address by Albert Einstein (1921), expanded

(Prussian Academy of Sciences, Jan 27th)

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hexagonal disk-packing projection of a finite spherical space onto an

infinite linear space

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 logical-formal from its objective 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.

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, self-evident, 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

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 relations 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

co-ordination of real objects of experience with the empty conceptual

frame-work 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 practically-rigid 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 non-inert systems we must

abandon Euclidean geometry. The decisive step in the transition to

general co-variant equations would certainly not have been taken if

the above interpretation had not served as a stepping-stone. If we

deny 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 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 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. Why is the equivalence of the

practically-rigid 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

measuring-rod and the idea of the clock co-ordinated 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 measuring-rod so

accurately that its behaviour relatively 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 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

practically-rigid body, 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 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 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 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 above-mentioned 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 four-dimensional

continuum of space-time.

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 practically-rigid 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 space-time under consideration are

diminished.

It is true that this proposed physical interpretation of geometry

breaks down when applied immediately to spaces of sub-molecular 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: –

1. 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.

2. 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 star-matter 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

mass-density of negative sign, distributed uniformly throughout space.

Since this factitious mass-density 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 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 three-dimensional 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

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

“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

practically-rigid 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 two-dimensional 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 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. 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.

hexagonal disk-packing

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 three-dimensional space, might discover, merely

by experimenting with discs, that their two-dimensional “space” is not

Euclidean, but spherical space.

From 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 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 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.

projection of a finite spherical space onto an infinite linear space

Now 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 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

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. Moreover, the plane is finite with respect to the

disc-shadows, since only a finite number of the shadows can find room

on the plane.

At 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 towards 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 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 disc-shadows, 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 disc-shadows. 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 three-dimensional 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 disc-shadows 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 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 environment of all points. * 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 stepping-stones 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 non-Euclidean

geometry.

footnote:

* This 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.