impossible if one could calculate future events, but the
methods of making them would have to be well defined. The scheme could
not be subject to the principle of simplicity since it would have to be
extremely complicated, but it may well be subject to the principle of
mathematical beauty.

I would like to put forward a suggestion as to how such a scheme might
be realized. If we express the present epoch, 2 x 109 years, in terms
of a unit of time defined by the atomic constants, we get a number of
the order 1039, which characterizes the present in an absolute sense.
Might it not be that all present events correspond to properties of this
large number, and, more generally, that the whole history of the
universe corresponds to properties of the whole sequence of natural
numbers? At first sight it would seem that the universe is far too
complex for such a correspondence to be possible. But I think this
objection cannot be maintained, since a number of the order 1039 is
excessively complicated, just because it is so enormous. We have a
brief way of writing it down, but this should not blind us to the fact
that it must have excessivly complicated properties.

There is thus a possibility that the ancient dream of philosophers to
connect all Nature with the properties of whole numbers will some day be
realized. To do so physics will have to develop a long way to establish
the details of how the correspondence is to be made. One hint for this
development seems pretty obvious, namely, the study of whole numbers in
modern mathematics is inextricably bound up with the theory of functions
of a complex variable, which theory we have already seen has a good
chance of forming the basis of the physics of the future. The working
out of this idea would lead to a connection between atomic theory and
cosmology.

* The recession velocities are not strictly proved, since one may
postulate some other cause for the spectral red-shift. However, the new
cause would presumably be equally drastic in its effect on cosmological
theory and would still need the introduction of a parameter of the order
2 x 109 years for its mathematical discussion, so it would probably not
disturb the essential ideas of the argument in the text.

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