by
Albert Einstein
The Herbert Spencer lecture, delivered at Oxford, June
10, 1933. Published in Mein Weltbild, Amsterdam: Querido Verlag, 1934.

If you want to find out anything from the theoretical physicists
about the methods they use, I advise you to stick closely to one
principle: don't listen to their words, fix your attention on
their deeds. To him who is a discoverer in this field, the
products of his imagination appear so necessary and natural that
he regards them, and would like to have them regarded by others,
not as creations of thought but as given realities.

These words sound like an invitation to you to walk out of this
lecture. You will say to yourselves, the fellow's a working physicist himself and ought therefore to leave all questions of
the structure of theoretical science to the epistemologists.

Against such criticism I can defend myself from the personal point
of view by assuring you that it is not at my own instance but at
the kind invitation of others that I have mounted this rostrum,
which serves to commemorate a man who fought hard. all his life
for the unity of knowledge. Objectively, however, my enterprise
can be justified on the ground that it may, after
all, be of interest to know how one who has spent a lifetime in striving with all his might to clear up and rectify its
fundamentals looks upon his own branch of science. The way in
which he regards its past and present may depend too much on what
he hopes for the future and aims at in the present; but that is
the inevitable fate of anybody who has occupied himself
intensively with a world of ideas. The same thing happens to him
as to the historian, who in the same way, even though perhaps
unconsciously, groups actual events round ideals which he has
formed for himself on the subject of human society.

Let us now cast an eye over the development of the theoretical
system, paying special attention to the relations between the
content of the theory and the totality of empirical fact. We are
concerned with the eternal antithesis between the two inseparable
components of our knowledge, the empirical and the rational, in
our department.

We reverence ancient Greece as the cradle of western science Here for the first time the world witnessed the miracle of a logi
cal system which proceeded from step to step with such precision that every single one of its propositions was absolutely
indubitableI refer to Euclid's geometry. This admirable triumph
of reasoning gave the human intellect the necessary confidence in
itself for its subsequent achievements. If Euclid failed to kindle
your youthful enthusiasm, then you were not born to be a
scientific thinker.

But before mankind could be ripe for a science which takes in the
whole of reality, a second fundamental truth was needed, which
only became common property among philosophers with the advent of
Kepler and Galileo. Pure logical thinking cannot yield us any
knowledge of the empirical world; all knowledge if reality starts
from experience and ends in it. Propositions arrived at by purely
logical means are completely empty as regards reality. Because
Galileo saw this, and particularly because he drummed it into the
scientific world, he is the father of modern physicsindeed, of
modern science altogether.

If, then, experience is the alpha and the omega of all our
knowledge of reality, what is the function of pure reason in
science?

A complete system of theoretical physics is made up of concepts,
fundamental laws which are supposed to be valid for those concepts and conclusions to be reached by logical deduction. It is these
conclusions which must correspond with our separate experiences;
in any theoretical treatise their logical deduction occupies
almost the whole book.

This is exactly what happens in Euclid's geometry, except that
there the fundamental laws are called axioms and there is no
question of the conclusions having to correspond to any sort of
experience. If, however, one regards Euclidean geometry as the
science of the possible mutUal relations of practically rigid
bodies in space, that is to say, treats it as a physical science,
without abstracting from its original empirical content, the
logical homogeneity of geometry and theoretical physics becomes
complete.

We have thus assigned to pure reason and experience their places
in a theoretical system of physics. The structure of the system is
the work of reason; the empirical contents and"their mUtual
relations must find their representation in the conclusions of the
theory. In the possibility of such a representation lie the sole
value and justification of the whole system, and especially of the
concepts and fundamental principles which underlie it. Apart from
that, these latter are free inventions of the human intellect,
which cannot be justified either by the natUre of that intellect
or in any other fashion a priori.

These fundamental concepts and postulates, which cannot be further
reduced logically, form the essential part of a theory, which
reason cannot touch. It is the grand object of all theory to make
these irreducible elements as simple and as few in number as
possible, without having to renounce the adequate representation
of any empirical content whatever.

The view I have just ou.tlined of the purely fictitious character
of the fundamentals of scientific theory was by no means the
prevailing one in the eighteenth and nineteenth centuries. BUt it
is steadily gaining ground from the fact that the distance in
thought between the fundamental concepts and laws on one side and,
on the other, the conclusions which have to be brought into
relation with our experience grows larger and
larger, the simpler the logical structure becomesthat is to say, the smaller the number of logically independent conceptual elements which are found necessary to support the structure.

Newton, the first creator of a comprehensive, workable system of
theoretical physics, still believed that the basic concepts and
laws of his system could be derived from experience. This is no
doubt the meaning of his saying, hypotheses non jingo.

Actually the concepts of time and space appeared at that time to
present no difficulties. The concepts of mass, inertia, and force,
and the laws connecting them, seemed to be drawn directly from
experience. Once this basis is accepted, the expression for the
force of gravitation appears derivable from experience, and it
was reasonable to expect the same in regard to other forces.

We can indeed see from Newton's formulation of it that the concept
of absolute space, which comprised that of absolute rest, made him
feel uncomfortable; he realized that there seemed to be nothing in
experience corresponding to this last concept. He was also not
quite comfortable about the introduction of forces operating at a
distance. But the tremendous practical success of his doctrines
may well have prevented him and the physicists of the eighteenth
and nineteenth centuries from recognizing the fictitious character
of the foundations of his system.

The natural philosophers of those days were, on the contrary,
most of them possessed with the idea that the fundamental concepts
and postulates of physics were not in the logical sense free
inventions of the human mind but could be deduced from experience
by "abstraction"that is to say, by logical means.

A clear recognition of the erroneousness of this notion really only came with the general theory of relativity, which showed that
one could take account of a wider range of empirical facts, and
that, too, in a more. satisfactory and complete manner, on a
foundation quite different from the Newtonian. But quite apart
from the question of the superiority of one or the other, the
fictitious character of fundamental principles is perfectly
evident from the fact that we can point to two essentially
different principles, both of which correspond with experience to a
large extent; this proves at the same time that every attempt at a
logical deduction of the basic concepts and postulates of
mechanics from elementary experiences is doomed to failure.

If, then, it is true that the axiomatic basis of theoretical
physics cannot be extracted from experience but must be freely
invented, can we ever hope to find the right way? Nay, more, has
this right way any existence outside our illusions? Can we hope to
be guided safely by experience at all when there exist theories
(such as classical mechanics) which to a large extent do justice
to experience, without getting to the root of the matter? I answer
without hesitation that there is, in my opinion, a right way, and
that we are capable of finding it. Our experience hitherto
justifies us in believing that nature is the realization of the
simplest conceivable mathematical ideas. I am convinced that we
can discover by means of purely mathematical constructions the
concepts and the laws connecting the.m with each other, which
furnish the key to the understanding of natural phenomena.
Experience may suggest the appropriate mathematical concepts, but
they most certainly cannot be deduced from it. Experience remains,
of course, the sole criterion of the physical utility of a
mathematical construction. But the creative principle resides in
mathematics. In a certain sense, therefore, I hold it true that
pure thought can grasp reality, as the ancients dreamed.

In order to justify this confidence, I am compelled to make use of
a mathematical concept. The physical world is represented as a
fourdimensional continuum. If I assume a Riemannian metric in it
and ask what are the simplest laws which such a metric can
satisfy, I arrive at the relativistic theory of gravitation in
empty space. If in that space I assume a vectorfield or an anti
symmetrical tensorfield which can be derived from it, and ask
what are the simplest laws which such a field can satisfy, I
arrive at Maxwell's equations for empty space.

At this point we still lack a theory for those parts of space in
which electrical charge density does not disappear. De Broglie
conjectured the existence of a wave field, which served to explain
certain quantum properties of matter. Dirac found in the spinors
fieldmagnitudes of a new sort, whose simplest
equations enable one to a large extent to deduce the properties of
the electron. Subsequently I discovered, in conjunction with my
colleague, Dr. Walter Mayer, that these spinors form a special
case of a new sort of field, mathematically connected with the
fourdimensional system, which we called "semivectors." The
simplest equations which such semivectors can satisfy furnish a
key to the understanding of the existence of two sorts of
elementary particles, of different ponderable mass and equal but
opposite electrical charge. These semivectors are, after ordinary
vectors, the simplest mathematical fields that are possible in a
metrical continuum of four dimensions, and it looks as if they
described, in a natural way, certain essential properties of
electrical particles.

The important point for us to observe is that all these
constructions and the laws connecting them can be arrived at by the
principle of looking for the mathematically simplest concepts and
the link between them. In the limited number of the mathematically
existent simple field types, and the simple equations possible
between them, lies the theorist's hope of grasping
the real in all its depth.

Meanwhile the great stumblingblock for a fieldtheory of this
kind lies in the conception of the atomic structure of matter and
energy. For the theory is fundamentally nonatomic in so far as it
operates exclusively with continuous functions of space,
in contrast to classical mechanics, whose most important element,
the material point, in itself does justice to the atomic structure
of matter.

The modern quantum theory in the form associated with the names of
de Broglie, Schrodinger, and Dirac, which operates with continuous
functions, has overcome these difficulties by a bold piece; of
interpretation which was first given a clear form by Max Born.
According to this, the spatial functions which appear in the
equations make no claim to be a mathematical model of the atomic
structure. Those functions are only supposed to determine the
mathematical probabilities to find such structures, if
measurements are taken, at a particular spot or in a certain state
of motion.

This notion is logically unobjectionable and has important
successes to its credit. Unfortunately, however, it compels one to
use a continuum the number of whose dimensions is not that
ascribed to space by physics hitherto (four) but rises
indefinitely with the number of the particles constituting the
system under consideration. I cannot but confess that I attach
only a transitory importance to this interpretation. I still
believe in the possibility of a model of realitythat is to say,
of a theory which represents things themselves and not merely the
probability of their occurrence.

On the other hand, it seems to me certain that we must give up the
idea of a complete localization of the particles in a theoretical
model. This seems to me to be the permanent upshot of Heisenberg's
principle of uncertainty. But an atomic theory in the true sense
of the word (not merely on the basis of an interpretation) without
localization of particles in a mathematical model is perfectly
thinkable. For instance, to account for the atomic character of
electricity, the field equations need only lead to the following
conclusions: A region of threedimensional space at whose boundary
electrical density vanishes every,vhere always contains a total
electrical charge whose size is represented by a whole number. In
a continuumtheory atomic characteristics would be satisfactorily
expressed by integral laws without localization of the entities
which constitute the atomic structure.

Not until the atomic structure has been successfully represented in such a manner would I consider the quantumriddle solved.
