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.