CHAPTER XI: IS THE UNIVERSE FINITE?

We have been dealing hitherto with matters that must be regarded as

acquired scientific results—not that they will never be found to need

improvement, but that further progress must be built upon them, as

Einstein is built upon Newton. Science does not aim at establishing

immutable truths and eternal dogmas: its aim is to approach truth by

successive approximations, without claiming that at any stage final and

complete accuracy has been achieved. There is a difference, however,

between results which are pretty certainly in the line of advance, and

speculations which may or may not prove to be well founded. Some very

interesting speculations are connected with the theory of relativity,

and we shall consider certain of them. But it must not be supposed that

we are dealing with theories having the same solidity as those with

which we have been concerned hitherto.

One of the most fascinating of the speculations to which I have been

alluding is the suggestion that the universe may be of finite extent.

Two somewhat different finite universes have been constructed, one by

Einstein, the other by De Sitter. Before considering their differences,

we will discuss what they have in common.

There are, to begin with, certain reasons for thinking that the total

amount of matter in the universe is limited. If this were not the

case, the gravitational effects of enormously distant matter would

make the kind of world in which we live impossible. We must therefore

suppose that there is some definite number of electrons and protons in

the world: theoretically, a complete census would be possible. These

are all contained within a certain finite region; whatever space lies

outside that region is, so to speak, waste, like unfurnished rooms in a

house too large for its inhabitants. This seems futile, but in former

days no one knew of any alternative possibility. It was obviously

impossible to conceive of an edge to space, and therefore, it was

thought, space must be infinite.

Non-Euclidean geometry, however, showed other possibilities. The

surface of a sphere has no boundary, yet it is not infinite. In

traveling round the earth, we never reach “the edge of the world,” and

yet the earth is not infinite. The surface of the earth is contained

in three-dimensional space, but there is no reason in logic why

three-dimensional space should not be constructed on an analogous plan.

What we imagine to be straight lines going on for ever will then be

like great circles on a sphere: they will ultimately return to their

starting point. There will not be in the universe anything straighter

than these great circles; the Euclidean straight line may remain as

a beautiful dream, but not as a possibility in the actual world. In

particular, light rays in empty space will travel in what are really

great circles. If we could make measurements with sufficient accuracy,

we should be able to infer this state of affairs even from a small part

of space, because the sum of the angles of a triangle would always be

greater than two right angles, and the excess would be proportional to

the size of the triangle. The suggestion we have to consider is the

suggestion that our universe may be spherical in this sense.

The reader must not confuse this suggestion with the non-Euclidean

character of space upon which the new law of gravitation depends. The

latter is concerned with small regions such as the solar system. The

departures from flatness which it notices are like hills and valleys

on the surface of the earth, local irregularities, not characteristics

of the whole. We are now concerned with the possible curvature of the

universe as a whole, not with the occasional ups and downs due to the

sun and the stars. It is suggested that on the average, and in regions

remote from matter, the universe is not quite flat, but has a slight

curvature, analogous, in three dimensions, to the curvature of a sphere

in two dimensions.

It is important to realize, in the first place, that there is not the

slightest reason _à priori_ why this should not be the case. People

unaccustomed to non-Euclidean geometry may feel that, even if such a

thing be _logically_ possible, the world simply _cannot_ be so odd

as all that. We all have a tendency to think that the world must

conform to our prejudices. The opposite view involves some effort of

thought, and most people would die sooner than think—in fact, they

do so. But the fact that a spherical universe seems odd to people

who have been brought up on Euclidean prejudices is no evidence that

it is impossible. There is no law of nature to the effect that what

is taught at school must be true. We cannot therefore dismiss the

hypothesis of a spherical universe as in any degree less worthy of

examination than any other. We have to ask ourselves the same two

questions as we should in any other case, namely: (1) Are the facts

consistent with this hypothesis? (2) Is this hypothesis the only one

with which the facts are consistent?

With regard to the first question, the answer is undoubtedly in the

affirmative. All the known facts are perfectly consistent with the

hypothesis of a spherical universe. A very slight modification of the

law of gravitation—a modification suggested by Einstein himself—leads

to a spherical space, without producing any measurable differences in a

small region such as the solar system. The known stars are all within

a certain distance from us. There is nothing whatever in the stellar

universe as we know it to show that space must be infinite. There can

therefore be no doubt whatever that, so far as our present knowledge

goes, the hypothesis of a finite universe _may_ be true.

But when we ask whether the hypothesis of a finite universe _must_

be true, the answer is different. It is obvious, on general grounds,

that we cannot, from what we know, draw conclusive inferences as to

the totality of things. A very slight change in the Newtonian formula

for gravitation would prevent masses beyond the limits of the visible

universe from having appreciable effects if they existed, and would

therefore destroy our reason for supposing that they do not exist.

All arguments as to regions which are too distant to be observed

depend upon extending to them the laws which hold in our part of

the world, and upon assuming that there is not, in these laws, some

inaccuracy which is inappreciable for observable distances, but fatal

to inferences in which very much greater distances are involved. We

cannot, therefore, say that the universe _must_ be finite. We can say

that it may be, and we can even say a little more than this. We can say

that a finite universe fits in better with the laws that hold in the

part we know, and that awkward adjustments of the laws have to be made

in order to allow the universe to be infinite. From the point of view

of choosing the best framework into which to fit what we know—best, I

mean, from a logico-æsthetic point of view—there is no doubt that the

hypothesis of a finite universe is preferable. This, I think, is the

extent of what can be said in its favor.

Let us now see what the two finite universes are like. The difference

between them is that in Einstein’s world it is only space that

is queer, whereas in De Sitter’s time is queer too. Consequently

Einstein’s world is less puzzling, and we will describe it first.

In Einstein’s world, light travels round the whole universe in a time

which is supposed to be something like a thousand million years. The

odd thing is that all the rays of light which start (say) from the sun

will meet again, after their enormous journey, in the place where the

sun was when they started. The case is exactly analogous to that of a

number of travelers who set out from London to go round the world in

great circles, all traveling at the same rate in different aeroplanes.

One starts due north, passes the North Pole, then the South Pole, and

finally comes home. Another starts due south, reaches the South Pole

first and then the North Pole. Another starts westward, but he must not

continue to travel due west, because then he would not be traveling on

a great circle. Another starts eastward, and so on. They all meet in

the antipodes of London, and then they all meet again in London. Now

if instead of aeronauts going round the earth you take rays of light

going round the universe, the same sort of thing happens: they all meet

first at the antipodes of their starting point, and then meet again at

their starting point. That means to say that a person who is near the

antipodes of the place where the sun was about five hundred million

years ago will see what is apparently a body as bright as the sun then

was (except for the small amount of light that has been stopped on the

way by opaque bodies), and having the same shape and size. And a person

who is near where the sun was a thousand million years ago will see

what is apparently a body just like what the sun was a thousand million

years ago. And the same applies to the antipodes of the sun fifteen

hundred million years ago, and to the place of the sun two thousand

million years ago, and so on. This series only ends when it carries us

back to a time before the sun existed.

But all these suns are only ghosts; that is to say, you could pass

through them without experiencing resistance, and they do not exert

gravitation. They are, in fact, like images in a mirror: they exist

only for the sense of sight, not for any other sense. It is rather

disturbing to reflect that, if this theory is true, any number of the

objects we see in the heavens may be merely ghosts. They are like

ghosts in their habit of revisiting the scenes of their past life.

Suppose a star had exploded at a certain place, as stars sometimes

will. Every thousand million years its ghost would return to the scene

of the disaster and explode again in the same place. There is, however,

considerable doubt whether rays of light could perform the journey with

sufficient accuracy to produce a clear image. Some would be stopped by

matter on the way, some would be turned out of the straight course by

passing near heavy bodies, as in the eclipse observations described in

Chapter IX, and for one reason or another their return would not be

punctual and exact.

There are various reasons for doubting whether Einstein’s universe can

be quite right.[12] Some of these are rather complicated. But there

is one objection which is easily appreciated: in Einstein’s theory,

absolute space and time re-enter by another door. The ghostly sun

is formed in the “place” where it was a thousand million years ago.

Both the “place” and the period of time are in a sense absolute. We

saw as early as Chapter I that “place” is a vague and popular notion,

incapable of scientific precision. It seems hardly worth while to go

through such a vast intellectual labor if the errors we set out to

correct are to reappear at the end.

[12] See Eddington, _Space, Time and Gravitation_, p. 162ff.

De Sitter’s world is even odder than Einstein’s, because time goes

mad as well as space. I despair of explaining, in non-mathematical

language, the particular form of lunacy with which time is afflicted,

but some of its manifestations can be described. An observer in this

world, if he observes a number of clocks, each of which is perfectly

accurate from its own point of view, will think that distant clocks

are going slow as compared with those in his neighborhood. They will

seem to go slower and slower, until, at a distance of one quarter of

the circumference of the universe, they will seem to have stopped

altogether. That region will seem to our observer a sort of lotus

land, where nothing is ever done. He will not be able to have any

cognizance of things farther off, because no light waves can get across

the boundary. Not that there is any real boundary: the people who live

in what our observer takes to be lotus land live just as bustling a

life as he does, but get the impression that he is eternally standing

still. As a matter of fact, you would never become aware of the lotus

land, because it would take an infinite time for light to travel from

it to you. You could become aware of places just short of it, but it

would remain itself always just beyond your ken. There will not be the

ghostly suns of Einstein’s world, because light cannot travel so far.

One of the oddest things about this state of affairs is that empirical

evidence for or against it is possible, and that there is actually

some slight evidence in its favor. If all “clocks” are slowed down at

a great distance from the observer, this will apply to the periodic

motions of atoms, and therefore to the light which they emit.

Consequently all rays of light emitted by distant objects ought, when

they reach us, to look rather more red or less violet than when they

started. This can be tested by the spectroscope. We can compare a

known line, as it appears in the spectrum of a spiral nebula, with

the same line as it appears in a terrestrial laboratory. We find, as a

matter of fact, that in a large majority of spiral nebulæ there is a

considerable displacement of spectral lines towards the red. The spiral

nebulæ are the most distant objects we can see: Eddington states that

their distances “may perhaps be of the order of a million light-years.”

(A light-year is the distance light travels in a year.) The usual

interpretation of a shifting of spectral lines towards the red is that

it is a “Doppler effect,” due to the fact that the source of light is

moving away from us. But one would expect to find the nebulæ just as

often moving towards us as moving away from us, if nothing operated but

the law of chances. If the world is such as De Sitter says it is, the

spectral lines of the spiral nebulæ will be displaced towards the red

owing to the slowing down of distant clocks, even if in fact they are

not moving away from us. This, for what it is worth, is an argument in

favor of De Sitter.

The same facts afford another argument in favor of De Sitter, for

another reason. If, at a given moment, a body is at rest relatively to

the observer, and at a distance from him, it will (in the absence of

counteracting causes) not remain at rest from his point of view, but

will begin to move away from him, and will continue to move away faster

and faster; the further it is from him, the more its retreat will be

accelerated. For bodies which are not too distant from each other,

gravitation may overcome this tendency; but as this tendency increases

with the distance, while gravitation diminishes, we should expect

to find very distant bodies receding from us if De Sitter’s theory

is right. Thus we have two reasons for the displacement of spectral

lines in spiral nebulæ: one, the slowing down of time; the other, the

movement away from us which we should expect at distances too great

for gravitation to be sensible. However, it cannot be said that the

argument, on either ground, is very strong. Eddington gives a list

of forty-one spiral nebulæ, of which five have their spectral lines

shifted towards the violet, not towards the red. Thus the material is

neither very copious nor quite harmonious.

Einstein’s and De Sitter’s hypotheses do not exhaust the possibilities

of a finite world: they are merely the two simplest forms of such a

world. There are arguments against each, and it hardly seems probable

that either is quite true. But it does seem probable that something

more or less analogous is true. If the universe is finite, it is

theoretically conceivable that there should be a complete inventory

of it. We may be coming to the end of what physics can do in the way

of stretching the imagination and systematizing the world. The period

since Galileo has been essentially the period of physics, as the age of

the Greeks was the period of geometry. It may be that physics will lose

its attractions through success: if the fundamental laws of physics

come to be fully known, adventurous and inquiring intellects will turn

to other fields. This may alter profoundly the whole texture of human

life, since our present absorption in machinery and industrialism is

the reflection in the practical world of the theorist’s interest in

physical laws. But such speculations are even more rash than those of

De Sitter, and I do not wish to lay any stress upon them.

CHAPTER XII: CONVENTIONS AND NATURAL LAWS

One of the most difficult matters in all controversy is to distinguish

disputes about words from disputes about facts: it ought not to be

difficult, but in practice it is. This is quite as true in physics as

in other subjects. In the seventeenth century there was a terrific

debate as to what “force” is; to us now, it was obviously a debate

as to how the word “force” should be defined, but at the time it was

thought to be much more. One of the purposes of the method of tensors,

which is employed in the mathematics of relativity, is to eliminate

what is purely verbal (in an extended sense) in physical laws. It is

of course obvious that what depends on the choice of co-ordinates is

“verbal” in the sense concerned. A man punting walks along the boat,

but keeps a constant position with reference to the river bed so

long as he does not pick up his pole. The Lilliputians might debate

endlessly whether he is walking or standing still: the debate would

be as to words, not as to facts. If we choose co-ordinates fixed

relatively to the boat, he is walking; if we choose co-ordinates

fixed relatively to the river bed, he is standing still. We want to

express physical laws in such a way that it shall be obvious when we

are expressing the same law by reference to two different systems

of co-ordinates, so that we shall not be misled into supposing we

have different laws when we only have one law in different words.

This is accomplished by the method of tensors. Some laws which seem

plausible in one language cannot be translated into another; these are

impossible as laws of nature. The laws that can be translated into

_any_ co-ordinate language have certain characteristics: this is a

substantial help in looking for such laws of nature as the theory of

relativity can admit to be possible. Combined with what we know of the

actual motions of bodies, it enables us to decide what must be the

correct expression of the law of gravitation: logic and experience

combine in equal proportions in obtaining this expression.

But the problem of arriving at genuine laws of nature is not to be

solved by the method of tensors alone; a good, deal of careful thought

is wanted in addition. Some of this has been done, especially by

Eddington; much remains to be done.

To take a simple illustration: Suppose, as in the hypothesis of the

Fitzgerald contraction, that lengths in one direction were shorter than

in another. Let us assume that a foot rule pointing north is only half

as long as the same foot rule pointing east, and that this is equally

true of all other bodies. Does such an hypothesis have any meaning?

If you have a fishing rod fifteen feet long when it is pointing west,

and you then turn it to the north, it will still measure fifteen feet,

because your foot rule will have shrunk too. It won’t “look” any

shorter, because your eye will have been affected in the same way. If

you are to find out the change, it cannot be by ordinary measurement;

it must be by some such method as the Michelson-Morley experiment, in

which the velocity of light is used to measure lengths. Then you still

have to decide whether it is simpler to suppose a change of length

or a change in the velocity of light. The experimental fact would be

that light takes longer to traverse what your foot rule declares to

be a given distance in one direction than in another—or, as in the

Michelson-Morley experiment, that it ought to take longer but doesn’t.

You can adjust your measures to such a fact in various ways; in any

way you choose to adopt, there will be an element of convention. This

element of convention survives in the laws that you arrive at after

you have made your decision as to measures, and often it takes subtle

and elusive forms. To eliminate the element of convention is, in fact,

extraordinarily difficult; the more the subject is studied, the greater

the difficulty is seen to be.

A more important example is the question of the size and shape of the

electron. We find experimentally that all electrons are the same size,

and that they are symmetrical in all directions. How far is this a

genuine fact ascertained by experiment, and how far is it a result of

our conventions of measurement? We have here a number of different

comparisons to make: (1) between different directions in regard to one

electron at one time; (2) in regard to one electron at different times;

(3) in regard to two electrons at the same time. We can then arrive

at the comparison of two electrons at different times, by combining

(2) and (3). We may dismiss any hypothesis which would affect all

electrons equally; for example, it would be useless to suppose that in

one region of space-time they were all larger than in another. Such a

change would affect our measuring appliances just as much as the things

measured, and would therefore produce no discoverable phenomena. This

is as much as to say that it would be no change at all. But the fact

that two electrons have the same mass, for instance, cannot be regarded

as purely conventional. Given sufficient minuteness and accuracy, we

could compare the effects of two different electrons upon a third;

if they were equal under like circumstances, we should be able to

infer equality in a not purely conventional sense. The question of

the symmetry of the forces exerted by an electron—_i.e._, that these

forces depend only upon the distance from the electron, and not upon

the direction—is more complicated. Eddington finally comes to the

conclusion that this, too, is a matter of convention. The argument

is difficult and I have not fully understood it; but I feel some

hesitation in accepting it as valid.

Eddington describes the process concerned in the more advanced portions

of the theory of relativity as “world-building.” The structure to be

built is the physical world as we know it; the economical architect

tries to construct it with the smallest possible amount of material.

This is a question for logic and mathematics. The greater our technical

skill in these two subjects, the more real building we shall do, and

the less we shall be content with mere heaps of stones. But before we

can use in our building the stones that nature provides, we have to

hew them into the right shapes: this is all part of the process of

budding. In order that this may be possible, the raw material must

have _some_ structure (which we may conceive as analogous to the

grain in timber), but almost any structure will do. By successive

mathematical refinements, we whittle away our initial requirements

until they amount to very little. Given this necessary minimum of

structure in the raw material, we find that we can construct from it a

mathematical expression which will have the properties that are needed

for describing the world we perceive—in particular, the properties

of conservation which are characteristic of momentum and energy (or

mass). Our raw material consisted merely of events; but when we find

that we can build out of it something which, as measured, will seem

to be never created or destroyed, it seems not surprising that we

should come to believe in “bodies.” These are really mere mathematical

constructions out of events, but owing to their permanence they are

practically important, and our senses (which were presumably developed

by biological needs) are adapted for noticing them, rather than the

crude continuum of events which is theoretically more fundamental. From

this point of view, it is astonishing how little of the real world is

revealed by physical science: our knowledge is limited, not only by the

conventional element, but also by the selectiveness of our perceptual

apparatus.

We assume that there is an “interval” between two events, in the

sense explained in Chapter VII, but we no longer assume that we can

unambiguously compare the length of an interval in one region with the

length of an interval in another. It is assumed by Weyl, who introduced

this limitation, that we can compare a number of small intervals which

all start from the same point; also that, in a very small journey,

our measuring rod will not alter its length much, so that there will

only be a small error if we compare lengths in neighboring places by

the usual methods. Weyl found that, by diminishing our assumptions as

to interval in this way, it was possible to bring electromagnetism

and gravitation into one system. The mathematics of Weyl’s theory is

complicated, and I shall not attempt to explain it. For the present,

I am concerned with a different consequence of his theory. If lengths

in different regions cannot be compared directly, there is an element

of convention in the indirect comparisons which we actually make. This

element will be at first unrecognized, but will be such as to simplify

to the utmost the expression of the laws of nature. In particular,

conditions of symmetry may be entirely created by conventions as to

measurement, and there is no reason to suppose that they represent any

property of the real world. The law of gravitation itself, according to

Eddington, may be regarded as expressing conventions of measurement.

“The conventions of measurement,” he says, “introduce an isotropy[13]

and homogeneity into measured space which need not originally have any

counterpart in the relation-structure which is being surveyed. This

isotropy and homogeneity is exactly expressed by Einstein’s law of

gravitation.”[14]

[13] “Isotropy” means being similar in all directions—_e.g._, that a

foot rule is as long when it points north as when it points east.

[14] _Mathematical Theory of Relativity_, p. 238.

The limitations of knowledge introduced by the selectiveness of our

perceptual apparatus may be illustrated by the indestructibility

of matter. This has been gradually discovered by experiment, and

seemed a well-founded empirical law of nature. Now it turns out

that, from our original space-time continuum, we can construct a

mathematical expression which will have properties causing it to appear

indestructible. The statement that matter is indestructible then ceases

to be a proposition of physics, and becomes instead a proposition

of linguistics and psychology. As a proposition of linguistics:

“Matter” is the name of the mathematical expression in question. As a

proposition of psychology: Our senses are such that we notice what is

roughly the mathematical expression in question, and we are led nearer

and nearer to it as we refine upon our crude perceptions by scientific

observation. This is much less than physicists used to think they knew

about matter.

The reader may say: What then is left of physics? What do we really

know about the world of matter? Here we may distinguish three

departments of physics. There is first what is included within the

theory of relativity, generalized as widely as possible. Next, there

are laws which cannot be brought within the scope of relativity.

Thirdly, there is what may be called geography. Let us consider each of

these in turn.

The theory of relativity, apart from convention, tells us that the

events in the universe have a four-dimensional order, and that,

between any two events which are near together in this order, there

is a relation called “interval,” which is capable of being measured

if suitable precautions are taken. We make also an assumption as to

what happens when a little measuring rod is carried round a closed

circuit in a certain manner; the consequences of this assumption are

such as to make it highly probable that it is true. Beyond this, there

is little in the theory of relativity that can be regarded as physical

laws. There is a great deal of mathematics, showing that certain

mathematically-constructed quantities must behave like the things we

perceive; and there is a suggestion of a bridge between psychology and

physics in the theory that these mathematically-constructed quantities

are what our senses are adapted for perceiving. But neither of these

things is physics in the strict sense.

The part of physics which cannot, at present, be brought within

the scope of relativity is large and important. There is nothing

in relativity to show why there should be electrons and protons;

relativity cannot give any reason why matter should exist in little

lumps. With this goes the whole theory of the structure of the atom.

The theory of quanta also is quite outside the scope of relativity.

Relativity is, in a sense, the most extreme application of what may

be called next-to-next methods. Gravitation is no longer regarded

as due to the effect of the sun upon a planet, but as expressing

characteristics of the region in which the planet happens to be.

Distance, which used to be thought to have a definite meaning however

far apart two points might be, is now only definite for neighboring

points. The distance between widely separated places depends upon the

route chosen. We may, it is true, define _the_ distance as the geodesic

distance, but that can only be estimated by adding up little bits,

that is to say, by the method we use in estimating the length of a

curve. What applies to distance applies equally to the straight line.

There is nothing in the actual world having exactly the properties

that straight lines were supposed to have; the nearest approach is the

track of a light ray. Straight lines have to be replaced by geodesics,

which are defined by what they do at each point, not all at once,

like Euclidean straight lines. Measurement, in Weyl’s theory, suffers

the same fate. We can only use a measuring rod to give lengths in one

place: when we move it to another region, there is no knowing how it

will alter. We do assume, however, that, if it alters, it alters bit

by bit, gradually, continuously, and not by sudden jumps. Perhaps

this assumption is unjustified. It belongs to the general outlook of

relativity, which is that of continuity. No doubt it is owing to this

outlook that relativity is unable to account for the discontinuities in

physics, such as quanta, electrons and protons. Perhaps relativity will

conquer these domains when it learns to dispense with the assumption of

continuity.

Finally we come to geography, in which I include history. The

separation of history from geography rests upon the separation of time

from space; when we amalgamate the two in space-time, we need one word

to describe the combination of geography and history. For the sake of

simplicity, I shall use the one word geography in this extended sense.

Geography, in this sense, includes everything that, as a matter of

crude fact, distinguishes one part of space-time from another. One

part is occupied by the sun, one by the earth; the intermediate

regions contain light waves, but no matter (apart from a very little

here and there). There is a certain degree of theoretical connection

between different geographical facts; to establish this is the purpose

of physical laws. It is thought that a sufficient knowledge of the

geographical facts of the solar system throughout any finite time,

however short, would enable an ideally competent physicist to predict

the future of the solar system so long as it remained remote from other

stars. We are already in a position to calculate the large facts about

the solar system backwards and forwards for vast periods of time. But

in all such calculations we need a basis of crude fact. The facts are

interconnected, but facts can only be inferred from other facts, not

from general laws alone. Thus the facts of geography have a certain

independent status in physics. No amount of physical laws will enable

us to infer a physical fact unless we know other facts as data for our

inference. And here when I speak of “facts” I am thinking of particular

facts of geography, in the extended sense in which I am using the term.

In the theory of relativity, we are concerned with _structure_, not

with the material of which the structure is composed. In geography,

on the other hand, the material is relevant. If there is to be any

difference between one place and another, there must either be

differences between the material in one place and that in another, or

places where there is material and places where there is none. The

former of these alternatives seems the more satisfactory. We might

try to say: There are electrons and protons, and the rest is empty.

But in the “empty” regions there are light waves, so that we cannot

say nothing happens in them. Some people maintain that the light

waves take place in the ether, others are content to say simply that

they take place; but in any case events are occurring where there are

light waves. That is all that we can really say for the places where

there is matter, since matter has turned out to be a mathematical

construction built out of events. We may say, therefore, that there

are events everywhere in space-time, but they must be of a somewhat

different kind according as we are dealing with a region where there is

an electron or proton or with the sort of region we should ordinarily

call empty. But as to the intrinsic nature of these events we can know

nothing, except when they happen to be events in our own lives. Our own

perceptions and feelings must be part of the crude material of events

which physics arranges into a pattern—or rather, which physics finds

to be arranged in a pattern. As regards events which do not form part

of our own lives, physics tells us the pattern of them, but is quite

unable to tell us what they are like in themselves. Nor does it seem

possible that this should be discovered by any other method.

CHAPTER XIII: THE ABOLITION OF “FORCE”

In the Newtonian system, bodies under the action of no forces move in

straight lines with uniform velocity; when bodies do not move in this

way, their change of motion is ascribed to a “force.” Some forces seem

intelligible to our imagination: those exerted by a rope or string,

by bodies colliding, or by any kind of obvious pushing or pulling. As

explained in an earlier chapter, our apparent imaginative understanding

of these processes is quite fallacious; all that it really means is

that past experience enables us to foresee more or less what is going

to happen without the need of mathematical calculations. But the

“forces” involved in gravitation and in the less familiar forms of

electrical action do not seem in this way “natural” to our imagination.

It seems odd that the earth can float in the void: the natural thing

to suppose is that it must fall. That is why it has to be supported on

an elephant, and the elephant on a tortoise, according to some early

speculators. The Newtonian theory, in addition to action at a distance,

introduced two other imaginative novelties. The first was, that

gravitation is not always and essentially directed what we should call

“downwards,” _i.e._, towards the center of the earth. The second was,

that a body going round and round in a circle with uniform velocity is

not “moving uniformly” in the sense in which that phrase is applied to

the motion of bodies under no forces, but is perpetually being turned

out of the straight course towards the center of the circle, which

requires a force pulling it in that direction. Hence Newton arrived at

the view that the planets are attracted to the sun by a force, which is

called gravitation.

This whole point of view, as we have seen, is superseded by relativity.

There are no longer such things as “straight lines” in the old

geometrical sense. There are “straightest lines,” or geodesics, but

these involve time as well as space. A light ray passing through

the solar system does not describe the same orbit as a comet, from

a geometrical point of view; nevertheless each moves in a geodesic.

The whole imaginative picture is changed. A poet might say that water

runs down hill because it is attracted to the sea, but a physicist or

an ordinary mortal would say that it moves as it does, at each point,

because of the nature of the ground at that point, without regard to

what lies ahead of it. Just as the sea does not cause the water to run

towards it, so the sun does not cause the planets to move round it. The

planets move round the sun because that is the easiest thing to do—in

the technical sense of “least action.” It is the easiest thing to do

because of the nature of the region in which they are, not because of

an influence emanating from the sun.

The supposed necessity of attributing gravitation to a “force”

attracting the planets towards the sun has arisen from the

determination to preserve Euclidean geometry at all costs. If we

suppose that our space is Euclidean, when in fact it is not, we shall

have to call in physics to rectify the errors of our geometry. We shall

find bodies not moving in what we insist upon regarding as straight

lines, and we shall demand a cause for this behavior. Eddington has

stated this matter with admirable lucidity. He supposes a physicist

who has assumed the formula for interval which is used in the special

theory of relativity—a formula which still supposes that the

observer’s space is Euclidean. He continues:

Since intervals can be compared by experimental

methods, he ought soon to discover that his

(formula for the interval) cannot be reconciled

with observational results, and so realize his

mistake. But the mind does not so readily get rid of

an obsession. It is more likely that our observer

will continue in his opinion, and attribute the

discrepancy of the observations to some influence

which is present and affects the behavior of his

test-bodies. He will, so to speak, introduce a

supernatural agency which he can blame for the

consequences of his mistake.... The name given to

any agency which causes deviation from uniform

motion in a straight line is _force_ according

to the Newtonian definition of force. Hence the

agency invoked through our observer’s mistake is

described as a “field of force.”... _A field of

force represents the discrepancy between the natural

geometry of a co-ordinate system and the abstract

geometry arbitrarily ascribed to it._[15]

[15] _Mathematical Theory of Relativity_, pp. 37-38. Italics in the

original.

If people were to learn to conceive the world in the new way, without

the old notion of “force,” it would alter not only their physical

imagination, but probably also their morals and politics. The latter

effect would be quite illogical, but is none the less probable on that

account. In Newton’s theory of the solar system, the sun seems like a

monarch whose behests the planets have to obey. In Einstein’s world

there is more individualism and less government than in Newton’s.

There is also far less hustle: we have seen that laziness is the

fundamental law of Einstein’s universe. The word “dynamic” has come to

mean, in newspaper language, “energetic and forceful”; but if it meant

“illustrating the principles of dynamics,” it ought to be applied to

the people in hot climates who sit under banana trees waiting for the

fruit to drop into their mouths. I hope that journalists, in future,

when they speak of a “dynamic personality,” will mean a person who

does what is least trouble at the moment, without thinking of remote

consequences. If I can contribute to this result, I shall not have

written in vain.

It has been customary for people to draw arguments from the laws of

nature as to what we ought to do. Such arguments seem to me a mistake:

to imitate nature may be merely slavish. But if nature, as portrayed by

Einstein, is to be our model, it would seem that the anarchists will

have the best of the argument. The physical universe is orderly, not

because there is a central government, but because every body minds

its own business. No two particles of matter ever come into contact;

when they get too close, they both move off. If a man were had up

for knocking another man down, he would be scientifically correct in

pleading that he had never touched him. What happened was that there

was a hill in space-time in the region of the other man’s nose, and it

fell down the hill.

The abolition of “force” seems to be connected with the substitution

of sight for touch as the source of physical ideas, as explained in

Chapter I. When an image in a looking glass moves, we do not think that

something has pushed it. In places where there are two large mirrors

opposite to each other, you may see innumerable reflections of the

same object. Suppose a gentleman in a top-hat is standing between the

mirrors, there may be twenty or thirty top-hats in the reflections.

Suppose now somebody comes and knocks off the gentleman’s hat with a

stick: all the other twenty or thirty top-hats will tumble down at the

same moment. We think that a force is needed to knock off the “real”

top-hat, but we think the remaining twenty or thirty tumble off, so to

speak, of themselves, or out of a mere passion for imitation. Let us

try to think out this matter a little more seriously.

Obviously something happens when an image in a looking glass moves.

From the point of view of sight, the event seems just as real as if it

were not in a mirror. But nothing has happened from the point of view

of touch or hearing. When the “real” top-hat falls, it makes a noise;

the twenty or thirty reflections fall without a sound. If it falls on

your toe, you feel it; but we believe that the twenty or thirty people

in the mirrors feel nothing, though top-hats fall on their toes too.

But all this is equally true of the astronomical world. It makes no

noise, because sound cannot travel across a vacuum. So far as we know,

it causes no “feelings,” because there is no one on the spot to “feel”

it. The astronomical world, therefore, seems hardly more “real” or

“solid” than the world in the looking glass, and has just as little

need of “force” to make it move.

The reader may feel that I am indulging in idle sophistry. “After all,”

he may say, “the image in the mirror is the reflection of something

solid, and the top-hat in the mirror only falls off because of the

force applied to the real top-hat. The top-hat in the mirror cannot

indulge in behavior of its own; it has to copy the real one. This shows

how different the image is from the sun and the planets, because _they_

are not obliged to be perpetually imitating a prototype. So you had

better give up pretending that an image is just as real as one of the

heavenly bodies.”

There is, of course, some truth in this; the point is to discover

exactly _what_ truth. In the first place, images are not “imaginary.”

When you see an image, certain perfectly real light waves reach your

eye; and if you hang a cloth over the mirror, these light waves cease

to exist. There is, however, a purely optical difference between an

“image” and a “real” thing. The optical difference is bound up with

this question of imitation. When you hang a cloth over the mirror,

it makes no difference to the “real” object; but when you move the

“real” object away, the image vanishes also. This makes us say that the

light rays which make the image are only reflected at the surface of

the mirror, and do not really come from a point behind it, but from

the “real” object. We have here an example of a general principle of

great importance. Most of the events in the world are not isolated

occurrences, but members of groups of more or less similar events,

which are such that each group is connected in an assignable manner

with a certain small region of space-time. This is the case with the

light rays which make us see both the object and its reflection in the

mirror: they all emanate from the object as a center. If you put an

opaque globe round the object at a certain distance, the object and

its reflection are invisible at any point outside the globe. We have

seen that gravitation, although no longer regarded as an action at a

distance, is still connected with a center: there is, so to speak, a

hill symmetrically arranged about its summit, and the summit is the

place where we conceive the body to be which is connected with the

gravitational field we are considering. For simplicity, common sense

lumps together all the events which form one group in the above sense.

When two people see the same object, two different events occur, but

they are events belonging to one group and connected with the same

center. Just the same applies when two people (as we say) hear the

same noise. And so the reflection in a mirror is less “real” than the

object reflected, even from an optical point of view, because light

rays do not spread in _all_ directions from the place where the image

seems to be, but only in directions in front of the mirror, and only so

long as the object reflected remains in position. This illustrates the

usefulness of grouping connected events about a center in the way we

have been considering.

When we examine the changes in such a group of objects, we find that

they are of two kinds: there are those which affect only some member

of the group, and those which make connected alterations in all the

members of the group. If you put a candle in front of a mirror, and

then hang black cloth over the mirror, you alter only the reflection

of the candle as seen from various places. If you shut your eyes,

you alter its appearance to you, but not its appearance elsewhere.

If you put a red globe round it at a distance of a foot, you alter

its appearance at any distance greater than a foot, but not at any

distance less than a foot. In all these cases, you do not regard the

candle itself as having changed; in fact, in all of them, you find that

there are groups of changes connected with a different center or with

a number of different centers. When you shut your eyes, for instance,

your eyes, not the candle, look different to any other observer: the

center of the changes that occur is in your eyes. But when you blow

out the candle, its appearance _everywhere_ is changed; in this case

you say that the change has happened to the candle. The changes that

happen to an object are those that affect the whole group of events

which center about the object. All this is only an interpretation of

common sense, and an attempt to explain what we mean by saying that the

image of the candle in the mirror is less “real” than the candle. There

is no connected group of events situated all round the place where the

image seems to be, and changes in the image center about the candle,

not about a point behind the mirror. This gives a perfectly verifiable

meaning to the statement that the image is “only” a reflection. And at

the same time it enables us to regard the heavenly bodies, although

we can only see and not touch them, as more “real” than an image in a

looking glass.

We can now begin to interpret the common sense notion of one body

having an “effect” upon another, which we must do if we are really to

understand what is meant by the abolition of “force.” Suppose you come

into a dark room and switch on the electric light: the appearance of

everything in the room is changed. Since everything in the room is

visible because it reflects the electric light, this case is really

analogous to that of the image in the mirror; the electric light is the

center from which all the changes emanate. In this case, the “effect”

is explained by what we have already said. The more important case is

when the effect is a movement. Suppose you let loose a tiger in the

middle of a Bank Holiday crowd: they would all move, and the tiger

would be the center of their various movements. A person who could

see the people but not the tiger would infer that there was something

repulsive at that point. We say in this case that the tiger has an

effect upon the people, and we might describe the tiger’s action upon

them as of the nature of a repulsive force. We know, however, that they

fly because of something which happens to _them_, not merely because

the tiger is where he is. They fly because they can see and hear him,

that is to say, because certain waves reach their eyes and ears. If

these waves could be made to reach them without there being any tiger,

they would fly just as fast, because the neighborhood would seem to

them just as unpleasant.

Let us now apply similar considerations to the sun’s gravitation. The

“force” exerted by the sun only differs from that exerted by the tiger

in being attractive instead of repulsive. Instead of acting through

waves of light or sound, the sun acquires its apparent power through

the fact that there are modifications of space-time all round the sun.

Like the noise of the tiger, they are more intense near their source;

as we travel away they grow less and less. To say that the sun “causes”

these modifications of space-time is to add nothing to our knowledge.

What we know is that the modifications proceed according to a certain

rule, and that they are grouped symmetrically about the sun as center.

The language of cause and effect adds only a number of quite irrelevant

imaginings, connected with will, muscular tension, and such matters.

What we can more or less ascertain is merely the formula according to

which space-time is modified by the presence of gravitating matter.

More correctly: we can ascertain what kind of space-time _is_ the

presence of gravitating matter. When space-time is not accurately

Euclidean in a certain region, but has a non-Euclidean character which

grows more and more marked as we approach a certain center, and when,

further, the departure from Euclid obeys a certain law, we describe

this state of affairs briefly by saying that there is gravitating

matter at the center. But this is only a compendious account of what

we know. What we know is about the places where the gravitating matter

is _not_, not about the place where it is. The language of cause

and effect (of which “force” is a particular case) is thus merely

a convenient shorthand for certain purposes; it does not represent

anything that is genuinely to be found in the physical world.

And how about matter? Is matter also no more than a convenient

shorthand? This question, however, being a large one, demands a

separate chapter.

CHAPTER XIV: WHAT IS MATTER?

The question “What is matter?” is of the kind that is asked by

metaphysicians, and answered in vast books of incredible obscurity.

But I am not asking the question as metaphysician: I am asking it as a

person who wants to find out what is the moral of modern physics, and

more especially of the theory of relativity. It is obvious from what we

have learned of that theory that matter cannot be conceived quite as it

used to be. I think we can now say more or less what the new conception

must be.

There were two traditional conceptions of matter, both of which have

had advocates ever since scientific speculation began. There were

the atomists, who thought that matter consisted of tiny lumps which

could never be divided; these were supposed to hit each other and then

bounce off in various ways. After Newton, they were no longer supposed

actually to come into contact with each other, but to attract and

repel each other, and move in orbits round each other. Then there

were those who thought that there is matter of some kind everywhere,

and that a true vacuum is impossible. Descartes held this view, and

attributed the motions of the planets to vortices in the ether. The

Newtonian theory of gravitation caused the view that there is matter

everywhere to fall into discredit, the more so as light was thought by

Newton and his disciples to be due to actual particles traveling from

the source of the light. But when this view of light was disproved, and

it was shown that light consisted of waves, the ether was revived so

that there should be something to undulate. The ether became still more

respectable when it was found to play the same part in electromagnetic

phenomena as in the propagation of light. It was even hoped that atoms

might turn out to be a mode of motion of the ether. At this stage, the

atomic view of matter was, on the whole, getting the worst of it.

Leaving relativity aside for the moment, modern physics has provided

proof of the atomic structure of ordinary matter, while not disproving

the arguments in favor of the ether, to which no such structure is

attributed. The result was a sort of compromise between the two views,

the one applying to what was called “gross” matter, the other to the

ether. There can be no doubt about electrons and protons, though, as we

shall see shortly, they need not be conceived as atoms were conceived

traditionally. As for the ether, its status is very curious: many

physicists still maintain that, without it, the propagation of light

and other electromagnetic waves would be inconceivable, but except in

this way it is difficult to see what purpose it serves. The truth is,

I think, that relativity demands the abandonment of the old conception

of “matter,” which is infected by the metaphysics associated with

“substance,” and represents a point of view not really necessary in

dealing with phenomena. This is what we must now investigate.

In the old view, a piece of matter was something which survived all

through time, while never being at more than one place at a given time.

This way of looking at things is obviously connected with the complete

separation of space and time in which people formerly believed. When we

substitute space-time for space and time, we shall naturally expect to

derive the physical world from constituents which are as limited in

time as in space. Such constituents are what we call “events.” An event

does not persist and move, like the traditional piece of matter; it

merely exists for its little moment and then ceases. A piece of matter

will thus be resolved into a series of events. Just as, in the old

view, an extended body was composed of a number of particles, so, now,

each particle, being extended in time, must be regarded as composed

of what we may call “event-particles.” The whole series of these

events makes up the whole history of the particle, and the particle is

regarded as _being_ its history, not some metaphysical entity to which

the events happen. This view is rendered necessary by the fact that

relativity compels us to place time and space more on a level than they

were in the older physics.

This abstract requirement must be brought into relation with the known

facts of the physical world. Now what are the known facts? Let us

take it as conceded that light consists of waves traveling with the

received velocity. We then know a great deal about what goes on in

the parts of space-time where there is no matter; we know, that is to

say, that there are periodic occurrences (light waves) obeying certain

laws. These light waves start from atoms, and the modern theory of

the structure of the atoms enables us to know a great deal about the

circumstances under which they start, and the reasons which determine

their wave lengths. We can find out not only how one light wave

travels, but how its source moves relatively to ourselves. But when I

say this I am assuming that we can recognise a source of light as the

same at two slightly different times. This is, however, the very thing

which had to be investigated.

We saw, in the preceding chapter, how a group of connected events can

be formed, all related to each other by a law, and all ranged about a

center in space-time. Such a group of events will be the arrival, at

various places, of the light waves emitted by a brief flash of light.

We do not need to suppose that anything particular is happening at the

center; certainly we do not need to suppose that we know _what_ is

happening there. What we know is that, as a matter of geometry, the

group of events in question are ranged about a center, like widening

ripples on a pool when a fly has touched it. We can hypothetically

invent an occurrence which is to have happened at the center, and set

forth the laws by which the consequent disturbance is transmitted. This

hypothetical occurrence will then appear to common sense as the “cause”

of the disturbance. It will also count as one event in the biography of

the particle of matter which is supposed to occupy the center of the

disturbance.

Now we find not only that one light wave travels outward from a center

according to a certain law, but also that, in general, it is followed

by other closely similar light waves. The sun, for example, does not

change its appearance suddenly; even if a cloud passes across it during

a high wind, the transition is gradual, though swift. In this way a

group of occurrences connected with a center at one point of space-time

is brought into relation with other very similar groups whose centers

are at neighboring points of space-time. For each of these other groups

common sense invents similar hypothetical occurrences to occupy their

centers, and says that all these hypothetical occurrences are part of

one history; that is to say, it invents a hypothetical “particle” to

which the hypothetical occurrences are to have occurred. It is only by

this double use of hypothesis, perfectly unnecessary in each case, that

we arrive at anything that can be called “matter” in the old sense of

the word.

If we are to avoid unnecessary hypotheses, we shall say that an

electron at a given moment is the various disturbances in the

surrounding medium which, in ordinary language, would be said to be

“caused” by it. But we shall not take these disturbances at what is,

for us, the moment in question, since that would make them depend upon

the observer; we shall instead travel outward from the electron with

the velocity of light, and take the disturbance we find in each place

as we reach it. The closely similar set of disturbances, with very

nearly the same center, which is found existing slightly earlier or

slightly later, will be defined as _being_ the electron at a slightly

earlier or slightly later moment. In this way, we preserve all the

laws of physics, without having recourse to unnecessary hypotheses or

inferred entities, and we remain in harmony with the general principle

of economy which has enabled the theory of relativity to clear away so

much useless lumber.

Common sense imagines that when it sees a table it sees a table. This

is a gross delusion. When common sense sees a table, certain light

waves reach its eyes, and these are of a sort which, in its previous

experience, has been associated with certain sensations of touch, as

well as with other people’s testimony that they also saw the table.

But none of this ever brought us to the table itself. The light waves

caused occurrences in our eyes, and these caused occurrences in the

optic nerve, and these in turn caused occurrences in the brain. Any one

of these, happening without the usual preliminaries, would have caused

us to have the sensations we call “seeing the table,” even if there had

been no table. (Of course, if matter in general is to be interpreted

as a group of occurrences, this must apply also to the eye, the optic

nerve, and the brain.) As to the sense of touch when we press the table

with our fingers, that is an electric disturbance in the electrons and

protons of our finger tips, produced, according to modern physics, by

the proximity of the electrons and protons in the table. If the same

disturbance in our finger tips arose in any other way, we should have

the same sensations, in spite of there being no table. The testimony

of others is obviously a second-hand affair. A witness in a law court,

if asked whether he had seen some occurrence, would not be allowed to

reply that he believed so because of the testimony of others to that

effect. In any case, testimony consists of sound waves and demands

psychological as well as physical interpretation; its connection with

the object is therefore very indirect. For all these reasons, when

we say that a man “sees a table,” we use a highly abbreviated form

of expression, concealing complicated and difficult inferences, the

validity of which may well be open to question.

But we are in danger of becoming entangled in psychological questions,

which we must avoid if we can. Let us therefore return to the purely

physical point of view.

What I wish to suggest may be put as follows. Everything that occurs

elsewhere, owing to the existence of an electron, can be explored

experimentally, at least in theory, unless it occurs in certain

concealed ways. But what occurs within the electron (if anything occurs

there) it is absolutely impossible to know: there is no conceivable

apparatus by which we could obtain even a glimpse of it. An electron is

known by its “effects.” But the word “effects” belongs to a view of

causation which will not fit modern physics, and in particular will

not fit relativity. All that we have a right to say is that certain

groups of occurrences happen together, that is to say, in neighboring

parts of space-time. A given observer will regard one member of the

group as earlier than the other, but another observer may judge the

time order differently. And even when the time order is the same for

all observers, all that we really have is a connection between the two

events, which works equally backwards and forwards. It is not true that

the past determines the future in some sense other than that in which

the future determines the past: the apparent difference is only due to

our ignorance, because we know less about the future than about the

past. This is a mere accident: there might be beings who would remember

the future and have to infer the past. The feelings of such beings

in these matters would be the exact opposite of our own, but no more

fallacious.

The moral of this is that, if an electron is only known by its

“effects,” there is no reason to suppose that anything exists except

the “effects.” In so far as these “effects” consist of light waves

and other electromagnetic disturbances, we may say that what is

called “empty space” consists of regions where these disturbances are

propagated freely. Every such disturbance, we find, has a center, and

when we get very near the center (though still at a finite distance

from it) we find that the law of propagation of the disturbance ceases

to be valid. This region within which the law does not hold is called

“matter”; it will be an electron or proton according to circumstances.

The region so defined is found to move relatively to other such

regions, and its movements follow the known laws of dynamics. So far,

this theory provides for electromagnetic phenomena and the motions of

matter; and it does so without assuming that “matter” is anything but

systems of electromagnetic phenomena. In order to carry out the theory

fully, it would no doubt be necessary to introduce many complications.

But it seems fairly clear that all the facts and laws of physics

can be interpreted without assuming that “matter” is anything more

than groups of events, each event being of the sort which we should

naturally regard as “caused” by the matter in question. This does not

involve any change in the symbols or formulæ of physics: it is merely

a question of interpretation of the symbols.

This latitude in interpretation is a characteristic of mathematical

physics. What we know is certain very abstract logical relations,

which we express in mathematical formulæ; we know also that, at

certain points, we arrive at results which are capable of being tested

experimentally. Take, for example, the eclipse observations by which

Einstein’s theory as to the bending of light was established. The

actual observation consisted in the careful measurement of certain

distances on certain photographic plates. The formulæ which were to

be verified were concerned with the course of light in passing near

the sun. Although the part of these formulæ which gives the observed

result must always be interpreted in the same way, the other part of

them may be capable of a great variety of interpretations. The formulæ

giving the motions of the planets are almost exactly the same in

Einstein’s theory as in Newton’s, but the meaning of the formulæ is

quite different. It may be said generally that, in the mathematical

treatment of nature, we can be far more certain that our formulæ are

approximately correct than we can be as to the correctness of this or

that interpretation of them. And so in the case with which this chapter

is concerned: the question as to the nature of an electron or a proton

is by no means answered when we know all that mathematical physics has

to say as to the laws of its motion and the laws of its interaction

with the environment. A definite and conclusive answer to our question

is not possible just because a variety of answers are compatible with

the truth of mathematical physics. Nevertheless some answers are

preferable to others, because some have a greater probability in their

favor. We have been seeking, in this chapter, to define matter so that

there _must_ be such a thing if the formulæ of physics are true. If we

had made our definition such as to secure that a particle of matter

should be what one thinks of as substantial, a hard, definite lump, we

should not have been _sure_ that any such thing exists. That is why

our definition, though it may seem complicated, is preferable from the

point of view of logical economy and scientific caution.

CHAPTER XV: PHILOSOPHICAL CONSEQUENCES

The philosophical consequences of relativity are neither so great nor

so startling as is sometimes thought. It throws very little light on

time-honored controversies, such as that between realism and idealism.

Some people think that it supports Kant’s view that space and time are

“subjective” and are “forms of intuition.” I think such people have

been misled by the way in which writers on relativity speak of “the

observer.” It is natural to suppose that the observer is a human being,

or at least a mind; but he is just as likely to be a photographic

plate or a clock. That is to say, the odd results as to the difference

between one “point of view” and another are concerned with “point of

view” in a sense applicable to physical instruments just as much as to

people with perceptions. The “subjectivity” concerned in the theory of

relativity is a _physical_ subjectivity, which would exist equally if

there were no such things as minds or senses in the world.

Moreover, it is a strictly limited subjectivity. The theory does

not say that _everything_ is relative; on the contrary, it gives a

technique for distinguishing what is relative from what belongs to a

physical occurrence in its own right. If we are going to say that the

theory supports Kant about space and time, we shall have to say that it

refutes him about space-time. In my view, neither statement is correct.

I see no reason why, on such issues, philosophers should not all stick

to the views they previously held. There were no conclusive arguments

on either side before, and there are none now; to hold either view

shows a dogmatic rather than a scientific temper.

Nevertheless, when the ideas involved in Einstein’s work have become

familiar, as they will when they are taught in schools, certain changes

in our habits of thought are likely to result, and to have great

importance in the long run.

One thing which emerges is that physics tells us much less about

the physical world than we thought it did. Almost all the “great

principles” of traditional physics turn out to be like the “great

law” that there are always three feet to a yard; others turn out to

be downright false. The conservation of mass may serve to illustrate

both these misfortunes to which a “law” is liable. Mass used to be

defined as “quantity of matter,” and as far as experiment showed it

was never increased or diminished. But with the greater accuracy of

modern measurements, curious things were found to happen. In the first

place, the mass as measured was found to increase with the velocity;

this kind of mass was found to be really the same thing as energy. This

kind of mass is not constant for a given body, but the total amount of

it in the universe is conserved, or at least obeys a law very closely

analogous to conservation. This law itself, however, is to be regarded

as a truism, of the nature of the “law” that there are three feet to a

yard; it results from our methods of measurement, and does not express

a genuine property of matter. The other kind of mass, which we may call

“proper mass,” is that which is found to be the mass by an observer

moving with the body. This is the ordinary terrestrial case, where

the body we are weighing is not flying through the air. The “proper

mass” of a body is very nearly constant, but not quite, and the total

amount of “proper mass” in the world is not quite constant. One would

suppose that if you have four one-pound weights, and you put them all

together into the scales, they will together weigh four pounds. This is

a fond delusion: they weigh rather less, though not enough less to be

discovered by even the most careful measurements. In the case of four

hydrogen atoms, however, when they are put together to make one helium

atom, the defect is noticeable; the helium atom weighs measurably less

than four separate hydrogen atoms.

Broadly speaking, traditional physics has collapsed into two portions,

truisms and geography. There are, however, newer portions of physics,

such as the theory of quanta, which do not come under this head, but

appear to give genuine knowledge of laws reached by experiment.

The world which the theory of relativity presents to our imagination

is not so much a world of “things” in “motion” as a world of _events_.

It is true that there are still electrons and protons which persist,

but these (as we saw in the preceding chapter) are really to be

conceived as strings of connected events, like the successive notes

of a song. It is _events_ that are the stuff of relativity physics.

Between two events which are not too remote from each other there is,

in the general theory as in the special theory, a measurable relation

called “interval,” which appears to be the physical reality of which

lapse of time and distance in space are two more or less confused

representations. Between two distant events, there is not any one

definite interval. But there is one way of moving from one event to

another which makes the sum of all the little intervals along the route

greater than by any other route. This route is called a “geodesic,” and

it is the route which a body will choose if left to itself.

The whole of relativity physics is a much more step-by-step matter than

the physics and geometry of former days. Euclid’s straight lines have

to be replaced by light rays, which do not quite come up to Euclid’s

standard of straightness when they pass near the sun or any other very

heavy body. The sum of the angles of a triangle is still thought to be

two right angles in very remote regions of empty space, but not where

there is matter in the neighborhood. We, who cannot leave the earth,

are incapable of reaching a place where Euclid is true. Propositions

which used to be proved by reasoning have now become either

conventions, or merely approximate truths verified by observation.

It is a curious fact—of which relativity is not the only

illustration—that, as reasoning improves, its claims to the power of

proving facts grow less and less. Logic used to be thought to teach

us how to draw inferences; now, it teaches us rather how not to draw

inferences. Animals and children are terribly prone to inference: a

horse is surprised beyond measure if you take an unusual turning. When

men began to reason, they tried to justify the inferences that they

had drawn unthinkingly in earlier days. A great deal of bad philosophy

and bad science resulted from this propensity. “Great principles,”

such as the “uniformity of nature,” the “law of universal causation,”

and so on, are attempts to bolster up our belief that what has often

happened before will happen again, which is no better founded than the

horse’s belief that you will take the turning you usually take. It is

not altogether easy to see what is to replace these pseudo-principles

in the practice of science; but perhaps the theory of relativity gives

us a glimpse of the kind of thing we may expect. Causation, in the

old sense, no longer has a place in theoretical physics. There is,

of course, something else which takes its place, but the substitute

appears to have a better empirical foundation than the old principle

which it has superseded.

The collapse of the notion of one all-embracing time, in which all

events throughout the universe can be dated, must in the long run

affect our views as to cause and effect, evolution, and many other

matters. For instance, the question whether, on the whole, there is

progress in the universe, may depend upon our choice of a measure of

time. If we choose one out of a number of equally good clocks, we may

find that the universe is progressing as fast as the most optimistic

American thinks it is; if we choose another equally good clock, we may

find that the universe is going from bad to worse as fast as the most

melancholy Slav could imagine. Thus optimism and pessimism are neither

true nor false, but depend upon the choice of clocks.

The effect of this upon a certain type of emotion is devastating. The

poet speaks of

One far-off divine event

To which the whole creation moves.

But if the event is sufficiently far off, and the creation moves

sufficiently quickly, some parts will judge that the event has already

happened, while others will judge that it is still in the future. This

spoils the poetry. The second line ought to be:

To which some parts of the creation move,

while others move away from it.

But this won’t do. I suggest that an emotion which can be destroyed by

a little mathematics is neither very genuine nor very valuable. But

this line of argument would lead to a criticism of the Victorian Age,

which lies outside my theme.

What we know about the physical world, I repeat, is much more

abstract, than was formerly supposed. Between bodies there are

occurrences, such as light waves; of the _laws_ of these occurrences,

we know something—just so much as can be expressed in mathematical

formulæ—but of their _nature_ we know nothing. Of the bodies

themselves, as we saw in the preceding chapter, we know so little

that we cannot even be sure that they are anything: they _may_ be

merely groups of events in other places, those events which we should

naturally regard as their effects. We naturally interpret the world

pictorially; that is to say, we imagine that what goes on is more or

less like what we see. But in fact this likeness can only extend to

certain formal logical properties expressing structure, so that all we

can know is certain general characteristics of its changes. Perhaps an

illustration may make the matter clear. Between a piece of orchestral

music as played, and the same piece of music as printed in the score,

there is a certain resemblance, which may be described as a resemblance

in structure. The resemblance is of such a sort that, when you know the

rules, you can infer the music from the score or the score from the

music. But suppose you had been stone deaf from birth, but had lived

among musical people. You could understand, if you had learned to speak

and to do lip-reading, that the musical scores represented something

quite different from themselves in intrinsic quality, though similar in

structure.[16] The value of music would be completely unimaginable to

you, but you could infer all its mathematical characteristics, since

they are the same as those of the score. Now our knowledge of nature is

something like this. We can read the scores, and infer just so much as

our stone-deaf person could have inferred about music. But we have not

the advantages which he derived from association with musical people.

We cannot know whether the music represented by the scores is beautiful

or hideous; perhaps, in the last analysis, we cannot be quite sure that

the scores represent anything but themselves. But this is a doubt which

the physicist, in his professional capacity, cannot permit himself to

entertain.

[16] For the definition of “structure,” see the present author’s

_Introduction to Mathematical Philosophy_.

Assuming the utmost that can be claimed for physics, it does not tell

us what it is that changes, or what are its various states; it only

tells us such things as that changes follow each other periodically,

or spread with a certain speed. Even now we are probably not at the

end of the process of stripping away what is merely imagination, in

order to reach the core of true scientific knowledge. The theory of

relativity has accomplished a very great deal in this respect, and in

doing so has taken us nearer and nearer to bare structure, which is

the mathematician’s goal—not because it is the only thing in which he

is interested as a human being, but because it is the only thing that

he can express in mathematical formulæ. But far as we have traveled in

the direction of abstraction, it may be that we shall have to travel

further still.

In the preceding chapter, I suggested what may be called a minimum

definition of matter, that is to say, one in which matter has, so

to speak, as little “substance” as is compatible with the truth of

physics. In adopting a definition of this kind, we are playing for

safety: our tenuous matter will exist, even if something more beefy

also exists. We tried to make our definition of matter, like Isabella’s

gruel in Jane Austen, “thin, but not too thin.” We shall, however, fall

into error if we assert positively that matter is nothing more than

this. Leibniz thought that a piece of matter is really a colony of

souls. There is nothing to show that he was wrong, though there is also

nothing to show that he was right: we know no more about it either way

than we do about the flora and fauna of Mars.

To the non-mathematical mind, the abstract character of our physical

knowledge may seem unsatisfactory. From an artistic or imaginative

point of view, it is perhaps regrettable, but from a practical point

of view it is of no consequence. Abstraction, difficult as it is, is

the source of practical power. A financier, whose dealings with the

world are more abstract than those of any other “practical” man, is

also more powerful than any other practical man. He can deal in wheat

or cotton without needing ever to have seen either: all he needs to

know is whether they will go up or down. This is abstract mathematical

knowledge, at least as compared to the knowledge of the agriculturist.

Similarly the physicist, who knows nothing of matter except certain

laws of its movements, nevertheless knows enough to enable him to

manipulate it. After working through whole strings of equations, in

which the symbols stand for things whose intrinsic nature can never be

known to us, he arrives at last at a result which can be interpreted

in terms of our own perceptions, and utilized to bring about desired

effects in our own lives. What we know about matter, abstract and

schematic as it is, is enough, in principle, to tell us the rules

according to which it produces perceptions and feelings in ourselves;

and it is upon these rules that the _practical_ uses of physics depend.

The final conclusion is that we know very little, and yet it is

astonishing that we know so much, and still more astonishing that so

little knowledge can give us so much power.

THE END