
"Geometry
and
Experience"
Albert Einstein
(1921)
An expanded form of
an Address to the Prussian Academy of Sciences in Berlin on January
27th, 1921
ONE
reason why mathematics enjoys special esteem, above all other
sciences, is that its laws are absolutely certain and indisputable,
while those of all other sciences are to some extent debatable and in
constant danger of being overthrown by newly discovered facts. In
spite of this, the investigator in another department of science
would not need to envy the mathematician if the laws of mathematics
referred to objects of our mere imagination, and not to objects of
reality. For it cannot occasion surprise that different persons
should arrive at the same logical conclusions when they have already
agreed upon the fundamental laws (axioms), as well as the methods by
which other laws are to be deduced therefrom. But there is another
reason for the high repute of mathematics, in that it is mathematics
which affords the exact natural sciences a certain measure of
security, to which without mathematics they could not attain.
At
this point
an enigma presents itself which in all ages has agitated inquiring
minds. How can it be that mathematics, being after all a product of
human thought which is independent of experience, is so admirably
appropriate to the objects of reality? Is human reason, then, without
experience, merely by taking thought, able to fathom the properties
of real things.
In
my opinion
the answer to this question is, briefly, this: – As far as the laws
of mathematics refer to reality, they are not certain; and as far as
they are certain, they do not refer to reality. It seems to me that
complete clearness as to this state of things first became common
property through that new departure in mathematics which is known by
the name of mathematical logic or "Axiomatics."
The
progress achieved by axiomatics consists in its having neatly
separated the logicalformal
from its objective or intuitive content; according to axiomatics the
logicalformal alone forms the subjectmatter of mathematics, which
is not concerned with the intuitive or other content associated with
the logicalformal.
Let
us for a
moment consider from this point of view any axiom of geometry, for
instance, the following: – Through two points in space there always
passes one and only one straight line. How is this axiom to be
interpreted in the older sense and in the more modern sense?
The
older
interpretation: – Every one knows what a straight line is, and what
a point is. Whether this knowledge springs from an ability of the
human mind or from experience, from some collaboration of the two or
from some other source, is not for the mathematician to decide. He
leaves the question to the philosopher. Being based upon this
knowledge which precedes all mathematics, the axiom stated above is,
like all other axioms, selfevident, that is, it is the expression of
a part of this a priori
knowledge.
The
more modern
interpretation: – Geometry treats of entities which are denoted by
the words straight line, point, etc. These entities do not take for
granted any knowledge or intuition whatever, but they presuppose only
the validity of the axioms , such as the one stated above, which are
to be taken in a purely formal sense, i.e. as void
of all content of
intuition or experience. These axioms are free creations of the human
mind. All other propositions of geometry are logical inferences from
the axioms (which are to be taken in the nominalistic sense only).
The matter of which geometry treats is first defined by the axioms.
Schlick in his book on
epistemology has
therefore characterised axioms very aptly as "implicit
definitions."
This view of
axioms, advocated by modern axiomatics, purges mathematics of all
extraneous elements, and thus dispels the mystic obscurity which
formerly surrounded the principles of mathematics. But a presentation
of its principles thus clarified makes it also evident that
mathematics as such cannot predicate anything about perceptual objects
or real
objects. In axiomatic geometry the words "point," "straight
line," etc., stand only for empty conceptual schemata. That
which gives them substance is not relevant to mathematics.
Yet
on the
other hand it is certain that mathematics generally, and particularly
geometry, owes its existence to the need which was felt of learning
something about the relations of real things to one another. The very
word geometry, which, of course, means earthmeasuring, proves this.
For earthmeasuring has to do with the possibilities of the
disposition of certain natural objects with respect to one another
namely, with parts of the earth, measuringlines, measuringwands,
etc. It is clear that the system of concepts of axiomatic geometry
alone cannot make any assertions as to the relations of real objects
of this kind, which we will call practicallyrigid bodies. To be able
to make such assertions, geometry must be stripped of its merely
logicalformal character by the
coordination of real objects of experience with the empty conceptual
framework of axiomatic geometry. To accomplish this, we need only
add the proposition: – Solid bodies are related, with respect to
their possible dispositions, as are bodies in Euclidean geometry of
three dimensions. Then the propositions of Euclid
contain affirmations as to the relations of practicallyrigid bodies.
Geometry thus
completed is evidently a natural science; we may in fact regard it as
the most ancient branch of physics. Its affirmations rest essentially
on induction from experience, but not on logical inferences only. We
will call this completed geometry "practical geometry," and
shall distinguish it in what follows from "purely axiomatic
geometry." The question whether the practical geometry of the
universe is Euclidean or not has a clear meaning, and its answer can
only be furnished by experience. All linear measurement in physics is
practical geometry in this sense, so too is geodetic and astronomical
linear measurement, if we call to our help the law of experience that
light is propagated in
a straight line, and indeed in a straight line in the sense of
practical geometry.
I
attach
special importance to the view of geometry which I have just set
forth, because without it I should have been unable to formulate the
theory of relativity. Without it the following reflection would have
been impossible: – In a system of reference rotating relatively to
an inert system, the laws of disposition of rigid bodies do not
correspond to the rules of Euclidean geometry on account of the
Lorentz contraction; thus if
we admit
noninert systems we must abandon Euclidean geometry. The decisive
step in the transition to general covariant equations would
certainly not have been taken if the above interpretation had not
served as a steppingstone. If we deny the relation between the body
of axiomatic Euclidean geometry and the practicallyrigid body of
reality, we readily arrive at the following view, which was
entertained by that acute and profound thinker,
H. Poincaré: – Euclidean
geometry is distinguished above all other imaginable axiomatic
geometries by its simplicity.
Now since axiomatic geometry by
itself contains no assertions as to the reality which can be
experienced, but can do so only in combination with physical laws, it
should be possible and reasonable – whatever may be the nature of
reality – to retain Euclidean geometry. For if contradictions
between theory and experience manifest themselves, we should rather
decide to change physical laws than to change axiomatic Euclidean
geometry. If we deny the relation between the practicallyrigid body
and geometry, we shall indeed not easily free ourselves from the
convention that Euclidean geometry is to be retained as the simplest.
Why is the equivalence of the practicallyrigid body and the body of
geometry – which suggests itself so readily – denied by Poincaré
and other investigators? Simply because under closer inspection the
real solid bodies in nature are not rigid, because their geometrical
behaviour, that is, their possibilities of relative disposition, depend
upon temperature, external forces,
etc. Thus the original, immediate relation between geometry and
physical reality appears destroyed, and we feel impelled toward the
following more general view, which characterizes Poincaré's
standpoint.
Geometry (G)
predicates
nothing about the relations of real things, but only geometry
together with the purport (P)
of
physical laws can do so. Using symbols, we may say that only the sum
of (G) + (P) is
subject to the control
of experience. Thus (G)
may be chosen
arbitrarily, and also parts of (P);
all
these laws are conventions. All that is necessary to avoid
contradictions is to choose the remainder of (P)
so that (G) and the
whole of (P)
are together in accord with experience.
Envisaged in this way,
axiomatic geometry and the part of natural law which has been given a
conventional status appear as epistemologically equivalent.
Sub
specie aeterni Poincaré
in my opinion, is right. The idea of the measuringrod and the idea
of the clock coordinated with it in the theory of relativity do not
find their
exact correspondence in the real world. It is also clear that the
solid body and the clock do not in the conceptual edifice of physics
play the part of irreducible elements, but that of composite
structures, which may not play any independent part in theoretical
physics. But it is my conviction that in the present stage of
development of theoretical physics these ideas must still be employed
as independent ideas; for we are still far from possessing such
certain knowledge of theoretical principles as to be able to give
exact theoretical constructions of solid bodies and clocks.
Further, as to
the objection that there are no really rigid bodies in nature, and
that therefore the properties predicated of rigid bodies do not apply
to physical reality, – this objection is by no means so radical as
might appear from a hasty examination. For it is not a difficult task
to determine the physical state of a measuringrod so accurately that
its behaviour relatively to other measuringbodies shall be
sufficiently free from ambiguity to allow it to be substituted for
the "rigid" body.
It is to measuringbodies of this
kind that statements as to rigid bodies must be referred.
All
practical
geometry is based upon a principle which is accessible to experience,
and which we will now try to realise. We will call that which is
enclosed between two boundaries, marked upon a practicallyrigid
body, a tract. We imagine two practicallyrigid bodies, each with a
tract marked out on it. These two tracts are said to be "equal
to one another" if the boundaries of the one tract can be
brought to coincide permanently with the boundaries of the other. We
now assume that:
If
two tracts
are found to be equal once and anywhere, they are equal always and
everywhere.
Not
only the
practical geometry of Euclid, but also
its nearest generalisation, the practical geometry of Riemann,
and therewith the general theory of relativity, rest upon this
assumption. Of the
experimental reasons which warrant this assumption I will mention
only one. The phenomenon of the propagation of light in empty space
assigns a tract, namely, the appropriate path of light, to each
interval of local time, and conversely.
Thence it follows that
the
above assumption for tracts must also hold good for intervals of
clocktime in the theory of relativity. Consequently it may be
formulated as follows: – If two ideal clocks are going at the same
rate at any time and at any place (being then in immediate proximity
to each other), they will always go at the same rate, no matter where
and when they again compared with each other at one place. – If this
law were not valid for real clocks, the proper frequencies for the
separate atoms of the same chemical element would not be in such
exact agreement as experience demonstrates. The existence of sharp
spectral lines is a convincing experimental proof of the
abovementioned principle of practical geometry. This is the ultimate
foundation in fact which enables
us to speak with of the mensuration, in Riemann's
sense of the word, of the fourdimensional continuum of spacetime.
The
question
whether the structure of this continuum is Euclidean, or in
accordance with Riemann's
general
scheme, or otherwise, is, according to the view which is here being
advocated, properly speaking a physical question which must: be
answered by experience, and not a question of a mere convention to be
selected on practical grounds. Riemann's
geometry will be the right thing if the laws of disposition of
practicallyrigid bodies are transformable into those of the bodies
of Euclid's geometry with an exactitude
which increases in proportion as the dimensions of the part of
spacetime under consideration are diminished.
It
is true that
this proposed physical interpretation of geometry breaks down when
applied immediately to spaces of submolecular order of magnitude.
But nevertheless, even in questions as to the constitution of
elementary particles, it
retains part of its importance. For even when it is a question of
describing the electrical elementary particles constituting matter,
the attempt may still be made to ascribe physical importance to those
ideas of fields which have been physically defined for the purpose of
describing the geometrical behaviour of bodies which are large as
compared with the molecule . Success alone can decide as to the
justification of such an attempt, which postulates physical reality
for the fundamental principles of Riemann's
geometry outside of the domain of their physical definitions. It
might possibly turn out that this extrapolation has no better warrant
than the extrapolation of the idea of temperature to parts of a body
of molecular order of magnitude.
It
appears less
problematical to extend the ideas of practical geometry to spaces of
cosmic order of magnitude. It might, of course, be objected that a
construction composed of solid rods departs more and more from ideal
rigidity in proportion as its spatial extent becomes greater.
But
it will hardly be possible, I think, to assign fundamental
significance to this objection. Therefore the question whether the
universe is spatially finite or not seems to me decidedly a pregnant
question in the sense of practical geometry.
I do not even
consider it impossible that this question will be answered before
long by astronomy. Let us call to mind what the general theory of
relativity teaches in this respect. It offers two possibilities: –
 The universe is
spatially
infinite. This can be so only if
the average spatial density of the matter in universal space,
concentrated in the stars, vanishes, i.e. if the ratio of the total
mass of the stars to the magnitude of the space through which they
are scattered approximates indefinitely to the value zero when the
spaces taken into consideration are constantly greater and greater.
 The universe is spatially finite.
This must be so,
if there
is a mean density of the ponderable matter in universal space
differing from zero. The smaller
that mean density, the greater is the volume of universal space.
I
must not fail
to mention that a theoretical argument can be adduced in favour of
the hypothesis of a finite universe.
The general theory of
relativity teaches that the inertia of a given body is greater as
there are more ponderable masses in proximity to it; thus it seems
very natural to reduce the total effect of inertia of a body to
action and reaction between it and the other bodies in the universe,
as indeed, ever since Newton's time,
gravity has been completely reduced to action and reaction between
bodies. From the equations of the general theory of relativity it can
be deduced that this total reduction of inertia to reprocial action
between masses – as required by E. Mach,
for example – is possible only if the universe is spatially finite.
On
many
physicists and astronomers this argument makes no
impression.
Experience alone can finally decide which of the
two
possibilities is realised in nature. How can experience furnish an
answer? At first it might seem possible to determine the mean density
of matter by observation of that part of the universe which is
accessible to our perception.
This hope is illusory. The
distribution of the visible stars is extremely irregular, so that we
on no account may venture to set down the mean density of starmatter
in the universe as equal, let us say, to the mean density in the
Milky Way. In any case, however great the space examined may be, we
could not feel convinced that there were no more stars beyond that
space. So it seems impossible to estimate the mean density
But
there is
another road, which seems to me more practicable, although it also
presents great difficulties. For if we inquire into the deviations
shown by the consequences of the general theory of relativity which
are accessible to experience, when these are compared with the
consequences of the Newtonian theory, we first of all find a
deviation which shows itself in close proximity to gravitating mass,
and has been confirmed in the case of the planet
Mercury. But if the universe is spatially finite there is a second
deviation from the Newtonian theory, which, in the language of the
Newtonian theory, may be expressed thus: – The gravitational field
is in its nature such as if it were produced, not only by the
ponderable masses, but also by a massdensity of negative sign,
distributed uniformly throughout space. Since this factitious
massdensity would have to be enormously small, it could make its
presence felt only in gravitating systems of very great extent.
Assuming
that
we know, let us say, the statistical distribution of the stars in the
Milky Way, as well as their masses, then by Newton's
law we can calculate the gravitational field and the mean velocities
which the stars must have, so that the Milky Way should not collapse
under the mutual attraction of its stars, but should maintain its
actual extent.
Now if the actual velocities of the stars,
which
can, of course, be measured, were smaller than the calculated
velocities, we should have a proof that the actual attractions
at great distances are smaller than by Newton's
's law. From such a deviation it could be proved indirectly that the
universe is finite. It would even be possible to estimate its spatial
magnitude.
Can
we picture
to ourselves a threedimensional universe which is finite, yet
unbounded?
The
usual
answer to this question is "No," but that is not the right
answer.
The purpose of the following remarks is to show that
the
answer should be "Yes." I want to show that without any
extraordinary difficulty we call illustrate the theory of a finite
universe by means of a mental image to which, with some practice, we
shall soon grow accustomed.
First of
all,
an observation of epistemological nature. A geometricalphysical
theory as such is incapable of being directly pictured, being merely
a system of concepts.
But these concepts serve the purpose of
bringing a multiplicity of real or imaginary sensory experiences into
connection in the mind "visualise" a theory or bring it
home to one's mind, therefore means to
give a representation to that abundance of experiences for which the
theory supplies the schematic arrangement. In the present case we
have to ask ourselves how we can represent that relation of solid
bodies with respect to their reciprocal disposition (contact) which
corresponds to the theory of a finite universe. There is really
nothing new in what I have to say about this; but innumerable
questions addressed to me prove that the requirements of those who
thirst for knowledge of these matters have not yet been completely
satisfied.
So, will the initiated please pardon me, if part of
what I shall bring forward has long been known?
What do
we wish
to express when we say that our space is infinite? Nothing more than
that we might lay any number whatever of bodies of equal sizes side
by side without ever filling space. Suppose that we are provided with
a great many wooden cubes all of the same size. In accordance with
Euclidean geometry we can place them above, beside, and behind one
another so as to fill a part of space of any dimensions; but
this construction would never be finished; we
could go on adding more and more cubes without ever finding that
there was no more room.
That is what we wish to express when
we
say that space is infinite. It would be better to say that space is
infinite in relation to practicallyrigid bodies, assuming that the
laws of disposition for these bodies are given by Euclidean geometry.
Another
example
of an infinite continuum is the plane. On a plane surface we may lay
squares of cardboard so that each side of any square has the side of
another square adjacent to it. The construction is never finished; we
can always go on laying squares – if their laws of disposition
correspond to those of plane figures of Euclidean geometry. The plane
is therefore infinite in relation to the cardboard squares.
Accordingly we say that the plane is an infinite continuum of two
dimensions, and space an infinite continuum of three dimensions. What
is here meant by the number of dimensions, I think I may assume to be
known.
Now
we take an
example of a twodimensional continuum which is finite, but
unbounded. We imagine the surface of a large globe and a quantity of
small paper discs, all of the same size. We place one of the discs
anywhere on the surface of the globe. If we move the disc about,
anywhere we like, on the surface of the globe, we do not come upon a
limit or boundary anywhere on the journey.
Therefore we say
that
the spherical surface of the globe is an unbounded
continuum.
Moreover, the spherical surface is a finite
continuum.
For if we stick the paper discs on the globe, so that no disc
overlaps another, the surface of the globe will finally become so
full that there is no room for another disc. This simply means that
the spherical surface of the globe is finite in relation to the paper
discs. Further, the spherical surface is a nonEuclidean continuum of
two dimensions, that is to say, the laws of disposition for the rigid
figures lying in it do not agree with those of the Euclidean
plane.
This can be shown in the following way. Place
a paper disc on the spherical
surface, and around it in a circle place six more discs, each of
which is to be surrounded in turn by six discs, and so on. If this
construction is made on a plane surface, we have an uninterrupted
disposition in which there are six discs touching every disc except
those which lie on the outside.
On
the spherical surface the construction also seems to promise success
at the outset, and the smaller the radius of the disc in proportion
to that of the sphere, the more promising it seems. But as the
construction progresses it becomes more and more patent that the
disposition of the discs in the manner indicated, without
interruption, is not possible, as it should be possible by Euclidean
geometry of the plane surface. In this way creatures
which cannot leave the spherical surface, and cannot even peep out
from the spherical surface into threedimensional space, might
discover, merely by experimenting with discs, that their
twodimensional "space" is not Euclidean, but spherical
space.
From the
latest
results of the theory of relativity it is probable that our
threedimensional space is also approximately spherical, that is,
that the laws of disposition of rigid bodies in it are not given by
Euclidean geometry, but approximately by spherical geometry, if only
we consider parts of space which are sufficiently great.
Now
this
is the place where the reader's imagination boggles. "Nobody can
imagine this thing" he cries indignantly. "It can be said,
but cannot be thought. I can represent to myself a spherical surface
well enough, but nothing analogous to it in three dimensions."
We
must try to
surmount this barrier in the mind, and the patient reader will see
that it is by no means a particularly difficult task. For this
purpose we will first give our attention once more to
the geometry of twodimensional spherical surfaces. In the
adjoining figure let K
be the spherical surface, touched at S by a plane, E,
which,
for facility of presentation, is shown in the drawing as a bounded
surface. Let L
be a disc on the
spherical surface. Now let us imagine that at the point N
of the spherical surface, diametrically opposite to S,
there is a luminous point, throwing a shadow L'of
the disc L
upon the
plane E.
Every point on the
sphere has its shadow on the plane. If the disc on the sphere K
is moved, its shadow L'
on the plane E also
moves. When
the disc L
is at S
it almost exactly coincides with its shadow. If it moves on the
spherical surface away
from S
upwards, the disc shadow
L'
on the plane also moves away
from S
on the plane outwards,
growing bigger and bigger. As the disc L
approaches the luminous point N
the shadow moves off to infinity, and becomes infinitely great.
Now
we put the
question, What are the laws of disposition of the discshadows L'
on the plane E?
Evidently they
are exactly the same as the laws of disposition of the discs L
on the spherical surface. For to each original figure on K
there is a corresponding shadow figure on E.
If two discs on K
are touching,
their shadows on E
also touch.
The shadowgeometry on the plane agrees with the discgeometry on the
sphere. If we call the discshadows rigid figures, then spherical
geometry holds good on the plane E
with respect to these rigid figures. Moreover, the plane is finite
with respect to the discshadows, since only a finite number of the
shadows can find room on the plane.
At
this point
somebody will say, "That is nonsense. The discshadows are not
rigid figures. We have only to move a twofoot rule about on the
plane E
to convince ourselves
that the shadows constantly increase in size as they move away from S
on the plane towards infinity." But what if the twofoot rule
were to behave on the plane E
in
the same way as the discshadows L'?
It would then be impossible to show that the shadows increase in size
as they move away from S;
such an
assertion would then no longer have any meaning whatever. In fact the
only objective assertion that can be made about the discshadows is
just this, that they are related in exactly the same way as are the
rigid discs on the spherical surface in the sense of Euclidean
geometry.
We
must
carefully bear in mind that our statement as to the growth of the
discshadows, as they move away from S towards
infinity has in itself no objective
meaning, as
long as we are unable to employ Euclidean rigid bodies which can be
moved about on the plane E
for
the purpose of comparing the size of the discshadows. In respect of
the laws of
disposition of the shadows L',
the point S
has no special
privileges on the plane any more than on the spherical surface.
The
representation given above of spherical geometry on the plane is
important for us, because it readily allows itself to be transferred
to the threedimensional case.
Let
us imagine
a point S
of our space, and a
great number of small spheres, L',
which can all be brought to coincide with one another. But these
spheres are not to be rigid in the sense of Euclidean geometry; their
radius is to increase (in the sense of Euclidean geometry) when they
are moved away from S
towards
infinity, and this increase is to take place in exact accordance with
the same law as applies to the increase of the radii of the
discshadows L'
on the plane.
After
having
gained a vivid mental image of the geometrical behaviour of our
spheres, let us assume that in our space there are no 'rigid' bodies
at all in the sense of Euclidean geometry, but only bodies having the
behaviour of our L'
spheres.
Then
we shall have a vivid representation of threedimensional spherical
space, or, rather of threedimensional spherical geometry. Here our
spheres must be called "rigid" spheres. Their increase in
size as they depart from S
is not
to be detected by measuring with measuringrods, any more than in the
case of the discshadows on E
because the standards of measurement behave in the same way as the
spheres. Space is homogeneous, that is to say, the same spherical
configurations are possible in the environment of all points. ^{[NOTE]}
Our space is finite, because, in consequence of the "growth"
of the spheres, only a finite number of them can find room in space.
In
this way, by
using as steppingstones the practice in thinking and visualisation
which Euclidean geometry gives us, we have acquired a mental picture
of spherical geometry. We may without difficulty impart more
depth and vigour to these ideas by carrying out special
imaginary constructions. Nor would it be difficult to represent the
case of what is called elliptical geometry in an analogous manner.
My
only aim today has been to show that the human faculty of
visualisation is by no means bound to capitulate to nonEuclidean
geometry.
Footnotes:
 This
is intelligible without calculation  but only for the twodimensional
case  if we revert once more to the case of the disc on the surface
of the sphere.

