Geometry and Experience
Lecture before the Prussian Academy of Sciences, January 27, 1921. The last part appeared first in a reprint by Springer, Berlin, 1921
reason why mathematics enjoys special esteem, above all
other sciences, is that its propositions 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 propositions 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 propositions
(axioms), as well as the methods by which other propositions
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 certainty, to which without mathematics they could not attain.
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
my opinion the answer to this question is, briefly, this: as
far as the propositions 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 clarity as to this state
of things became common property only through that trend in mathematics, which
is known by the name of "axiomatics." The progress achieved by
axiomatics consists in
its having neatly separated the logical-formal from its
or intuitive content; according to axiomatics
the logical-formal alone forms the subject
matter of mathematics, which is not
concerned with the intuitive or other content
associated with the logical-formal.
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?
older interpretation: everyone 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
cooperation 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, self-evident, that is, it is the expression of a part
of this a
more modern interpretation: geometry treats of objects, which are denoted by
the words straight line, point, etc. No knowledge or intuition of these
objects is assumed but 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 axioms define
objects of which geometry treats. Schlick in his book on epistemology has
axioms very aptly as "implicit definitions."
view of axioms, advocated by modern axiomatics, Purges
mathematics of all extraneous elements, and thus dispels
the mystic obscurity, which formerly surrounded the basis
of mathematics. But such an expurgated
exposition of mathematics makes it
also evident that mathematics as
such cannot predicate anything about
objects of our intuition or real
objects. In axiomatic geometry the words "point," "straight
line," etc., stand only for empty conceptual schemata.
That which gives them content is not relevant to mathematics.
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 behavior
of real objects. The very word geometry, which, of course, means
earth-measuring, proves this. For earth-measuring has
to do with the possibilities of the disposition of certain natural
objects with respect to one another, namely, with parts
of the earth, measuring-lines, measuring-wands, etc. It
is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real
objects of this kind, which we will call practically-rigid bodies. To
be able to make such assertions, geometry must be
stripped of its merely logical-formal character by the coordination of
real objects of experience with the empty conceptual
schemata 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 behavior of practically-rigid bodies.
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 length-measurements in physics
constitute practical geometry in this
sense, so, too, do geodetic and astronomical length measurements,
if one utilizes the empirical law that light is propagated in a
straight line, and indeed in a straight line in the sense
of practical geometry.
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 inertial 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 non-inertial systems on an equal footing,
we must abandon Euclidean geometry. Without the above interpretation the decisive step in the transition to generally
covariant equations would certainly not have been taken.
If we reject the relation between the body of axiomatic Euclidean
geometry and the practically-rigid 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 conceivable
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 reject the relation between the practically-rigid
body and geometry, we shall indeed not easily free ourselves from the
convention that Euclidean geometry is to be retained
as the simplest.
is the equivalence of the practically-rigid body and
specie aeterni Poincaré,
in my opinion, is right. The idea of the measuring-rod 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 must not play any independent part
in theoretical physics. But it is my conviction that in the present
stage of development of theoretical physics these concepts must still be employed as independent concepts; for we are still
far from possessing such certain knowledge of the theoretical
principles of atomic structure as to be able to construct solid bodies
and clocks theoretically from elementary concepts.
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
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 measuring-body so accurately that its behavior
relative to other measuring-bodies shall be sufficiently
free from ambiguity to allow it to be substituted for the "rigid"
body. It is to measuring-bodies of this kind that statements about rigid bodies must be referred.
practical geometry is based upon a principle which is accessible to
experience, and which we will now try to realize.
Suppose two marks have been put upon a practically-rigid
body. A pair of two such marks we shall call a tract. We imagine two
practically-rigid bodies, each with a tract marked
out on it. These two tracts are said to be "equal to one
another" if the marks of the one tract can be brought to coincide
permanently with the marks of the other. We now assume
two tracts are found to be equal once and anywhere, they
are equal always and everywhere.
only the practical geometry of Euclid, but also its nearest generalization,
the practical geometry of Riemann, and
therewith the general theory of relativity, rest upon this assumption.
Of the experimental reasons that 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 clock-time 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 are again compared with each
other at one place. If this law were not valid for natural
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 above-mentioned principle of practical geometry. This, in the
last analysis, is the reason that enables us to speak meaningfully
of a Riemannian metric of the four-dimensional space-time
to the view advocated here, the question whether
this continuum has a Euclidean, Riemannian, or any other
structure is a question of physics proper, which must be answered by experience, and not a question of a convention to be
chosen on grounds of mere expediency. Riemann's geometry will hold if the laws of disposition of practically-rigid
bodies approach those of Euclidean geometry the more closely
the smaller the dimensions of the region of space-time
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 significance. For even when it is a question of
describing the electrical elementary particles constituting matter,
the attempt may still be made to ascribe physical meaning
to those field concepts which have been physically defined
for the purpose of describing the geometrical behavior 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 concept of temperature to parts of a body of molecular
order of magnitude.
appears less problematical to extend the concepts of practical
geometry to spaces of cosmic order of magnitude. It might,
of course, be objected that a construction composed of
solid rods departs the more from ideal rigidity the greater its
spatial extent. 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 an entirely meaningful 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 is possible only if
in the universe the average spatial density of matter, concentrated
in the stars, vanishes, i.e.,
if the ratio of the total mass of the stars
to the volume of the space through which they
are scattered indefinitely approaches zero as greater and greater volumes are considered.
The universe is spatially finite. This must be so, if there
exists an average density of the ponderable matter in the universe that is
different from zero. The smaller that average
density, the greater is the volume of the universe.
must not fail to mention that a theoretical argument can be
adduced in favor 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
inertia of a body to interaction between it and the other bodies
in the universe, as indeed, ever since Newton's time, gravity
has been completely reduced to interaction between bodies. From the equations of the general theory of relativity it
can be deduced that this total reduction of inertia to interaction
demanded by E. Mach, for example—is
possible only if the universe is spatially finite.
physicists and astronomers are not impressed by this
argument. In the last analysis, experience alone can decide
which of the two possibilities is realized in nature. How can experience furnish an answer? At first it might seem
possible to determine the average density of matter by observation
of that part of the universe that is accessible to
our observation. This hope is illusory. The distribution of
the visible stars is extremely irregular, so that we on no account
may venture to set the average density of star-matter
in the universe equal to, let us say, the average density in
the Galaxy. In any case, however great the space examined may be, we
could not feel convinced that there were any
more stars beyond that space. So it seems impossible to
estimate the average density.
there is another road, which seems to me more practicable,
although it also presents great difficulties. For if we inquire
into the deviations of the consequences of the general theory
of relativity which are accessible to experience, from the consequences of the
Newtonian theory, we first of all find a
deviation which manifests 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 such as if it were produced, not only by
the ponderable masses, but in addition by a mass-density of
negative sign, distributed uniformly throughout space. Since this fictitious mass-density would have to be extremely small,
it would be noticeable only in very extensive gravitating
that we know, let us say, the statistical distribution
and the masses of the stars in the Galaxy, then by Newton's
law we can calculate the gravitational field and the average
velocities which the stars must have, so that the Galaxy
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 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 law. From such a deviation it could be proved indirectly that the universe is finite. It would even be possible
to estimate its spatial dimensions.
we visualize a three-dimensional universe which is finite,
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 can illustrate the theory
of a finite universe by means of a mental picture to which,
with some practice, we shall soon grow accustomed.
of all, an observation of epistemological nature. A geometrical-physical
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.
To "visualize" a theory therefore means to bring to mind
that abundance of sensible experiences for which the theory
supplies the schematic arrangement. In the present case
we have to ask ourselves how we can represent that behavior of solid
bodies with respect to their mutual disposition
(contact) that 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 curiosity of those who are interested in these matters has
not yet been completely satisfied. So, will the initiated please
pardon me, in that part of what I
shall say has long been known?
do we wish to express when we say that our space is
infinite? Nothing more than that we might lay any number
of bodies of equal sizes side by side without ever filling space.
Suppose that we are provided with a great many cubic boxes
all of the same size. In accordance with Euclidean geometry
we can place them above, beside, and behind one another
so as to fill an arbitrarily large part of space; 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 practically-rigid bodies, assuming that the laws of
disposition for these bodies are given by Euclidean geometry.
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.
we take an example of a two-dimensional continuum that
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 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 means exactly
that the spherical surface of the globe is finite in relation
to the paper discs. Further, the spherical surface is a non-Euclidean
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. Take a disc and surround it in a circle by
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 obtain an uninterrupted arrangement in which
there are six discs touching every disc except those that 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 arrangement
of the discs in the manner indicated, without interruption, is not possible,
as it should be possible by the Euclidean
geometry of the plane. In this way creatures which cannot
leave the spherical surface, and cannot even peep out from the
spherical surface into three-dimensional space, might
discover, merely by experimenting with discs, that their
two-dimensional "space" is not Euclidean, but spherical
the latest results of the theory of relativity it is probable
that our three-dimensional 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
extended. 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 imagine 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 two-dimensional 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.
we put the question: what are the laws of disposition of
the disc-shadows L' on
the plane E? Evidently they are exactly
the same as the laws of disposition of the discs L
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
shadow-geometry on the plane agrees with the disc-geometry on the sphere. If
we call the disc-shadows rigid figures, then spherical
geometry holds good on the plane E
with respect to these
rigid figures. In particular, the plane is finite with respect
to the disc-shadows, since only a finite number of the shadows can find room on the plane.
this point somebody will say, "That is nonsense. The disc-shadows
are not rigid figures. We have only to move a two-foot
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 toward infinity." But what if the two foot
rule were to behave on the plane E in
the same way as the disc-shadows 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 disc-shadows is just this, that they are related
in exactly the same way as the rigid discs on the spherical
surface in the sense of Euclidean geometry.
must carefully bear in mind that our statement as to the
growth of the disc-shadows, as they move away from S toward infinity, has in itself no objective
meaning, as long as we are unable to compare
the disc-shadows with Euclidean rigid bodies that can be moved about on the
plane E. 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.
representation given above of spherical geometry on the
plane is important for us, because it readily allows itself to
be transferred to the three-dimensional case.
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 a rd infinity; it is to increase according to the same
law as the radii of the disc-shadows L'
having gained a vivid mental image of the geometrical behavior of our L'
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 behavior of our L" spheres.
Then we shall have a clear picture of three-dimensional
spherical space, or, rather of three-dimensional 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 measuring-rods, any more than in the case of the
disc-shadows 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 neighborhood of every
Our space is finite, because, in consequence of the "growth" of the spheres, only a finite number of them can
find room in space.
this way, by using as a crutch the practice in thinking and
visualization which Euclidean geometry gives us, we have
acquired a mental picture of spherical geometry. We may
without difficulty impart more depth and vigor 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 visualization
is by no means bound to capitulate to non-Euclidean geometry.
is intelligible without calculation—but only for the two-dimensional case—if
we revert once more to the case of the disc on the surface of the sphere.