The Philosophy Of Science

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THE PHILOSOPHY OF SCIENCE

By the same Author

THE PLACE OF REASON IN ETHICS

(Cambridge University Press)

THE PHILOSOPHY OF
SCIENCE

An Introduction

by
STEPHEN TOULMIN, M.A., PH.D.

LECTURER IN THE PHILOSOPHY OF SCIENCE
UNIVERSITY OF OXFORD

HUTCHINSON'S UNIVERSITY LIBRARY

Hutchinson House, London, W.i.
New York Toronto Melbourne Sydney Cape Town

First Published - 1953

Printed in Great Britain by

William Brendan and Son, Ltd.

The Mayflower Press (late of Plymouth)

at Bushey Mill Lane

Watford, Herts.

CONTENTS

Preface Page vii

Chapter i Introductory 9

II Discovery , 17

in Laws of Nature 57

iv Theories and Maps 105

v Uniformity and^Deteninism 140

Suggested Reading 171

Index 173

PREFACE

SCIENCE and philosophy meet at innumerable points, and
are related in countless ways. The philosophy of science has,
accordingly, been taken to cover a wide variety of things,
ranging from a branch of symbolic logic to the propagation of
secularist gospels. Writing a brief introduction to such an
amorphous subject is a task of some delicacy, since, in order to
avoid being completely superficial, one is forced to limit one's
field of attention, and so to set up landmarks where at present
none are to be found. In making my own selection, I have
particularly kept in mind the audience for which this series is
intended: the topics chosen and the manner of treatment are
primarily designed to meet the needs of University students in
philosophy, and assume no special knowledge either of
mathematics or of natural science. At the same time, I hope that
the book will have its interest for the general reader.

The knot of problems on which I have'concentrated seems
to me to underlie the whole range of topics constituting "the
philosophy of science": without some understanding of these
issues one can, for instance, neither assess the relevance of
mathematical logic to the sciences, nor appreciate the true
status of those "religions without revelation" sometimes
built upon them.

At any rate, I have tried wherever possible to deal with the
problems the layman finds puzzling when he reads about the
exact sciences.

I owe a special debt to the late Professor Ludwig Wittgen-
stein, and to Professor W. H. Watson, whose book On Under-
standing Physics I have found a continual stimulus. Others
whose ideas I have adopted from time to time without specific
acknowledgement include J. J. C. Smart, D. Taylor and John
Wisdom. Professor H. J. Paton and Professor Gilbert Ryle
have read the completed book and made valuable suggestions,
which I have in most cases adopted. If other friends with

vii

Vlll PREFACE

whom I have talked over the problems here discussed recognize
their own ideas in the text, I hope they will forgive me for
borrowing them, and take the credit themselves.

S.E.T.
October 1952

CHAPTER I

INTRODUCTORY

NOT everyone can be an expert physicist, but everybody likes
to have a general grasp of physical ideas. The learned journals
and treatises which record the progress of the physical sciences
are open only to trained readers the Proceedings of the Royal
Society are less readable nowadays than they were in the Royal
Society's early days, when Pepys, Dryden and Evelyn were
Fellows. In consequence, there have grown up two classes of
writings, less needed in^those days, on which the non-scientific
reader has to rely for his understanding of the physical sciences.
For the ordinary man, there are works of popular science, in
which the theoretical advances in physics are explained in a
way designed to avoid technicalities; and for students of
philosophy there are, in addition, books and articles on logic,
in which the nature and problems of the physical sciences are
discussed under the heading 'Induction and Scientific Method*.
There are, however, certain important questions which
both these classes of work leave undiscussed; and, as a result,
the defenceless reader tends to get from them a distorted
picture of the aims, methods and achievements of the physical
sciences. These are questions for which the phrase 'the
philosophy of science* has come to be used: it is the task of
this book to draw attention to them, to show in part at least
how they are to be answered, and to indicate the kinds of mis-
conception which have been generated in the past by leaving
them unconsidered.

1.1 Logic and the physical sciences

Notice first the topics one finds discussed in books of logic.
Induction, Causality, whether the results of the sciences are true
or only highly probable, the Uniformity of Nature, the accumu-
lation of confirming instances, Mill's Methods and the prob-

9

10 THE PHILOSOPHY OF SCIENCE

ability-calculus: such things form the staple of most expositions.
But to anyone with practical experience of the physical sciences
there is a curious air of unreality about the results. Lucid,
erudite and carefully argued they may be; yet somehow they
seem to miss the mark. It is not that the things that are said
are untrue or fallacious, but rather that they are irrelevant:
the questions which are so impeccably discussed have no bear-
ing on physics. Meanwhile the actual methods of argument
physical scientists employ are only rarely examined. French
writers on the philosophy of science, Poincare for instance,
at any rate recognize that in this field one must not take too
much for granted. English and American writers on the subject
tend nowadays, by contrast, to set off on their work assuming
that we are all familiar with the things that scientists say and do,
and can therefore get on to the really interesting philosophical
points that follow.

This attitude exposes one to serious dangers. For if one
has too simple an idea of what scientific arguments are like,
one may regard as serious philosophical problems questions
having no application to the practice of physicists at all. If one
takes it for granted, for instance, that laws of nature can be
classed for logical purposes with generalizations like "Women
are bad drivers" and "Ravens are black", one may conclude
that all appeal to such laws must rest on some presupposition
about the reliability of generalizations. But unless one sees in
some detail what the status of laws of nature in practice is,
one cannot decide whether this is a proper conclusion or no.
In fact, laws of nature will not easily fit into the traditional
array of logical categories, and their discussion calls for a more
refined logical classification. Similarly, one can continue to
write about 'Causation and its Place in Modern Science*
indefinitely, if one fails to notice how rarely the word 'cause 1
appears in the writings of professional scientists. Yet there are
good reasons for this rarity, and to ignore them is again to
divorce the philosophical discussion of scientific arguments
from the reality.

The student of philosophy therefore needs an introductory
guide to the types of argument and method scientists in actual

INTRODUCTORY 11

practice employ: in particular, he needs to know how far
these arguments and methods are like those which logicians
have traditionally considered. How far do the problems the
logic books discuss have any bearing on the things working
scientists do? Do we want to attack these problems in the
customary fashion, and attempt to propound some novel
solution; or should we rather see the problems themselves
as arising from an over-naive conception of what the sciences
are like? How do physicists in fact decide that an explanation
is acceptable? What sort of job must an expression perform to
qualify for the title of 'law of nature' ; and how do laws of nature
differ from hypotheses? Is the difference a matter of our degrees
of confidence in the two classes of propositions, or is the dis-
tinction drawn on other grounds? Again, how does mathematics
come to play so large a part in the physical sciences? And as
for those new entities scientists talk so much about genes,
electrons, meson fields and so on how far are they thought of
as really existing, and how far as mere explanatory devices?
These are all questions about whose answers it is easy to be
mistaken, unless one pays sufficient attention to the actual
practice of scientists: one aim of what follows will be to present
those features of the physical sciences which must be under-
stood before we can settle such questions.

1.2 Popular physics and the layman

The difficulties that arise over books on popular science are
rather different. Here there is no doubt that authentic science
is being discussed; but the terms in which it is presented are
not as explanatory as they at first seem. There is a tendency for
a writer in this field to tell us only about the models and con-
ceptions employed in a novel theory; instead of first giving us a
firm anchor in the facts which the theory explains, and after-
wards showing us in what manner the theory fits these facts.
The best the layman can then hope for is a misleadingly
unbalanced picture of the theory; while, at worst, he is liable to
put the book down more mystified than when he began it.

Recall, for instance, the way in which Sir James Jeans
and Sir Arthur Eddington set about popularizing the theories

12 THE PHILOSOPHY OF SCIENCE

of modern physics. Too often they did what was compara-
tively inessential, that is, introduced us to the particular
conceptions and models used in the theories, while failing to
do what is essential, namely, explain in detail the function of
these models, theoretical conceptions and the rest. Eddington's
well-known account of 'the two tables' is a case in point: to
be told that there is not only a common-sense, solid table,
but also a scientific one, mostly consisting of empty space,
does not particularly help one to understand the atomic
theory of matter. The whole reason for accepting the atomic
model is that it helps us to explain things we could not explain
before. Cut off from these phenomena, the model can only
mislead, raising unreal and needless fears about what will
happen when we put the tea-tray down. The same also goes,
regrettably, for many of those pretty pictures which captured
our imaginations: the picture of the electrons in an atom as
like bees in a cathedral, the picture of the brain as a telephone
exchange, and the rest. Regrettably, it can be said, because as
literary devices they certainly have a value and, if they were
not left to stand on their own feet, might genuinely help us to
understand. As things are, however, they act like a searchlight
in the darkness, which picks up here a pinnacle, here a chimney,
and there an attic window: the detail it catches is lit up dazzlingly,
but everything around is thrown into even greater obscurity
and we lose all sense of the proportions of the building.

But this is not the worst that happens. At times the attempt
to popularize a physical theory may even end by unpopular-
izing it. Jeans, for instance, relied on finding a happy analogy
which would by itself bring home to his readers the chief
features of the General Theory of Relativity. And how did
he invite them to think of the Universe? As the three-dimensional
surface of a four-dimensional balloon. The poor layman, who
had been brought up to use the word 'surface' for two-
dimensional things alone, now found himself instructed to
visualize what for him was a self-contradiction, so it was no
wonder if he agreed to Jeans' calling the Universe -a mysterious
one. This mystification was also unnecessary. There is no reason
why the principles of the. Theory of Relativity should not be

INTRODUCTORY 13

explained in terms the ordinary reader can make something
of Einstein himself does this very well. But Jeans' method
defeated its own end: by trying to make the subject too easy
and to do with a simile what no simile alone can do, he led
many readers to conclude that the whole thing was utterly
incomprehensible, and so must be put aside as not for them.

This might suggest that Jeans was just careless, but there
is more to it than that. For the fact that he picked on a mode of
expression which to the outsider is self-contradictory points
to something which the layman needs to be told about the
language of physical theories. When a theory is developed, all
kinds of phrases which in ordinary life are devoid of meaning
are given a use, many familiar terms acquire fresh meanings,
and a variety of new terms is introduced to serve the purposes
of the theory. A scientist, who learns his physics the hard way,
gradually becomes accustomed to using the novel technical
terms and the everyday-sounding phrases in the way required ;
but he may only be half-aware of what is happening as Pro-
fessor Born remarks, the building of the language of the sciences
is not entirely a conscious process. This has its consequences
when the scientist comes to explain some new theory to the
layman. For then he may unwittingly use in his exposition
terms and turns of phrase which can be understood properly
only by someone already familiar with the theory. To a man
trained in the use of sophisticated kinds of geometry the phrase
'three-dimensional surface' may no longer be a self-contra-
diction, but for him to use it in talking to a non-mathematician
is to invite incomprehension. And what applies to 'three-
dimensional surfaces' applies equally to 'invisible light' and
the like: when scientific notions are being popularized, it is
necessary to explain the point of such phrases, instead of mak-
ing an unexplained use of them.

To introduce a distinction we shall find important later:
the adoption of a new theory involves a language-shift, and one
can distinguish between an account of the theory in the new
terminology in 'participant's language' and an account in
which the new terminology is not used but described an
account in 'onlooker's language'. 'Suppose', as Wittgenstein

14 THE PHILOSOPHY OF SCIENCE

once said, 'that a physicist tells you that he has at last discovered
how to see what people look like in the dark, which no one had
ever before known. Then you should not be surprised. If he
goes on to explain to you that he has discovered how to photo-
graph by infra-red rays, then you have a right to be surprised
if you feel like it. But then it is a different kind of surprise,
not just a mental whirl. Before he reveals to you the discovery
of infra-red photography, you should not just gape at him;
you should say, "I do not know what you mean".'

An analogy will help to explain how misconceptions may
follow if we attempt to popularize the physical sciences in
this way. When we tell children stories at bed-time, we talk
to them about all kinds of people by which is meant not just
rich and poor, white and black, beggars and kings, but logically
different kinds of people. Some nights we tell them stories
from history, other nights ancient myths; sometimes legends,
sometimes fables, sometimes accounts of things that we our-
selves have done, sometimes stories by contemporary authors.
So in bed-time stories Julius Caesar, Hercules, Achilles, the
Boy who cried "Wolf!", Uncle George and Winnie-the-Pooh
all appear, at first sight, on the same footing. A clever child,
no doubt, soon learns to spot from internal evidence what kind
of story tonight's story is; and what sort of people its characters
are fabulous, legendary, or historical. But to begin with we
have to explain, in asides, what the logical status of each
character and story is, saying, "No, there aren't really any talk-
ing bears: this is just a made-up story", or "Yes, this really did
happen, when my father's father was a boy." Unless the child is
told these things in addition to the stories themselves, he may
not know how to take them; and thus he may get quite false
ideas about the world into which he has been born, about its
history, its inhabitants, and the kinds of thing he might encounter
one day as he turned the corner of the street. If entertainment
alone were needed, the story alone might do. But the risks of
misunderstanding are serious, and for real understanding more
is needed.

So also in popular science: the layman is not just ignorant
of the theories of science, but also unequipped to understand

INTRODUCTORY 15

the terms in which a scientist will naturally begin to explain
them. To explain the sciences to him by giving him only
potted theories and vivid analogies, without a good number
of logical asides, is accordingly like telling a child all the sorts
of stories we do tell children and not warning him how very
different they are: he will not know how much weight to put
on the various things that are said, which of the statements
about physics are to be taken at their face value, and which of
the characters in the stories he could ever hope to meet.

Perhaps the nub of the difficulty is this, that the popularizer
has a double aim. For the layman wants to be told about the
theories of the sciences in language he can understand ; and he
also wants to be told about them briefly, 'in a nutshell'. These
two demands are bound in practice to conflict. For a major
virtue of the language of the sciences is its conciseness. It is
always possible to say what a scientific theory amounts to with-
out using the technical terms which scientists introduce to
serve the purposes of the theory, but one can do so only by
talking at very much greater length. If the popularizer is to
explain a theory in everyday terms, and at the same time put
it in a nutshell, something must be sacrificed: usually the logical
asides are the first things to go, and drastic cuts follow in the
account of the phenomena the theory is employed to explain.
Once this has happened, the layman is given no real entrance
to the subject; for unless he is told a good deal about the
phenomena a theory is introduced to explain, and what is even
more important, just how much further on we are when this
'explanation' has been given, he might as well have been left
quite in the dark. Even a real key is of little use if we do not
know what rooms it will let us into. And there is no point at
all in being told that Einstein has discovered the metaphorical
Key to the Universe if we are not also told what sort of thing
counts as opening a door with this Key.

Something can be done, however, to remedy this state of
affairs. With the help of a few elementary examples, it should
be possible to show the common reader some of the more im-
portant things he needs to know about the logic of the physical
sciences. There is no reason why he need rest content with the

16 THE PHILOSOPHY OF SCIENCE

idea that physics is a conglomeration of self-contradictions,
like 'invisible light' and 'three-dimensional surfaces', and
mysteries like 'the curvature of space': armed with the right
questions, he can penetrate behind this screen of words to the
living subject. For the words of scientists are not always what
they seem, and may be misleading taken out of their original
context. The vital thing to know is, what sorts of questions need
to be asked, if one is to get a satisfactory account of a theory;
and this, fortunately, is something which can be shown as
well with simple as with sophisticated examples. To show,
with illustrations, what these questions are is the principal
aim of this book; and it will require us, not so much to quote
the things that scientists say, as to see what sort of things they
do with the words they employ. As Einstein has said, 'If you
want to find out anything from the theoretical physicists about
the methods they use, I advise you to stick closely to one prin-
ciple: don't listen to their words, fix your attention on their
deeds.'

CHAPTER II

DISCOVERY

IF we are to know what questions to ask about physical theories,
we must be clear to begin with what kinds of things count as
discoveries in the physical sciences. What is it for something
to be 'discovered* in physics? When a physicist announces
that it has been discovered that heat is a form of motion, or
that light travels in straight lines, or that X-rays and light-
waves are varieties of electro-magnetic radiation, what
kind of discovery is this? What does such a discovery
amount to?

The question can be put in another way: if, in physics,
someone claims to have discovered something, what sort of
demonstration will justify us in agreeing that, whereas this
was not previously known, it can now be regarded as known?
Is it like that required when an explorer discovers a new
river, or when a botanist discovers a new variety of flower,
or when a doctor discovers what is wrong with a patient, or
when an engineer discovers how to bridge a hitherto-
unbridgeable river, or when a man doing a crossword-puzzle
discovers the word that has been eluding him? Or is it like none
of these?

2. 1 Physics presents new ways of regarding old phenomena

This question will best be answered with the help of
examples. Let us look first at a discovery so elementary that it
may hardly seem nowadays ever to have needed discovering, or
to be anything more than a piece of common sense: the discovery
that light travels in straight lines. This example, for all its
appearance of obviousness, displays many of the features
characteristic of discoveries in the exact sciences. Its very com-
monsensicality is indeed a merit, reminding us how the sciences
grow out of our everyday experience of the world, and what

17

B

18 THE PHILOSOPHY OF SCIENCE

people mean who speak of science, epigrammatically, as
'organized common sense'.

To recognize just what was discovered when it was first
announced that 'light travels in straight lines*, we must think
ourselves back to the way things were before the discovery.
This is not entirely easy to do, for we tend nowadays to grow
up completely familiar with the idea that sunlight, shadows
and the like are the effects of light travelling: it is only by an
effort that one can throw off the habit, and look at optical
phenomena once again with the eyes of those who knew nothing
of geometrical optics, to whom this would be a novel,
revolutionary suggestion. Yet the effort is worth making. So
let us ask, for a start, what would have been the data on which
this discovery was based?

There are three sources of material which we can think of
as the backing for it: first, our experience of everyday phenomena
like those of light and shade ; second, the practical skills and
techniques which have been developed as a result of this ex-
perience; and third, those regularities in optical phenomena
which are not stated but taken for granted and enshrined in
our everyday language. We know very well, for instance, that
the higher the sun rises in the sky, the shorter are the shadows
cast by the objects it illuminates; and that, as it moves across
the sky, so do the shadows turn with it. Out of this knowledge,
and exploiting it, have grown the techniques used in the design
of sundials: the sundial-maker in the course of his trade
develops a familiarity with optical phenomena which provides
a second starting-point for optics. And there is a further range
of physical regularities, with which everyone becomes familiar
at an early age, but which are rarely stated. It is harder work
running uphill than down ; the shortest way to get to the opposite
corner of a field is to 'follow your nose 5 ; put your hand in the
fire and it will burn you these are things which any child,
and many animals too, may be said to know, yet they seem
almost tautologous when put into words; for our recognition
of them comes before, rather than after, the development of
our everyday language. The way we ordinarily use the word
'straight', for instance, takes it for granted that the shortest

DISCOVERY 19

and the straightest road are both the one you can see straight
along; and our manner of using words like 'up* and 'down',
'fire' and 'burns' likewise links together things we commonly
find going together.

The question that faces us is the question, what kind of
step is taken when we pass from these data to the conclusion
that 'light travels in straight lines'. What type of inference is
this? Or is the very word 'inference' a misleading name for
such a step?

Let us, as a preliminary, try setting this step alongside a
couple of inferential steps, which at first sight it resembles.
Robinson Crusoe, we are told, found a footprint on the sandy
beach of his island, and concluded that a man had been walking
there. Again, a naturalist studying the migration of swallows
might find, by plotting the observed tracks of a large number of
flocks, that they all flew along 'great circles'. In these cases, too,
one can speak of discoveries being made, which can be put in
the words "A man has been walking along the beach" and
"Migrating swallows always travel along great circles". Let
us contrast these discoveries with the discovery that "light
travels in straight lines": how does the step from our obser-
vations on shadows to this discovery compare with Crusoe's
step from the footprint to a man walking, and the naturalist's
step from the bird-watcher's reports to his generalization
about migrating swallows?

Two important differences spring to the eye at once.

(i) To compare first the step from shadows to light and the
step from the footprint to a man. One might turn a corner and
come face to face with the man who was responsible for the
footprint this, in fact, was what Robinson Crusoe was terrified
of doing. But telling from our study of shadows that light
travels in straight lines is quite unlike telling from a footprint
that a man has been walking on a beach. To hint at the difference,
there is nothing in the case which would count as 'coming face
to face with' the light which was 'responsible for' the shadows:
no single happening could establish the optical theory once
and for all, in the way Crusoe's conclusion could be established.
For Crusoe reached his novel conclusion by applying a familiar

20 THE PHILOSOPHY OF SCIENCE

type of inference to fresh data: "Footprint! Footprints mean
man. Therefore man." But in geometrical optics it is not the
data which are fresh, for we have known about shadows for
a very long time. The novelty of the conclusion comes, not
from the data, but from the inference: by it we are led to look
at familiar phenomena in a new way, not at new phenomena in
a familiar way.

The discovery that light travels in straight lines was not,
therefore, the discovery that, where previously nothing had
been thought to be, in any ordinary sense, travelling, there
turned out on closer inspection to be something travelling
namely, light: to interpret the optical statement in this way
would be to misunderstand its point. We can call this the 'Man
Friday fallacy'.

(ii) Nor is it the discovery that whatever is travelling,
in the everyday sense, is doing so in one way rather than another,
along great circles rather than parallels of latitude, or straight
lines rather than spirals. Often enough, as we soon find out,
light does not travel strictly in straight lines, but is diffracted,
refracted or scattered; yet, in practice, this in no way affects
the point of the principle that light travels in straight lines (the
Principle of the Rectilinear Propagation of Light). In this
respect the optical discovery is quite unlike the naturalist's
discovery about swallows, which was precisely that they always
migrate thus, and not otherwise. Rather, the optical discovery
is, in part at any rate, the discovery that one can speak at all
profitably of something as travelling in these circumstances,
and find a use for inferences and questions suggested by this
way of talking about optical phenomena the very idea that
one should talk about anything as travelling in such circum-
stances being the real novelty.

These differences are, however, only pointers towards a
larger difference, and this we must now try to state. In Robinson
Crusoe's discovery, and in the naturalist's also, the language
in which the conclusion is expressed, like that in which the
data would be reported, is the familiar language' of everyday
life: there is no question of giving new senses to any of the words
involved, or of using them in a way which is at all out of the

DISCOVERY 21

ordinary. Yet in the optical case, both the key words in our
conclusion 'light* and 'travelling* are given new uses in
the very statement of the discovery. Before the discovery is
made, the word 'light* means to us such things as lamps
the 'light* of "Put out the light**; and illuminated areas the
'light* of "The sunlight on the garden**. Until the discovery,
changes in light and shade, as we ordinarily use the words
(i.e. illuminated regions which move as the sun moves), remain
things primitive, unexplained, to be accepted for what they are.
After the discovery, we see them all as the effects of something,
which we also speak of in a new sense as 'light', travelling from
the sun or lamp to the illuminated objects. A crucial part of the
step we are examining is, then, simply this: coming to think
about shadows and light-patches in a new way, and in con-
sequence coming to ask new questions about them, questions
like "Where from?**, "Wheie to?** and "How fast?*', which
are intelligible only if one thinks of the phenomena in this
new way.

It is worth emphasizing how far the physicist's way of look-
ing at optical phenomena is a new way, and how far by accepting
it we are required to extend the notions of light and travelling.
Until one has been introduced to the fundamental ideas of
geometrical optics, there is no way of understanding what it
means for a physicist to talk of light travelling: he clearly does
not mean 'sending lanterns by rail', nor can he mean 'cloud-
shadows drifting across the grass' for he talks of light travelling
equally whether the patches of light are moving or still. Indeed,
it would be somewhat queer, in the sort of situations with which
the physicist is concerned, to talk in the ordinary sense of the
word of anything 'travelling' at all.

An example will bring this queerness out. Suppose that
we are sitting on a hillside, gazing across the country, and you
ask, "Is anything on the move?": the appropriate answer will
be some such thing as "Clouds and larks overhead, down below
two men on horseback and a wagon of hay, and there in the
distance a railway-train" and this answer may be, from the
everyday point of view, an exhaustive one. Taking your question
in the sense in which it was asked, I could neither give nor you

22 THE PHILOSOPHY OF SCIENCE

accept, as an answer to it, such a reply as "Photons". It is
true that I might say, "Light": but, were I to do so, I could
only be understood to mean, e.g., patches of sunlight moving
across the heather on the far hillside, and this is certainly not
what the physicist means when he speaks of 'light travelling'.
And if I were to answer "Photons", all you could do would be
to wonder whether I did so out of plain misunderstanding ; or
whether, as the expression of a poetical fancy, I was choosing
to borrow a term from physics to suggest, with Heraclitus and
Walt Whitman, that even when so few things are, literally,
on the move, the world still 'teems with flux'. At any rate
and this is all that it is essential to recognize the introduction
of the notion of 'light' as something 'travelling' is not the simple,
literal discovery of something moving, like the detection of
frogs in a flower-bed or boys in an apple tree: rather it is an
extension of the notion of travelling to do a new job in the
service of physics.

Not only is it an extended application of the word: it is
also rather a thin one. Somehow, indeed, the use of the par-
ticular word 'travelling' does not seem to be of central import-
ance. One finds it being used alongside other words which,
from a non-scientist's point of view, are quite incompatible
with it: light will be spoken of in the same book sometimes as
'travelling', but at other times as 'being propagated'. Yet there
is certainly something of central importance about the kind of
word whose meaning it is found natural to extend in this way. 1
So, in answer to the question, "What sort of discovery is this?",
we can already give something of a hint: the discovery that
light travels in straight lines is, in part at least, the discovery
that the phenomena from which we started (shadow-casting
and the rest), can be regarded as consequences of something
(it matters not yet what) travelling, or being propagated, or
something of the kind, from the light-source to the surrounding
objects, except where it is cut off by intervening bodies of the
kind we call 'opaque*.

x The sort of word chosen must reflect such familiar facts as this: that
by lighting a lamp in one corner of a room one can produce patches of light
in another.

DISCOVERY 23

2.2 New points of view come with new inferring techniques

The next question to be asked is this: What does it mean to
say that these phenomena can be regarded in this way? Still
more, what could it mean for a physicist to say, as he might
do, that they must be so regarded? For, as we have seen, to
say this is not like saying that a certain kind of depression in
the sand must be the effect of a man standing on it. Since there
is nothing quite like meeting Man Friday which would oblige
us to accept the new optical theory, how is it that we must do
so? May we not decline to look at the phenomena in this new
way?

Certainly we can. We are not compelled unconditionally
to think of the phenomena in the physicist's way. We can, if
we choose, refrain from asking any scientific questions about
them. If we prefer, we can think of the phenomena of sight
as the Greeks did, regarding the eye not as a kind of sensitive
plate, but as the source of antennae or tentacles which stretch
out and seize on the properties of the objects it surveys. Not
only can we look at it in this way; we quite frequently do so,
or talk as though we did as, for instance, when we speak
of Able Seaman Jones, seated in the crow's-nest, 'sweeping
the horizon' with his eagle eyes. Outside physics, the way we
think and talk about light is not greatly changed by the optical
discovery, nor is there much reason why it should be. Novelists
can continue to write as they would have done before: "As
the first sunbeams lit the snow-capped peaks, and the peach-
coloured glow spread down the mountain-side, chasing away
the shadows and restoring their colour to the sleeping villages
below, Charles awoke with a groan." Nor need our everyday
instructions be rephrased: "Keep this bottle away from strong
light" need not be replaced by "Do not allow light of high
energy-density to be propagated on to this bottle."

Something, however, would be lost if we never did as the
physicist recommends. There is a familiar sense in which we
must accept the new picture of optical phenomena, for certain
of the purposes of physics at any rate. And so far we have not
seen what it is that obliges us to do so.

24 THE PHILOSOPHY OF SCIENCE

To see this, we must examine in more detail how the
Principle of Rectilinear Propagation enters into a physicist's
explanations: only a close examination will show us clearly
where it comes in. For the physicist will say, fairly, that the
reason why we must regard shadows in the way he recommends
is that only in this way can their occurrence and movement
be explained: it is in the service of his explanation that the
principle, and with it the new way of thinking about shadow-
casting and the like, are to be accepted.

Consider, then, a specific situation of the kind in which the
physicist will be interested: notice how he sets about explaining
an optical phenomenon, and in particular where the principle
comes into his account. Suppose therefore that the sun, from
an angle of elevation of 30, is shining directly on to a
six-foot-high wall, casting a shadow ten and a half feet deep
on the level ground behind the wall. Why, we may ask, do we
find that the shadow is just ten and a half feet deep: why not
fifty feet, or two? How are we to explain this fact?

"Well, that's easy enough," the physicist will say. "Light
travels in straight lines, so the depth of the shadow cast by a
wall on which the sun is directly shining depends solely on the
height of the wall and the angle of elevation of the sun. If
the wall is six feet high and the angle of elevation of the
sun is 30, the shadow must be ten and a half feet deep.
In the case described, it just follows from the Principle of the
Rectilinear Propagation of Light that the depth of the shadow
must be what it is."

Now we must not jump to conclusions about the logical
form of this explanation. We must ask, first, how it can be said
to follow from anything that the depth of a shadow must be
just ten feet six and nothing else. What sort of inference, what
sort of following is this? Not a bare inference from one straight-
forward matter-of-fact to a different one, for, as Hume rightly
insisted, there can be no 'must' about any such inference
only a 'usually does'. Not a deduction from a generalization
to an instance either for, considered as a generalization, the
principle is just not true: in diffraction, refraction and scatter-
ing light ceases to travel in straight lines. Further, there is

DISCOVERY

25

nothing in the principle about all shadows being ten feet six
ins. deep, rather than fifty feet or two feet, so the only infer-
ence of a syllogistic kind one could look for would be "All
light travels in straight lines; what we have here is light; so
what we have here travels in a straight line", and this leaves
the substantial step unaccounted for. In any case, if the in-
ference were of a syllogistic kind, it would be open to the objec-
tion that logicians have always said it was, that of circularity
since one would be justified in saying only, "Light always has
travelled in straight lines; what we have here is light; so what
we have here will almost certainly travel in a straight line".
Somehow none of the kinds of inference we are accustomed
to from the logic-books seems to fit the case.

This should not surprise us. The fact of the matter is that
we are faced here with a novel method of drawing physical
inferences one which the writers of books on logic have not
recognized for what it is. The new way of regarding optical
phenomena brings with it a fresh way of drawing inferences
about optical phenomena.

This will become evident if we look and see what a physicist
does when asked to set his explanation out in more detail,
and make its form explicit. For the natural thing for him to
do at this point will be to draw a diagram: in this diagram, the
ground will be represented by a horizontal line, the wall by

- x d-

/ \ "

10' 6"-

30^,

26 THE PHILOSOPHY OF SCIENCE

a vertical line, and a third line will be added at 30 to
the horizontal, touching the top of the line representing the
wall, and intersecting that representing the ground. This dia-
gram plays a logically indispensable part in his explanation.

"Here", says our physicist, pointing to the third line,
"we have the bottom ray of light which can get past the wall
without being cut off. All the lower ones are intercepted, which
explains why the ground behind the wall is in shadow. And if
you measure the depth of the shadow on the diagram, you'll
find that it is one and three-quarter times the height of the
wall: that is to say, if the wall is six feet high, the shadow must
be ten feet six deep."

Given the height of the wall and the sun, the physicist
is in a position to discover by these means what depth the shadow
of the wall will have ; but he is able to do so only because he
accepts the new account of optical phenomena and the infer-
ring techniques that come with it. The view of optical pheno-
mena as consequences of something travelling and the diagram-
drawing techniques of geometrical optics are introduced hand-
in-hand: to say that we must regard light as travelling is to
say that only if we do so can we use these techniques to account
for the phenomena being as they are. Neither the everyday
nor the ancient way of talking and thinking about 'light' and
'sight' will make sense of the geometrical method of represent-
ing optical phenomena. And if the novel techniques of infer-
ence-drawing here used have not been recognized by logicians
for what they are, that is probably because in geometrical
optics one learns to draw inferences, not in verbal terms, but
by drawing lines.

Of course, the fact that our physicist draws his diagram
exactly as we have supposed, or draws any diagram rather
than resorting to trigonometry, may not be important. But
resort to some mathematical symbolism or other representational
device is essential. As for the question, how the physicist's
Principle of Rectilinear Propagation enables him to argue
from the conditions of the phenomenon the height of the wall
and the angle of elevation of the sun to his conclusion about
the depth of the shadow: it does this, in practice, through the

DISCOVERY 27

part it plays in the representation of the phenomenon concerned.
In such a case as this, appeal to the principle means to the
physicist something like the following that the optical
phenomena to be expected in this situation can be represented
and so explained by drawing a straight line at the appropriate
angle to the line representing the wall ; that this line will mark
the boundary between light and shade; and that one can read
off such things as the depth of the shadow from the resulting
diagram, confident that the result will be found to agree with
observation within limits of accuracy greater than at present
interest us.

The particular example here chosen may seem trivial,
especially as we are limiting ourselves for the moment to cir-
cumstances in which there are no complicating phenomena
such as refraction; but the steps we have gone through are
of the very stuff of geometrical optics, and so in miniature
of the exact sciences generally. Two things about it are worth
noticing. First, the importance for physics of such a principle
as that of the Rectilinear Propagation of Light comes from the
fact that, over a wide range of circumstances, it has been found
that one may confidently represent optical phenomena in this
sort of way. The man who comes to understand such a principle
is not just presented with the bare form of words, for these
we have already seen to be on a naive interpretation quite false:
he learns rather what to do when appealing to the principle in
what circumstances and in what manner to draw diagrams or
perform calculations which will account for optical phenomena,
what kind of diagram to draw, or calculation to perform, in
any particular case, and how to read off from it the information
he requires.

Secondly, when a physicist has once drawn such a diagram
of the 'optical state of affairs', he can use it not only to explain
the original phenomenon, namely, the fact that the shadow was
ten feet six ins. deep, but also to answer any number of other
questions. It may, for instance, be asked what depth the shadow
of our wall will have at a height of four feet from the ground.
A horizontal line drawn two-thirds of the way up the line
representing the wall intersects the line representing the 'light-

28

THE PHILOSOPHY OF SCIENCE

ray* three and a half units out: answer, 3 feet 6 ins. Again,
suppose that later in the year the sun shines directly on to the
wall from an angle of 15, instead of 30. What will the depth
of the shadow be then? A fresh line drawn at 15 to the

\ /

-O:

-lO' 6"-

horizontal will cut the ground-line about thirty units from
the wall-line: answer, about thirty feet. There is no limit
to the number of such questions which a single ray-diagram
can be used to answer.

2.3 Inferring techniques and models are the core of discoveries

At this point we can reconsider the question from which we
began: the question what such a discovery as that light travels
in straight lines amounts to. For we can see now that a vital
part of the discovery is the very possibility of drawing 'pictures*
of the optical state-of-affairs to be expected in given circum-
stances or rather, the possibility of drawing them in a way
that fits the facts.

Two things need saying to qualify this statement. To begin
with, it is not necessary that the particular techniques we are
here concerned with should be applicable in all circumstances.
The way shadows fall and move, the patterns of light and shade
cast by lamps, the places from which lights are visible or eclipsed

DISCOVERY 29

it is enough that all these things can be accounted for, over
a wide range of circumstances, in the way we have been study-
ing. If under some circumstances refraction, diffraction and
other such phenomena limit the use of these techniques, or
require them to be supplemented, that does not destroy their
value within the wide region to which they are applicable.
Secondly, what is or is not to count as 'fitting the facts* has to
be decided: there must be standards of accuracy. It can always
be asked to what degree of accuracy a given method of repre-
sentation can be used to account for a set of phenomena ; and
the best that we need demand of a theory is that it should fit
the facts as to high a degree of accuracy as we yet have the means
of measuring.

If these qualifications are borne in mind, we can answer
our original question. The discovery that light travels in straight
lines the transition from the state-of-affairs in which this
was not known to that in which it was known was a double
one: it comprised the development of a technique for represent-
ing optical phenomena which was found to fit a wide range of
facts, and the adoption along with this technique of a new
model, a new way of regarding these phenomena, and of under-
standing why they are as they are.

These are the core of the discovery. Compared with them,
the particular words in which the discovery is expressed are
a superficial matter: whether we speak of light as travelling
or as being propagated is hardly important, for either is an
equally good interpretation of the geometrical picture at this
stage, only so much of each notion matters as is common to
them both. Further, the very notions in terms of which we
state the discovery, and thereafter talk about the phenomena,
draw their life largely from the techniques we employ. The
notion of a light-ray, for instance, has its roots as deeply in the
diagrams which we use to represent optical phenomena as in
the phenomena themselves: one might describe it as our
device for reading the straight lines of our optical diagrams into
the phenomena. We do not find light atomized into individual
rays: we represent it as consisting of such rays.

As for the Principle of the Rectilinear Propagation of

30 THE PHILOSOPHY OF SCIENCE

Light, the doctrine that light travels in straight lines, which
figures in our sample explanation: we are now in a position to
reconsider its status. We saw from the start that it could not
be regarded as an empirical generalization of the kind logicians
have so often discussed, since when so interpreted it is untrue.
By itself, the principle tells us no additional facts over and above
the phenomena it is introduced to explain, and if read as a
factual generalization it would have to be qualified by some
such clause as 'in general' or 'other things being equal' or
'except when it doesn't.' The point of the doctrine is in fact
quite otherwise: its acceptance marks the introduction of the
explanatory techniques which go to make up geometrical
optics, namely, the model of light as something travelling from
the source to the illuminated objects and the use of geometrical
diagrams to infer what phenomena are to be expected in any
given circumstances.

The doctrine is, so to speak, parasitic on these techniques:
separated from them it tells us nothing, and will be either
unintelligible or else misleading. For, as a discovery, it is
opposed neither to the hypothesis that nothing is travelling,
nor to the hypothesis that light is travelling differently in
both of which hypotheses the term 'travelling' must already
have a sense. It is opposed rather to the use of a completely
different model: to our thinking of optical phenomena for
purposes of physics in wholly different terms for instance, in
terms of antennae from the eye seizing on the properties of
the object opposed, that is to say, to ways of thinking about
light such that to talk of light travelling would not even be in
place, ways which would lead us to formulate quite different
questions and hypotheses about optical phenomena, in fact
different kinds of question and hypothesis. As such, one might
almost as well call the principle a 'law of our method of repre-
sentation' as a 'law of nature': its role is to be the keystone
of geometrical optics, holding together the phenomena which
can be explained by that branch of science and the
symbolism which, when interpreted in the way suggested
by the model, is used by physicists to account for these
phenomena.

DISCOVERY 31

2.4 The place of mathematics and of models in physics

How far are the things we have found in this particular
example peculiar to it, and how far are they characteristic of
discovery and explanation in the physical sciences generally?

In many respects the sample will be seen to be representa-
tive, once its extreme simplicity is allowed for. For in every
branch of the physical sciences, the questions we have come to
ask here can be asked again. Each branch is developed in order
to account for a range of physical phenomena, and in each we
can ask about the methods of representation and the models
employed in doing so.

(i) Consider first the phenomena accounted for. In the case
we have looked at, these will be such things as the changes
in the distribution of light and shade as the sun moves across
the sky, the times of eclipses and so on. But, as it stands, the
range of the new principle is bounded. Any one branch of
physics, and more particularly any one theory or law, has only
a limited scope: that is to say, only a limited range of phenomena
can be explained using that theory, and a great deal of what a
physicist must learn in the course of his training is concerned
with the scopes of different theories and laws. It always has
to be remembered that the scope of a law or principle is not
itself written into it, but is something which is learnt by scientists
in coming to understand the theory in which it figures. Indeed,
this scope is something which further research is always liable
to, and continually does modify ; and it is a measure of economy,
apart from anything else, to state theories and laws in a manner
which does not need to be changed whenever a fresh applica-
tion of them is encountered.

(ii) Second, we have to consider the techniques of repre-
sentation employed in the different branches of physics.
In our sample, we are concerned solely with primitive mathe-
matical techniques of a geometrical kind, including constructions
with ruler and pencil and, at the most refined, the use of
trigonometrical tables. It is from these techniques that this
branch of optics gets its name of 'geometrical' optics. In it, we
deal with optical phenomena by the use of geometrical pictures

32 THE PHILOSOPHY OF SCIENCE

pictures in which straight lines represent the paths along
which light is to be thought of as travelling and work out
rules for manipulating the straight lines of our figures so that
they shall reflect as far as possible the observed behaviour of
light, i.e. the optical phenomena concerned.

In some respects, our example is not characteristic, since
the method by which the problems are handled is more than
usually pictorial, giving the physicist what we have in fact
called a 'picture' of the optical state-of-affairs. This vividness
will make the example especially intelligible to the non-
mathematician, but should not be allowed to mislead. For,
though one can speak of this diagram as a picture, it is as well
to remind oneself that such a picture would never appear in
an art exhibition, however representational the tastes of the
Hanging Committee there being more than one kind of
representation. The physicist's diagram is not valued for what
the man-in-the-street would regard as a likeness, since the
physicist's notion of light departs in important respects from
the everyday one: still less is it valued on aesthetic grounds.
Its point is a more prosaic one, that by the use of diagrams
of this kind it has been found possible to show, and so explain,
over a wide range of circumstances and to a high degree of
accuracy, what optical phenomena are to be expected.

Wherever possible, physicists would like to be able to
represent the phenomena they are studying pictorially: where
this is possible, one can 'see' the force of their explanations
in a specially convincing way. For the same reason, geometry
seemed to mathematicians in the seventeenth century to be
superior to algebra: they felt that algebra provided only a short-
cut to truths which geometry displayed. But this can rarely
be done to anything like the degree to which it can in geo-
metrical optics. Only in a very few branches of physics does
the drawing of diagrams play a logically central part. In most
branches the logical role played in geometrical optics by
diagrammatic techniques is taken over by other less primitive
kinds of mathematics; and these are often of a complexity and
sophistication far greater than could ever be handled dia-
grammatically. Yet however sophisticated and complex these

DISCOVERY 33

may become, they play a part comparable to that of picture-
drawing in geometrical optics: they serve, that is to say,
as techniques of inference-drawing. In dynamics, for
instance, the counterparts of our geometrical diagram are the
equations of motion of the system of bodies under investigation.
Given a suitable description of a system, a physicist who has
learnt Newtonian dynamics will be in a position to write
down its equations of motion: these equations can then be
thought of as providing, in a mathematical form, a 'picture'
of the motions of the system, logically parallel to that which
our diagram gives of optical phenomena. Using the equations,
he will be able to compute such things as the velocity a particular
body will have when it has risen to such-and-such a height
from the ground, and the height at which it will cease to rise;
just as, from our diagram, we can discover the depth of the
wall's shadow at different heights from the ground.

This is a point worth emphasizing, for the place of mathe-
matics in the physical sciences is something people tend to
find mystifying. One is even told at times that physicists work
in two worlds, the 'world of facts' and the 'world of mathe-
matics', which makes one wonder how it can be that the world
around us is, as they imply, interpenetrated by this other,
unseen 'mathematical world'. But there is no point in talking
about a separate 'world of mathematics', unless to remind our-
selves not to look for all the features of, e.g., light-rays in sun-
beams and shadows alone; the world in which our theoretical
concepts belong being as much the paper on which our
computations are performed as the laboratory in which our
experiments are conducted. If mathematics has so large a place
in the physical sciences nowadays, the reason is simple: it
is that all such complex sets of exact inferring-techniques as
we have need of in physics can be, and tend to be, cast in a
mathematical form.

Certainly none of the substantial inferences that one comes
across in the physical sciences is of a syllogistic type. This is
because, in the physical sciences, we are not seriously interested
in enumerating the common properties of sets of objects, but
are concerned with relations of other kinds. This point will

34 THE PHILOSOPHY OF SCIENCE

be taken up again later, when we consider the differences be-
tween the physical sciences and natural history. The operations
we perform and the observations we make in physics are not
just head-counting; the logical form of the conclusions we
reach is not that of a simple generalization; and the kinds of
inference we can draw as a result are not syllogistic inferences.
Indeed, the inferences of physics are substantial just because
they are so much more than transformations of our observation-
reports. If one has counted over all As and checked that they
are all Bs, one has thereby checked that any particular A one
selects will be a B: subsequent inferences from "All As are
Bs" to "This A is a B" are automatic. On the other hand,
if one has measured the height of a wall and the angle of elevation
of the sun, one has not thereby measured the depth of the
shadow cast by the wall: yet this is something which the tech-
niques of geometrical optics enable one to infer, providing
the circumstances are of a kind in which physicists have found
the techniques reliable.

The same is true more generally. The heart of all major
discoveries in the physical sciences is the discovery of novel
methods of representation, and so of fresh techniques by which
inferences can be drawn and drawn in ways which fit the
phenomena under investigation. The models we use in physical
theories, which tend to be featured in popular accounts as
though they were the whole of the theories, are of value to
physicists primarily as ways of interpreting these inferring
techniques, and so of putting flesh on the mathematical
skeleton. The geometrical diagram used in our optical example
is lifeless unless we think of light as something travelling
'down the dotted line': only so shall we be able to see how it
is that the diagram explains the phenomena it does. But equally
the model of light travelling, remote as it is from our non-
scientific way of thinking about light and shade, is pointless
without the diagram. To present a theory simply in terms of
the models employed is to forget the thing that matters above
all, and to leave the use of the model completely unexplained.

In practice, then, a theory is felt to be entirely satisfactory
only if the mathematical calculus is supplemented by an

DISCOVERY 35

intelligible model. It is not enough that one should have ways
of arguing from the circumstances of any phenomenon to its
characteristics, or vice versa: the mathematical theory may be an
excellent way of expressing the relations we study, but to under-
stand them to 'see the connection' between sun-height and
shadow-depth, say one must have also some clearly intelligible
way of conceiving the physical systems we study. This is the
primary task of models: for know-how and understanding
both mathematics and models are wanted. The impossibility
of providing a single model by which to interpret the mathe-
matical theories of quantum mechanics has accordingly been
felt by many to be a drawback and even spoken of, frivolously
or confusedly, as showing that 'God must be a mathematician'.
Previously, it had always been possible to match one inferring
technique over its whole range of application with a single
model: it is this which, for demonstrable reasons, cannot be
done in the case of quantum mechanics, so that while a wave-
model may be of use in some applications of the theory a
particle-model is more suitable in others.

(iii) Let us next look at the notion of a model a little more
closely. Consider once again our example: in that sample
explanation, the diagram provides, as we have seen, something
in the nature of a picture of the optical state-of-affairs ; a picture
with the help of which we can infer things about the shadows
and other optical phenomena to be observed under the cir-
cumstances specified. But to understand how the explanation
works, it is not enough to point to the phenomena on the one
hand and the physicist's diagram on the other. For the physicist
uses other terms, having at first sight nothing to do either with
shadows or with diagrams, which nevertheless constitute in
some ways the heart of the explanation. He talks, for instance,
of light 'travelling', of rays of light 'getting past the wall'
or 'being intercepted by it', and declares that this interception
of light by the wall is what fundamentally explains the
existence of the shadow.

A point which we made earlier is worth repeating here.
In developing geometrical optics, we have passed from regard-
ing the phenomena of light and shade as primitive phenomena,

36 THE PHILOSOPHY OF SCIENCE

which have just to be accepted and left unexplained, to seeing
them as the common effects of something, for which 'light' is
again the word we use, travelling from the sun to the objects
lit by it. This step means coming to speak and think about the
phenomena in a new way, asking questions which before would
have been unintelligible, and using all the words in our ex-
planations 'light', 'travel', 'propagated', 'intercept' and the
rest in quite novel and extended senses. Later on, of course,
we come to feel that these are the most natural extensions in
the world; so much so, in fact, that we forget that they ever
had to be made.

Since these uses of the words are extended ones, only
some of the questions which ordinarily make sense of things
we can describe as travelling are applicable to the novel
traveller, the physicist's new entity, 'light'. Some of the
questions which we do not ask in the new application are ones
which anyone would feel to be obviously irrelevant, some
of them are ones which in the everyday application are
central. Thus we find it natural enough not to ask of 'light'
whether it travels by road, rail or air, or whether it has a single
or return ticket though remember that the discredited
'ether' was meant in part as an answer to the question "By
what means does light travel?" But it is stranger to find that
nothing in geometrical optics gives us any occasion to discuss
the question what it is that 'travels'. So far as geometrical
optics is concerned, it is enough that we have as the gram-
matical subject of our sentences the bare substantive 'light',
and it does not matter whether or no we can say any more about
it.

This point is worth following up. No doubt it is an import-
ant feature of the new way of thinking about optics that we
are prompted to ask such questions as "What travels?" There
are indeed many phenomena in accounting for which we
come to think of the grammatical subject as having a physical
counterpart: these are the phenomena with which we are
concerned in physical optics. Nevertheless, the questions with
which physical and geometrical optics are concerned are logically
independent. We know that light starts off from lamps,

DISCOVERY 37

stars and other shining bodies, and ends up on illuminated
surfaces: all we need ask, therefore, in geometrical optics are
the questions, "Where from? Where to? And by what path?"
The whole of geometrical optics could have been, and much
in fact was developed, without there being real backing for
any particular answer to the question "What is it that travels?"
Even the question "How fast?" was answered by Romer in
1676 from observations on the eclipses of the satellites of
Jupiter, before any substance had been given to the bare
grammatical substantive 'light'.

This is something which one quite often finds in the physical
sciences. At the stage at which a new model is introduced, the
data that we have to go on, the phenomena which it is used to
explain, do not justify us in prejudging, either way, which of
the questions that must normally make sense when asked of
things which, say, travel will eventually be given a meaning
in the new theory also. The acceptance of the model is justified
in the first place by the way in which it helps us to explain,
represent and predict the phenomena under investigation.
Which of the questions that it suggests will be fertile and what
hypotheses will prove acceptable are things which can be found
out only in the course of later research, in a manner which we
shall have to examine later.

One might speak of models in physics as more or less
'deployed'. So long as we restrict ourselves to geometrical
optics, the model of light as a substance travelling is deployed
only to a small extent; but as we move into physical optics,
exploring first the connexions between optical and electro-
magnetic phenomena, and later those between radiation and
atomic structure, the model is continually further deployed.

The process by which, as we go along, fresh aspects of
the model are exploited and fresh questions given a meaning
is a complicated one, and one which needs to be studied in
detail for each fresh branch of physical theory if the logic of
that theory is to be clearly understood. At the moment, all
we need to note is this: although only some of the questions
which ordinarily apply to things which, e.g., travel do so in
the extended use, one cannot say beforehand which questions

38 THE PHILOSOPHY OF SCIENCE

will and which will not apply, and it has to be discovered as
time goes on how far the old questions can be given a meaning
in the new type of context. Some of the most important steps
in physics have in fact consisted in giving to more of these
questions interpretations they did not have before (e.g. the
development of physical optics, and the introduction of the
notion of sub-atomic structure); others in doing something
which was in many ways more difficult to do, namely, giving
up hope of answering questions which up to that time had
seemed perfectly natural and legitimate (e.g. Leibniz on the
mechanism of gravity, and the nineteenth-century disputes
about the luminiferous ether).

The unlimited deployability of physical models seems to
be one of the things Planck and Einstein have in mind when
they insist that electrons and gravitational fields are as real as
tables and chairs and omnibuses. 1 For there is no denying the
differences, in logical status as well as in physical properties,
between such theoretical entities and notions as 'electrons',
'genes', 'potential gradients' and 'fields', and everyday objects
like buses and tables. But what physicists are entitled to insist
is, that their models need not necessarily be spoken of, de-
precatingly or otherwise, as theoretical fictions; for to regard
them all equally as fictions would imply that there is no hope
of deploying any of them very far, and would suggest that it was
risky following up for any distance the questions which they
prompt us to ask.

This would be a great mistake. It is in fact a great virtue
of a good model that it does suggest further questions, taking
us beyond the phenomena from which we began, and tempts
us to formulate hypotheses which turn out to be experimentally
fertile. Thus the model of light as a substance in motion is
a good model, not only because it provides us with an easily
intelligible interpretation of the diagrams of geometrical optics
though this is a sine qua non but also because it carries us
beyond the bare picture of something unspecified travelling,
no matter what, and leads us to speculate about light-particles
or light-waves as the things which travel, or are propagated:

1 This topic will be taken up in more detail in Sec. 4.7 below.

DISCOVERY 39

these speculations have borne fruit. Correspondingly, the
models of thermal and gravitational phenomena as the effects
of caloric and gravitational fluids were bad models, since the
questions they prompt one to ask turned out in fact to be
as unprofitable as those which the Greek antennae-model led
one to ask in optics.

Certainly it is this suggestiveness, and systematic deploy-
ability, that make a good model something more than a simple
metaphor. When, for instance, we say that someone's eyes
swept the horizon, the ancient model of vision as the action
of antennae from the eye is preserved in our speech as a
metaphor; but when we talk of light travelling our figure of
speech is more than a metaphor. Consequently, when people
say that to talk of light travelling in some sense reflects the
nature of the world in a way in which to talk of eyes as sweeping
the horizon does not, they have some justification. For to say
that "Light travels'' reflects the nature of reality, in a way
in which "His eyes swept the horizon" does not, is to point
to the fact that the latter remains at best a metaphor. The optical
theory from which it came is dead. Questions like "What sort
of broom do eyes sweep with?" and "What are the antennae
made of ?" can be asked only frivolously. The former does more:
it can both take its place at the heart of a fruitful theory and
suggest to us further questions, many of which can be given
a sense in a way in which the questions suggested by "His eyes
swept the horizon" never could.

2.5 Theories and observations are not deductively connected

One can, therefore, afford to speak of physical theories as
drawing their life from the phenomena they are used to explain.
If the layman is told only that matter consists of discrete
particles, or that heat is a form of motion, or that the Universe
is expanding, he is told nothing or rather, less than nothing.
If he were given a clear idea of the sorts of inferring techniques
the atomic model of matter, or the kinetic model for thermal
phenomena, or the spherical model of the Universe is used to
interpret, he might be on the road to understanding; but with-
out this he is inevitably led into a cul-de-sac.

40 THE PHILOSOPHY OF SCIENCE

It is as though we were to show a brand-new gas geyser,
still lying in its box, to a man who was quite unfamiliar with
the mechanical devices of Western life, and were to say to
him, "That heats water". We should have no right to be sur-
prised if he thought that we were showing him a robot cook
this is the counterpart of the Man Friday fallacy. The least
we can do for him is to say, more lucidly, "This is something
which can be used for the heating of water", and indicate
roughly the way in which it would have to be assembled in
order to do what it was designed to do. The sentence "This heats
water", uttered in such a context, is a condensed form of words
intelligible only to those familiar with the kind of device in
question. No geyser heats water or anything else so long as it
is left lying in its box surrounded by shavings: it must be joined
up to the mains in the way the makers specify before there is
any hope of it doing its job. The same holds of sentences
like "The atomic model explains all known chemical pheno-
mena". Once again, the atomic model by itself can do nothing
at all; but it can be used, in the way in which it was designed
to be used, in explaining the changes and processes that chemists
study. As for "Heat is a form of motion", this leaves almost
everything unsaid. Light, as we ordinarily understand the
word, is not something which can be spoken of as travelling:
so too, heat is no more a form of motion than dampness
is a form of departure.

One philosopher of science who saw the importance of
this point was Ernst Mach. He, too, used to insist that the
adoption of new theories and models was justified only by the
observational and experimental results which led up to them;
but he overstated his case in an interesting way. For the con-
clusion he came to was that the statements of theoretical
physics were abridged descriptions of the experimental results,
comprehensive and condensed reports on our observations, and
nothing more. He thought that we should be justified in accept-
ing our theoretical conclusions only if these were logically
constructed out of the reports of our experiments; that is,
related to them in a deductive way, as strictly as statements
about 'the average Englishman' and data about individual

DISCOVERY 41

Englishmen. Only in this way, he concluded, could one avoid
either anthropomorphism or what we have called 'the Man
Friday fallacy'. All talk about explanation, especially in terms
of 'insight into causal connexions', seemed to him to run into
these difficulties: causal connexions were in his view as my-
thical as the personage, Light, whom a complete novice might
suppose us to regard as 'responsible for making shadows'.

The confusion of thought which led Mach and the Pheno-
menalist School to this conclusion is not entirely easy to sort
out, and we shall have to return to the matter in later chapters.
But it is essential to see at the outset that there can be no
question of observation-reports and theoretical doctrines being
connected in the way Mach thought: the logical relation between
them cannot be a deductive one. This comes out clearly from
our example: however many statements you collect of the
form, "When the sun was at 30 and the wall six feet
high, the shadow was ten feet six ins. deep", you will not be
able to demonstrate from them in a deductive manner the
necessity of the conclusion, "Ergo, light travels in straight
lines". Not that this is anything to worry about; for, given on
the one hand statements about everyday things, like lamps,
the sun, shadows and walls, and, on the other hand, theoretical
statements in terms of the physicist's concept, light, how can
we even imagine finding deductive connexions between them?
The types of sentence which are deductively related are always
taken out of roughly the same drawer and stated in similar
terms for instance, "Fish are vertebrates", "Mullet are fish"
and "Mullet are vertebrates". But the two classes of sentence
now under consideration are stated in quite dissimilar terms,
and in them language is being used in radically different ways.

To say "Light travels in straight lines" is, therefore,
not just to sum up compactly the observed facts about shadows
and lamps: it is to put forward a new way of looking at the pheno-
mena, with the help of which we can make sense of the observed
facts about lamps and shadows. But this is not the same as
to say, "One can represent the phenomena thus: . . .", or
"Physicists now regard light and shade thus: . . .". Rather it is
to play the physicist, to speak the words of one who regards

42 THE PHILOSOPHY OF SCIENCE

them in the new way. In view of this, we can see how misleading
it might be to say, without qualification, that "Light travels in
straight lines 1 ' is a law as much of our method of representation
as of nature. For the discovery that light travels in straight
lines was certainly not a discovery about physicists i.e. that
they can choose, or do choose, to represent optical phenomena
in a geometrical manner. Not at all: if they did not mind about
the consequences, they could choose to represent them anyhow
they pleased. There is an additional discovery, beyond the
fact that they do so choose, which alone shows the importance
of the principle for physics: namely, that if one does so
represent them, it is possible to explain optical phenomena
of a wide range of types light and shade, eclipses and so on
with certain restrictions (no refraction etc.) but to a high degree
of accuracy; and further, as we shall see, that these techniques
can, with the aid of simple rules, be extended to situations
involving refraction and reflection and other phenomena so
far ruled out.

Still, the difficulty Mach felt is one that we are all liable
to feel when we first notice the logical differences between
theoretical statements, like "Light travels in straight lines",
and observation-reports like "The shadow was ten feet six
ins. deep". It is natural for a logician to suppose that, in order
to justify a theoretical conclusion, one must collect sufficient
experimental material to entail it; and that, if one does any-
thing less, the theoretical conclusion will assert something more
than the experimental data warrant. Mach, at any rate, was
very keen to show that laws of nature 'contain nothing more
than* the facts of observation for which they account. But this
is a mistake. For it is not that our theoretical statements ought
to be entailed by the data, but fail to be, and so assert things the
data do not warrant: they neither could be nor need to be
entailed by them, being neither generalizations from them nor
other logical constructs out of them, but rather principles in
accordance with which we can make inferences about pheno-
mena. This point will be made clearer in the next chapter.
To justify the conclusion that light travels in straight lines,
we do not have to make observations which entail this con-

DISCOVERY 43

elusion: what we have to do is to show how the data we have
can be accounted for in terms of this principle. The absence
in this case of a deductive connexion is not to be thought
of as a lack of connexion, any more than a hammer need be
thought of as lacking a screw-thread: justification here calls
for something other than a demonstrative proof.

The real difficulty is to avoid stating the obvious in a mis-
leading way. Einstein, for instance, objects to Mach's doctrine
but almost tips over backwards in his effort to rebut it: he
speaks of physical theories as 'free products' of the human
imagination. Granted that discoveries in theoretical physics
are not such things as could be established either by deductive
argument from the experimental data alone, or by the type of
logic-book 'induction' on which philosophers have so often
concentrated, or indeed by any method for which formal
rules could be given. 1 Granted that discoveries in the physical
sciences consist in the introduction of fresh ways of looking
at phenomena and in the application of new modes of repre-
sentation, rather than in the discovery of new generalizations.
Perhaps, too, the recognition of fresh and profitable ways of
regarding phenomena is, in part at least, a task for the imagina-
tion, so that Einstein can say of them, as he says of the axiomatic
basis of theoretical physics, that they "cannot be abstracted
from experience but must be freely invented. . . . Experience
may suggest the appropriate [models and] mathematical
concepts, but they most certainly cannot be deduced from it."
But we must not be tempted to go too far. This is not work
for the untutored imagination. It may be an art, but it is one
whose exercise requires a stiff training. Though there is nothing
to tell just what new types of model and mode of representa-
tion scientists may not in time find it profitable to adopt, nor
any formal rules which can be demanded for discovering
profitable new theories, theoretical physicists have to be
taught- their trade and cannot afford to proceed by genius alone.

lr This is why it is so unfortunate that logicians have come to speak of
scientific discovery as 'inductive inference': where no rule of inference could
ever be given, the very notion of inference loses its point. Discovery is,
rather, a prerequisite of inference, since it includes the introduction of novel
techniques of inference-drawing.

44 THE PHILOSOPHY OF SCIENCE

One cannot teach a man to be imaginative ; but there are certain
kinds of imagination which only a man with a particular
training can exercise.

The situation is rather like that in which, as we are some-
times told, unbreakable glass or saccharin or radio-activity
or blotting-paper was discovered 'by accident'. Again this is a
misleading way to talk: such discoveries are not made by
accident, even though they may be made as a result of an
accident. Most people, if they knocked a glass jar on to a stone
floor and it did not break, would pick it up, thank their lucky
stars and leave things at that: only a scientist with the right
initial training would know just how odd a happening this was,
and would be equipped to find out what had happened to the
jar beforehand that prevented it from shattering. It might be
a piece of luck that one scientist rather than another first
noticed the phenomenon ; but it would not be luck which guided
the rest of his investigation. It may, likewise, be a fertile
imagination which first leads one physicist rather than another
to explore the possibilities of some novel theory; but again,
it is trained skill quite as much as imagination which guides
him in the exploration once it is begun.

2.6 Physics is not the natural history of the inert

There is one final point about the sorts of things which
count as discoveries in the physical sciences which must be
emphasized at the outset: this will help us to understand the
differences between explanatory sciences, such as physics, and
descriptive sciences, such as natural history. The point can be
put concisely by saying: physicists do not hunt out regularities
in phenomena, but investigate the form of regularities whose
existence is already recognized. As it stands, this may seem
rather a dark saying; so let us take another look at some
examples.

It must have been recognized that there was some regularity
in the way in which shadows were cast long before this fact
was scientifically explained: the development of geometrical
optics made clear and explicit the nature of a regularity which
had previously been appreciated only roughly. Again, it was

DISCOVERY 45

known that the planets moved in a regular way, and these
regularities had been studied, for many centuries before there
was any dynamical theory to make sense of them: the develop-
ment of dynamics once again made intelligible regularities
whose existence was previously known, but whose exact nature
and limits had not been understood.

The consequences can be seen if one looks at the starting-
point of the physical sciences, and at the scientist's opening
moves. For the regularities of everyday experience, with which
we are all familiar, provide him with a natural point of attack:
and the questions he will begin by asking are not "Are
there laws of motion, optics or chemical combination?" but
"What are the forms of these laws?" With such a starting-
point, one question does not need to be asked: namely, whether
there is any connexion between, say, the slope of a hill and the
way a stone moves when placed on it, or between the position
of the sun in the sky and the length of shadows. Like the rest
of us, the scientist knows very well that these things are, in
some way to be discovered, interdependent: the form of his
first question will therefore be not "Are these things inter-
dependent?" but "How do they depend on one another?"

Philosophers have sometimes talked as though science
could be divorced from common experience, and as though
the scientist had a completely free choice of starting-point.
Now it is true that, once his subject is established, a scientist
will choose what experiments to perform and how to perform
them on the basis of scientific considerations alone we shall
see later how closely the conditions of an experiment are
determined by the nature of the theoretical problem on which
the experiment is designed to throw light. But it does not follow
that, at the very beginning of a science, the investigator can
start just anywhere. Though we can hardly speak of the ordin-
ary man having theories about natural phenomena, it is never-
theless such everyday regularities as we have been concerned
with in our optical example, and the departures from them,
that pose to the scientist his first theoretical problems.

To point to the very beginning of a science is, in fact, to
make an artificial division. Current theoretical problems in,

46 THE PHILOSOPHY OF SCIENCE

say, the dynamics of fast-moving particles arise out of the
limitations of the Newtonian theory; the Newtonian theory
of motion was the solution of problems posed by the limitations
of the Aristotelian theory, since it was the failure of Aristotle's
dynamics to deal with acceleration that focused attention on
that phenomenon during the sixteenth and seventeenth cen-
turies; Aristotle's dynamics in its turn was an attempt to
systematize and extend our ordinary ideas about motion;
and where exactly in this sequence are we going to draw the
line? At each stage, the centre of interest depends on the current
background of ideas about motion. These provide the standard
of what is normal, of what is to be expected, and it is primarily
departures from this standard which are spoken of as
'phenomena', that is, as happenings requiring explanation.
When we go back to the stage in any science at which the
first systematic attempts were made to theorize, to connect up
the phenomena in that field, it is the notions of contemporary
common sense which provide the background of ideas by refer-
ence to which phenomena arc chosen for investigation. And,
since common sense in this context means 'recognizing the
regularities with which we are familiar from everyday experi-
ence,' it is natural that these should play a prominent part in
the early stages of most of the sciences so that it was, for
instance, from a study of breathing and burning ('respiration'
and 'combustion') that the savants of the late eighteenth century
first began to understand the nature of chemical reactions, and
gave Dalton his chance to make of chemistry something more
than a collection of isolated industrial techniques and conjuring
tricks.

From this we can see the source of one of the differences
between the physical sciences and natural history. In physics
we cannot afford to begin where we like. Rather, as Newton
puts it, we must trace out the laws from the phenomena in
a few simple cases ; and apply what we discover in these cases,
as principles, when we turn to more involved cases. "It would
be endless and impossible to bring every particular to direct
and immediate observation"; so the physicist only has time to
investigate in detail the behaviour of the simpler systems.

DISCOVERY 47

If you bring a physicist or chemist a box containing an
unidentified assemblage of things, he may be perfectly entitled
to brush aside your request to be told how it works and what
will happen if you do different things to it: the contents of
your box will probably not be a suitable object of study.
He may possibly, given time, discover what it is that you have
brought him, and so be able to answer your questions at any
rate, in certain respects, and to a limited degree- of accuracy.
But unless the assemblage is a particularly simple one, the
task of identification will be lengthy, and the scientist will be
within his rights if he regards you as having interrupted, not
contributed to, the progress of his work.

In natural history, things are quite otherwise. Whatever
kind of living creature we come across, it will be equally fair
to ask the naturalists what it is, and what its habits are. Any
kind of animal is a 'suitable object of study* for the natural
historian; and if at a particular stage in history one species
has been studied more than others that will not be for theoretical
reasons, but for practical ones for instance, because it is easy
to feed and is not afraid of humans, so that it can be watched
without the need for elaborate hides. All living creatures equally
may be subjects for the natural historian, but, for theoretical
as well as practical reasons, observation and experiment in
the physical sciences have to be highly selective.

This, however, is a comparatively minor difference between
the descriptive and explanatory sciences. The larger differences
have a more subtle origin, and we must try to get clear about
it. Notice for a start, then, that the kinds gf regularity we
encounter in everyday life, which form the starting-points of
the physical sciences, are hardly ever invariable ; and corre-
spondingly, the degree of system in everyday language is
limited. Only rarely can one infer from an everyday description
of the circumstances of a phenomenon just what form it will
take.

Some small amount of system there is, reflecting the familiar
regularities that every child soon discovers. This is most
clearly to be seen in the use we make of law-like statements:
"Don't hit the window: glass is brittle (i.e. breaks if hit)". But

48 THE PHILOSOPHY OF SCIENCE

this system is not particularly reliable. All such inferences
in ordinary language are open to qualification: "This is made
of wood, so it must float unless it's lignum vitae or is water-
logged", "You can see the road's straight, so that must be the
shortest way unless we're up against some optical illusion".
These inferences depend on physical or natural-historical
regularities of whose scope we have only a vague idea, and they
are therefore liable to exceptions. We should not be very much
surprised, e.g., to find another kind of wood besides lignum
vitae which refused to float.

Many of the delights of childhood, indeed, consist in
defeating these regularities. It may be fun to roll a stone down-
hill; but it is much more fun to fill a balloon with gas, and watch
it float up to the ceiling. We only expect these regularities to
hold on the whole, and we are not particularly disconcerted
when we encounter the exceptional case.

Nor need these limitations matter for most practical pur-
poses. A carpenter need be no physicist to know that, in the
main, the way two planks look is a good guide to the way they
will fit, and that if the foot of a plank is in water the look
of it will no longer be such a good guide. The ability to explain
why a plank looks bent in water would not simplify his tasks
as a carpenter: his professional attitude to this phenomenon
will accordingly be one of indifference. So long as he is able
to tell in practice when look will and will not be a good indica-
tion of fit, he need not be particularly interested in the optical
theories required to explain these facts.

It is the mafk of the physical scientist, on the other hand,
to be interested in such regularities and their limitations for
their own sakes. It is a matter of professional concern to him
to find out what exactly they amount to, why they hold and fail
to hold when they do, under what conditions departures are
and are not to be expected and, if possible, to develop a
theory which will explain all these things. The questions which
are of importance to him are, accordingly, these: "What form
does the regularity take, in the cases in which it occurs?"
and "Under what circumstances are we to expect it to occur?"
To put the point briefly, the physicist seeks the form and the

DISCOVERY 49

scope of regularities which are found to happen, not universally,
but at most on the whole.

This point has been consistently misunderstood in text-
book discussions of scientific method. Starting with a study
of the syllogism, the probability calculus and the calculus of
classes, and then coming to the physical sciences, logicians
have been misled by their earlier preoccupations and interests,
vested as they are in formal systems of considerable refinement
and elaboration, into looking for the wrong things. One form
of statement alone has commonly been examined, the universal
empirical generalization ; and only the more detailed treatments
of the subject have even succeeded in passing on from "All
As are Bs" to "The probability of an A being a B is 3/5"
and "Conditions C I9 C 2 and C 3 being fulfilled, all As are
Bs." The consequences have been unfortunate. Laws of
nature have been confounded with generalizations, such
sentences as "All swans are white" and "All ravens are black",
being gravely discussed under this heading. Hypotheses have
been treated as though they were simply laws of which we
are not yet confident, since they have not been checked in a
sufficient number of instances. As for experiments, these have
been presented as first cousins of the Gallup poll concerned
only with how often different pairs of properties are found
to go together.

But to accept such an account is to treat physics as though
it were a kind of natural history, and so to waste one's labour.
Natural historians may be interested enough in discussing
whether or no all ravens are black, and whether all mice like
cheese. But so long as one remains within natural history there
is little scope for explaining anything: "Chi-chi is black, because
Chi-chi is a raven and all ravens are black" is hardly the kind
of thing a scientist calls an explanation. Indeed, among
scientists, to say that a newly fledged subject is still in 'the
natural-history stage* is a way of depreciating it: natural history
and the like are felt to lack many of the essential features of a
full-grown science, and to be entitled to the name of sciences
only conditionally and out of courtesy.

This practice is not entirely fair to natural history, since

50 THE PHILOSOPHY OF SCIENCE

as soon as an observer suggests, e.g. how the colouring of
some sub-species of rat may be explained in terms of its envi-
ronment, he is promoted from 'natural historian' to the more
respectable rank of 'zoologist'. But the feeling has some
justification. For, if explanatory power is regarded as the stamp
of a science, then the shallow explanations which are all that
we can demand of natural history take us little beyond the point
which, in dynamics, every child has reached: "This rolls down-
hill, because this is a stone, and stones generally do roll down
hills." How different are the sorts of conclusion aimed at in
the physical sciences: "Light travels in straight lines", "The
hydrogen atom consists of one proton and one electron"
the very point of such statements lies in their explanatory
fertility; and in hardly a single respect are they comparable
with the generalizations about habits or plumage which are
all that natural historians can announce.

2.7 The crucial differences between physics and natural history

The reason for the differences between generalizations
about habits, plumage, etc. ('habit-statements') and what, by
contrast, may be called 'nature-statements', will become
evident as we go along. But there is one point of general
importance that requires to be touched on here. This has to
do with the question, what sorts of subject-matter the two types
of statement can have i.e. what sorts of grammatical subject
they can contain. Here at last we shall begin to see how the
logical differences between the two classes of statement spring
from differences between the two kinds of scientific activity.

The subject-matter of the natural historian's habit-state-
ments is the same as that of everyday speech and affairs: at
most, the natural historian will sub-divide the everyday
classification in ways we would not normally bother to do,
distinguishing, for example, between the Spotted Woodpecker,
Dryobates major anglicus, and the Northern Spotted Wood-
pecker, Dryobates major major. The task of identifying to what
class a given subject belongs will not in general be a highly
technical one; although there may be difficult or borderline
cases which have to be left to the expert, in the main, as

DISCOVERY 51

Wittgenstein has remarked, 'what is or is not a cow is for the
public to decide*.

Being tied, in its essentials, to the everyday classification,
the natural historian has left to him to discover such things
as what breeding-habits are common to all gulls, and what
proportion of the North Sea herring shoals passes through the
Straits of Dover in the average summer. In consequence,
his conclusions are, from a logician's point of view, both quite
straightforwardly factual and open to logical analysis in the
traditional way: they will fit without appreciable distortion
into the familiar patterns, "All As are Bs" "All As which are
also Cs are Bs", "The proportion of As which are Bs is
3/5", and so on.

Furthermore, since the classification of his subject-matter
is made along everyday lines, it is not open to the natural
historian to modify its principles in the light of his discoveries.
Were he to find that half the house-mice in England were
herbivorous and half carnivorous, and that these two sets of
mice did not interbreed, he could and would distinguish
between the two classes and, if the circumstances made this
appropriate, might come to speak of them as two different
species of mouse; but he would not be at liberty to say either
"One half lives on lettuces, so they can't be mice after all",
or "Only the ones that live on lettuces are to be regarded as
mice." Or rather, if he did insist on doing so, the agreement of
the public would be a sign, not of his expert knowledge, but
of his prestige like the agreement never to call whales 'fish'.

When one turns from natural history, with its habit-
statements, to the nature-statements of the physical sciences, one
finds that the situation is markedly different. In talking about
the phenomena they study, physicists need no more confine
themselves to the everyday classification of the things they
encounter than they do to the more elementary logical forms.
Reclassification of subject-matter in the light of discovery
is the rule in the physical sciences: 1 the decision, what is or is

l lt is in this way, for instance, that the classification of kinds of matter
by origin and the like, i.e. as 'wood', 'water', 'stone', etc., comes to be
supplemented by the classification into kinds of chemical substance, as
'hydrogen', 'carbon dioxide', etc.

52 THE PHILOSOPHY OF SCIENCE

not to be spoken of as 'a purely gravitational phenomenon'
as opposed to 'a cow* becomes therefore a highly technical
matter, and the grounds on which it is made change as the
theories of science develop.

This fact has important consequences for the logic of the
things the physicist says. In natural history, one can distinguish
sharply between two stages in any piece of research: the
initial step of identifying an animal unnecessary, of course,
if it was bred in the laboratory and the subsequent
process of studying its habits. In the physical sciences,
there is no such sharp division: the things that come to
light as one goes along will frequently lead one to
relabel the system being studied. The statement, "This
can't be classified as a mouse, for it eats lettuce", may be
inadmissible, but its physical counterpart is quite conceivable:
"This can't be classified as a purely gravitational phenomenon,
for the orbit is nutating as well as precessing." Now we can
account for something we noticed earlier, namely, the impossi-
bility of treating the statements of theoretical physics as
universal empirical generalizations. The reason why the form
"All As are Bs" does not fit the statements of physics is this:
only where one can ask separately, first, "What are these?"
(Answer: As), and then, "What common properties have
they?" (Answer: being Bs), is "All As are Bs" the natural
form in which to couch one's conclusions. One can make this
separation in natural history; but in the physical sciences
the two questions are interdependent, and in consequence the
simple generalization is out of place.

What is the point of the physicist's reclassification? To
see this, recall that it is his aim to find ways of inferring the
characteristics of phenomena from a knowledge of their cir-
cumstances. This aim is one which ordinary language, being
largely devoid of system, does not serve very well. To speak
of something as a blackboard', for example, implies hardly
anything about how it will behave. No doubt, if it explodes,
or crumbles into dust, or vanishes without warning, we shall
be very much surprised, and try to find an explanation; but
it cannot be said to be implied by one's description of it as a

DISCOVERY 53

blackboard that these things either will or will not happen to
it, however unexpected and inconvenient they may be. If the
manufacturers of blackboards found that their products could
not be guaranteed against disintegration in fact, that they
were all liable to crumble away at an unpredictable time after
manufacture, like radioactive nuclei that would not stop us
talking of them as blackboards, any more than the finite
life of the filament stops us calling electric lamp-bulbs 'lamp-
bulbs'. No reclassification or other linguistic steps would be
called for. We should simply have to lay in a stock of replace-
ments; and, if things became too bad, school-teachers would
take care to say, "I'll leave this graph on the board, and we'll
talk about it next time, with luck."

Once again, how different is the situation in the physical
sciences. There the specification of a system carries rigorous
implications about its behaviour. The chemist analysing a
specimen, for example, will not be satisfied until he can account
for its observed chemical properties by reference to its con-
stitution as strictly as we accounted for the depth of the wall's
shadow; and if two specimens, both from the same source,
have quite different properties, he will not be satisfied to regard
them as being of the same substance or as having the same
structure. His classification must take account of the differences
between their properties: if it does not allow for these differ-
ences, so much the worse for his classification. Indeed, the
classification-system scientists employ changes as time goes
on, and the way in which it does so shows what their ideal is:
that, from a complete specification of the nature of any system
they have under investigation, it should be possible to infer
how it will behave, in as many respects and to as high a degree
of accuracy as possible.

2.8 Description and explanation in science

Natural historians, then, look for regularities of given forms ;
but physicists seek the form of given regularities. In natural
history, accordingly, the sheer accumulation of observations can
have a value which in physics it could never have. This is one of
the things which the sophisticated scientist holds against natural

54 THE PHILOSOPHY OF SCIENCE

history: it is 'mere bug-hunting' a matter of collection, rather
than insight.

Now there is something important in this way of putting
the difference, which is reflected in the sorts of thing that
could be accepted as observations in physics and natural history
respectively. As one cannot start doing physics just anywhere, so
also there are very definite limits to what will count in physics
as an observation. Gilbert White was able to make valuable
contributions to natural history by keeping a diary of the things
he noticed as he went around the Hampshire countryside, for
in natural history all facts about fauna are logically on a par.
But, as Popper has pointed out, one could not hope to
contribute to physics in this way. However full a note-book
one kept of the phenomena one came across in the ordinary
course of one's life, it would in all probability be of no value
to physicists at all. In physics, it is no use even beginning to
look at things until you know exactly what you are looking
for: observation has to be strictly controlled by reference to
some particular theoretical problem. Just how close the connex-
ion has to be, we shall see in the next chapter.

On this point, Mach and his followers again tend to be
misleading. One finds them, for instance, identifying 'observa-
tions' on the one hand, and 'sense-data' on the other, which
suggests that we are for ever making observations. This is a
confusing practice, for it entangles the logical problems of
physics with the philosophical problems which have to do
with perception and material objects. Furthermore, it does
this needlessly, since it is not difficult to keep the terms
'sensation' and 'observation' sorted out: as though one only
had to open one's eyes to 'make observations'. This tendency
is probably connected with Mach's desire to show that all
sciences are equally descriptive, and to avoid the terms 'in-
sight', 'causal connexion' and the like, which he found so
obnoxious. But whatever the explanation, the result is that he
talks about physics almost as though it were the Natural
History of Sensations, describing 'the habits of sensations' in
the way that zoologists describe the habits of zebras.

The conclusion that the sciences tell us only how things

DISCOVERY 55

happen, not why they do, and that all science is really an elabor-
ate mode of description, is one that has been seized on as a
lifebelt by various interested parties. Some theologians, for
instance, have welcomed it as providing them with a hope of
survival: if science does not aim at explaining why things happen,
then they can continue to do so themselves, without fear of
challenge from that most dangerous quarter. Their welcome
has, however, been both premature and misplaced. Certainly
it is no longer regarded as part of a scientist's job to say what
God had in mind when He created refractive substances;
so, if that is what a theologian means by 'explaining why
refraction happens', a theory of refraction is not required to
tell us why. But the fact of the matter is, not that physicists
leave the question "What is the purpose of refraction?" to be
answered by others, but that as a result of their work they no
longer see this as a question which needs asking. Since the
failure of Leibniz's attempts to prove that neither atoms nor
a vacuum could possibly exist, 'since it would have been un-
reasonable of God to create them', questions about the purpose
of physical phenomena have come to seem particularly fruit-
less which is not the same as saying that scientists now regard
physical phenomena as purposeless. In any case, the premise
that all the sciences are alike descriptive is hardly acceptable
anymore. The manifest differences between the physical sciences
and natural history show that this is, at best, an exaggeration,
for how different are scientific explanations of the physical
type from anything we could ordinarily speak of as descriptions;
and how little can one think of, say, the doctrine that light
travels in straight lines as 'reporting a fact* or 'describing a
state-of-affairs'.

Instead of treating all sciences as equally descriptive, and
explanation as metaphysically disreputable, it would be more
interesting to consider how far the aims of any particular
science are explanatory and how far they are descriptive.
Most of the sciences which are of practical importance are,
logically speaking, a mixture of natural history and physics.
The nearer one is to natural history, in the agricultural sciences,
for instance, the better the traditional logic-book account

56 THE PHILOSOPHY OF SCIENCE

fits: the nearer one is to physics, the more unsatisfactory it
becomes. In some subjects, such as geology and pathology, the
strands are interwoven in a way which is complicated and needs
examining. But the issues involved could not help being some-
what technical, and this is not the place to deal with them.

CHAPTER III

LAWS OF NATURE

FROM our study of the Principle of Rectilinear Propagation,
we have seen how necessary it is always to understand a physical
principle in the context of its use. Looked at against this back-
ground, its force will be clear enough: divorced from all prac-
tical contexts and left to stand on its own, its meaning will be
far from clear, and it will be open to all sorts of misunderstand-
ing and misapplications. The same is true of laws of nature;
and in this chapter we must try to see what the tasks of such
laws are that is to say, how they contribute to the fulfilment
of the programme of the physical sciences.

3.1 How laws of nature help one to explain phenomena

Up to this point in our discussion, we have not come across
anything that a scientist would speak of as a lav/ of nature,
for the doctrine that light travels in straight lines is not so much
a 'law' as a 'principle' the force of this distinction we shall
see later. Nor have we encountered a situation in which a
scientist would go in for any very elaborate experiments, so
that we have yet to see the place of the laboratory in the develop-
ment of the physical sciences. Nor, again, have we allowed
ourselves to go beyond the kinds of phenomenon which, in
the twentieth century, it does not take a scientist to explain:
the study of shadow-casting hardly taxes the resources of
physics. These three facts are related. It is only when we go
beyond the simplest everyday phenomena to a study of more
sophisticated things that resort to the laboratory becomes
necessary ; and it is in the form of laws of nature that the scientist
ordinarily aims to express the results of the experiments he
then undertakes.

We need not look far for an example to consider. When
we discussed shadow-casting, we found that certain restrictions

57

58 THE PHILOSOPHY OF SCIENCE

had to be placed on the circumstances in which the principle
that light travels in straight lines was applied. One restriction
was 'no refraction': we can use our principle confidently to
argue from the height of a wall and the sun to the depth of the
wall's shadow, only when there is, e.g., no glass tank of water
just behind the wall, and no bonfire to produce currents of
warm air and blur the shadow. It should be noticed, incidentally,
that one cannot give an exhaustive list of such conditions,
which does not begin with an 'e.g.' or end with the phrase
'and so on . . .', since the number of different kinds of situations
in which refraction may occur is indefinitely large. Only in the
absence of water, glass and the like are the techniques of
geometrical optics applicable in their simplest form. So, in
order to get clear about the techniques first of all we confined
ourselves to everyday things, showing how the physicist's
picture of optical phenomena introduces precision and system
into the everyday field, and makes it possible to argue from one
set of exact measurements (e.g. wall-height, 6 ft.: sun-height,
30) to others (e.g. shadow-depth, 10 ft. 6 in.). But can
we now extend the techniques of geometrical optics so as to
explain also the optical phenomena we encounter in the presence
of water, glass, warm air currents and the rest? This is where
Snell's Law comes in.

it is worth remarking, before we go any further, that the
terms in which we are here describing the investigation are
not those which a scientist himself would use. What we call
'extending the range of application of the theories and tech-
niques of geometrical optics to situations in which water, glass
or other such transparent substances intervene between the
lamp, or the sun, and the illuminated objects' he would call
'investigating the optical properties of transparent media'.
The difference between these two ways of stating the problem
arises partly from a desire for compactness, but it reflects also
the differences between the attitudes which the logician
who is an onlooker and the scientist who is a participant
will adopt towards the symbolism of the sciences, and towards
their subject-matter. Naturally enough, the scientist will
always use his theoretical terminology in describing what he

LAWS OF NATURE

59

is doing. For the logician, however, the way the scientist uses
his theories and symbolism is itself a part of the activity under
examination: from his place in the grandstand, therefore, he
will prefer to give a more cumbersome description, in which
the roles of the scientist's symbolic techniques are not left
unexamined, but stated explicitly.

What is Snell's Law?, Let us state it first as a physicist
would state it, and then go on to see how it serves to solve
our problem. To use participant's language for the moment,
what Snell discovered was this: that, if one measures the angles
at which a ray of light is inclined as it strikes the surface of
a piece of glass, water, or other transparent substance, and after
passing into it, there is a simple relation between these two
angles.

AIR

i 1

/GLASS'

/
i 1

/

///

-ft

i
i

.(f''1

i i

i

\y

1 : '
i ' /

//

1

/

/ / /

/

If the angle t, at which the specimen is set askew to the light
striking it, is called the 'angle of incidence', and the correspond-
ing angle r, at which the light travels after entering the glass,
is called the 'angle of refraction', then Snell's Law states that
"whenever any ray of light is incident at the surface which
separates two media, it is bent in such a way that the ratio
of the sine of the angle of incidence to the sine of the angle of
refraction is always a constant quantity for those two media." 1

J The 'sine' of an angle is a simple trigonometrical function, varying from
for an angle of to 1 for an angle of 90, which can be found tabulated
in any book of mathematical tables.

60 THE PHILOSOPHY OF SCIENCE

With a wide range of transparent substances, and under
similar conditions, the phenomena again obey the same law,

-: =const., only with a different 'constant quantity' for each
sin T

substance. In the case of a few substances difficulties arise,
and in these cases the refraction is said to be anomalous;
but wherever the law holds in this simple manner we speak of
the constant quantity for refraction out of air into the substance 1
as the 'refractive index* of the substance.

It is easy to see in outline how this law helps us. If, for
instance, we find that a light-ray striking a piece of glass at
an angle of incidence (i) of 60 is inclined after refraction at an
angle (r) of 45, we can at once work out what the angle of
refraction will become if the angle of incidence is changed to 45.
For the ratio of the sines will, according to Snell's Law, be
the same in both cases; and a little arithmetic will show
that, when i is 45, r will be about 36. This application of
Snell's Law is like inferring what the length of the shadow
of a wall will be when the sun has dropped to 15, knowing
what the length of the shadow is when the sun is at 30.

Our example is, however, still stated in participant's
language, and uses terms like 'light-ray', which themselves
form part of the theory we are examining. Can we, as logicians,
restate the law in a way which will avoid doing this? This is
what we must next attempt to do.

Previously, when producing the picture of the optical state
of affairs needed to explain shadow-casting, eclipses and the
like, we thought of light as propagated in straight lines (rays)
from the source of light to the objects lit up, drew straight lines
to represent the direction of travel of these light-rays, and
remarked how they were cut off by opaque obstacles. This
technique was all very well for shadow-casting, but did not
explain refraction. Now we can add a new rule. When, in our
picture, the straight line representing a light-ray impinges on
the line representing the surface of a transparent obstacle, we

1 Strictly speaking, this should read 'out of a vacuum into the substance',
but in the case of most transparent solids and liquids the difference is
trifling.

LAWS OF NATURE 61

are to change its direction where it passes through the surface,
and the amount of the change is to be calculated using Snell's
formula. It is necessary to say 'in our picture', so that we keep
in mind the fact that the lines we draw in the diagram do not
necessarily stand for individual 'things* in the state of affairs
represented: as we have seen, the notion of a light-ray is a
theoretical ideal, which derives its meaning as much from our
diagrams as from the phenomena represented, and this fact
is reflected, as we shall soon discover, in the practical diffi-
culties which limit the extent to which we can get light to travel
in ever-narrower beams.

This new rule allows us to extend the inferring techniques of
geometrical optics in the way we aimed to do. It also shows how
the model of light as a substance travelling has to be extended
to cover this new application: just as, to understand about
shadows, we had to begin thinking of sunlight as travelling in
straight lines from the sun to the objects it shines on, so now, to
understand about refraction, we must think of the light as chang-
ing direction when it enters transparent media such as glass.

Using this new rule, we can account not only for observa-
tions made in the laboratory. We can explain also many optical
phenomena which had simply to be ruled out of consideration,
so long as we could employ only the more primitive techniques
needed for dealing with shadow-casting. For instance, we can
account for that King Charles' Head of philosophy, the stick
which looks bent when its end is dipped into water.

Furthermore, when one says 'account for j the phenomenon,
this does not mean coming down on one side or the other in the
vacuous dispute as to whether 'in ultimate reality* the stick is
bent or not: it means that, given the angle of viewing of the
stick and the refractive index of water, one can actually construct,
in a diagram of the kind given overleaf, the 'apparent position*
of the stick, and so confirm that it is to be expected, light
travelling as it does, that the stick will appear as it is in fact
found to appear. 1

l The diagram in the text has been simplified for the sake of clarity.
The construction shewn in fact determines only the degree of foreshorten-
ing: the exact angle at which the stick appears to be bent could be found
by drawing a rather more complex diagram.

62

THE PHILOSOPHY OF SCIENCE

DIRECTION
OF VIEW

POSITION APPARENT
OF STICK \ ACTUAL

Since this construction is as strict an application of Snail's
Law as our shadow-diagram was of the principle that light
travels in straight lines, one can properly say that it follows
from the theory that the phenomenon must be what it is.
Provided that the appropriate conditions are fulfilled, the theory
can be said in these circumstances to imply the occurrence of
this particular phenomenon. Arguing in accordance with the
Law, that is, one can infer what will happen: unless one
disputes the adequacy of the theory, therefore, one will
be bound to foretell just that phenomenon in those circum-
stances. We can also argue in reverse. In fact, observations of
a phenomenon very similar to that of the 'bent stick' are used
when measuring the 'refractive index' of a substance. This
fact reminds us of the virtues of the view that the physical
sciences form 'deductive systems': the defects in this view we
shall see shortly.

As onlookers, then, we can regard the discovery of Snell's
Law as the discovery of how the optical phenomena encountered

LAWS OF NATURE 63

in a specifiable rangfe of situations are to be represented, and
so explained to such-and-such a degree of precision, and with
certain provisos, which we shall have to consider in a moment.
This may seem to be stated vaguely, but it is inevitable that
it should be: if you try to say exactly and explicitly what is
involved in the discovery, with all the conditions and limitations
put in, hoping to 'make an honest fact of it*, you will succeed
only in producing a tautology. For to cover yourself you will
have either to employ at some point an omnibus phrase, like
'all relevant factors' or 'other similar situations', the nature
of whose relevance or similarity cannot be independently
specified, or else introduce into your provisos a circularly
defined technical term like 'optically homogeneous', i.e.
'having a uniform refractive index'. But this does not mean,
as some have thought, that laws of nature themselves are treated
by scientists as tautologies, or as conventions: rather, it shows
us one of the reasons why, in practice, the scope of a law is
stated separately from the law itself why Snell's Law,
for instance, has to be supplemented by a set of statements of
the form, "Snell's Law has been found to hold under normal
conditions for most non-crystalline materials of uniform
density." This is a distinction which will receive a more detailed
examination later in the chapter.

3.2 Establishing a law of nature (I)

The discovery of SnelTs^aw has several features in com-
mon with the discovery we studied in the last chapter the
discovery that light travels in straight lines. To begin with,
the transition from the stage at which it was not known that
light travels in straight lines to the stage at which this had
become known, was a double one: it involved the introduction
of novel techniques for drawing inferences about shadows,
eclipses and the like, and also of a novel way of thinking about
the situations in which these phenomena occur one that
makes the new inferring techniques seem natural and int^l-
ligible. So here, the change which takes place when Snell's Law
becomes known is also a double one: we are given a rule for
extending the inferring techniques of geometrical optics to

64 THE PHILOSOPHY OF SCIENCE

cover refractive phenomena, and the model of light as a
substance in motion is deployed a little further.

Again, we found that, logically speaking, the Rectilinear
Propagation Principle belonged in quite a different box from
the data which are taken as establishing it ; so that there can be
no question of its being deductively related to these data, nor
any point in looking for, or bewailing the absence of such a con-
nexion. The transition from the everyday to the physicist's view
of light involves not so much the deduction of new corollaries
or the discovery of new facts as the adoption of a new approach.
So now, the step from the experimental observations on which
Snell's Law is based to the Law itself cannot be thought of
as a matter of natural history, as a summing-up of the obser-
vations in terms with which we are already familiar. Once again,
there is no question of our conclusion being either deductively
related to, or a plain generalization of the observations we write
down in our laboratory note-books. One might manipulate
experimental apparatus for a lifetime, and accumulate all the
observations one cared to, without ever spotting what form
the law should take. For many centuries, indeed, scientists
were with striking distance, but failed to discover it: Ptolemy,
about A.D. 100, had already made many important obser-
vations on the subject but, like Roger Bacon and Kepler
later on, failed narrowly to hit on the law which, in 1621,
Snell at last formulated.

These things are connected with the fact that what Snell
discovered was, again, the form of a regularity whose existence
was already recognized. Ptolemy, Bacon and Kepler could not
have studied refraction in the way they did unless they had been
sure that there was some regularity to be discovered: indeed,
it will be clear to anyone who studies the phenomena concerned
that they are of a sort that cries out for an explanation. But
though the existence of a regularity was clear to them, at any
rate so long as one kept away from Iceland Spar and other
anomalous materials, it remained to be found out what form
the regularity took. This was what their experiments were
designed to reveal, or rather, what they hoped to be able to
spot from the results of their experiments.

LAWS OF NATURE 65

To bring out the force of these points, consider how one
might set about establishing SnelPs Law. Let us discuss, there-
fore, what kind of apparatus we might assemble in order to
collect suitable data. There are several important morals
which we can illustrate in the course of this examination;
first, about the place of experiments in the physical sciences,
and secondly, about the relation between concepts of theory
(such as 'light-ray') and the phenomena they are used to
explain.

The question we have to ask is, in participant's language,
"What happens to light-rays when they enter refracting
media?" or, to put the same thing in onlooker's language,
"How are we to extend the techniques of geometrical optics
to account for the optical phenomena we meet in the presence
of glass, water and the like?" This is very much the sort of
explicit and limited problem that we can hope to tackle experi-
mentally. But a number of things require to be done, if we are
to achieve anything:

(i) The theoretical notion of a light-ray must be given
some more definite practical realization. Means are needed
for producing beams of light, in the everyday sense of the phrase,
which will approximate as nearly as need be to the Euclidean
ideal of breadthlessness, and which will therefore be of a
kind that we can accurately represent by geometrically straight
lines. Until this is done, we shall have nothing that we can
confidently treat as light-rays, and so nothing to study in our
attempt to extend the theory and techniques of geometrical
optics to the new field.

(ii) We must find out under what circumstances the pheno-
mena of refraction will be reproducible and steady: whatever
apparatus we assemble must provide us with phenomena worth
investigation.

(iii) We must so arrange our apparatus that we can make
measurements on it comparable with those we made when
studying shadows. Only if we do so, shall we have any way of
choosing how to extend the techniques of geometrical optics
to the new field: otherwise the techniques will have nothing
precise to explain.

66 THE^PHILOSOPHY OF SCIENCE

These considerations are worth setting out in detail,
for they can be used to illustrate an important fact. No com-
petent scientist does pointless or unplanned experiments. There
is no place in science for random observations, and only in the
rarest cases have scientists made experiments whose results
were of any value, without knowing very well what they were
about. Before the scientist enters his laboratory at all, he must
therefore have guidance about the kind of state of affairs worth
investigation, the type of apparatus worth assembling, and
the sort of measurements worth making. This guidance can
come only from a careful statement of his theoretical problem,
and if one looks at the conditions of the experiment he performs
one will find that they are tailor-made to suit this theoretical
problem.

In the present case, for instance, what is required is for
the scientist to pass extra-narrow beams of light in precisely
measurable directions through carefully ground prisms or
lenses of unusually homogeneous glass. By arranging for the
light-beams to be as narrow as possible we satisfy condition (i)
the narrower they are, the nearer they become a physical
realization of the theoretical ideal of a light-ray. By demanding
that our lenses or prisms be carefully ground from glass of
greater than usual homogeneity we satisfy condition (ii); for
only if we take some such precautions shall we find that our
phenomena are sufficiently steady and reproducible to be worth
studying. And by noting precisely the directions of the narrow
beams of light both outside and inside the glass, we provide
ourselves with observations comparable with those that we are
used to dealing with in the more restricted circumstances
which we have been studying up to now. Here as elsewhere,
if you want to understand why a scientist is performing a
particular experiment, ask how his problem came to be posed
and what it was in his theory which led up to it. If you under-
stand the theoretical problem, the reasons for the conditions
of the experiment will almost certainly be clear to you: unless
you understand the problem, they certainly will not.

Here again we must recognize the great differences between
the physical sciences and natural history. The naturalist can

LAWS OF NATURE 67

afford to keep his eyes skinned from the start: it is never too
soon to notice some fact of interest about the birds and animals
around him. In physics, by contrast, it may easily be too soon
to make any observations: until your theoretical problem has
been carefully thought out, experiments will be premature.
The naturalist goes about the world with an open eye and mind,
prepared to notice anything of interest that may occur in his
path. But the physicist does not enter his laboratory until
he has some completely specific question to answer; and his
apparatus will be carefully designed to extort the material he
needs for an answer to this question.

Let us consider next how an experimental apparatus might
be designed in order to fit our particular theoretical problem.
First, there is the problem of getting light to travel in suffi-
ciently straight and narrow beams, and in sufficiently precise
directions. Normally light fans out as the origin of the word
'ray', the Latin radius, reminds us, our first exemplars are
the sun's rays spreading out in all directions. The difficulties
one encounters when one tries to get a beam sufficiently narrow
for experimental purposes are instructive, and illustrate well
the nature of our theoretical concepts.

The first difficulty is a purely practical one, which raises
no theoretical problems. One might begin by thinking that all
one needed was a bright lamp and a single screen having a
narrow slit in it:

This, however, will not be satisfactory, however narrow we
make the slit in the screen. Since the glowing filament of the
lamp will be at least a millimetre or two across, we shall obtain
not a narrow beam of light, but a fan diverging from an angle

68

THE PHILOSOPHY OF SCIENCE

(a) of several degrees, quite unsuitable for precise measure-
ments. This, of course, is to be expected even on the principles
of geometrical optics.

The natural next suggestion, which is the basis of all the
equipment used in experiments of this kind, is to employ two
screens (S l9 S 2 ) each with an adjustable slit in it, the slit in
the first acting as the source of light for the second.

Given this set-up, there seems no reason, on the principles
of geometrical optics, why we should not make the angle of
divergence (/?) of the resultant fan as small as we please, and
so obtain as narrow a beam as near an approximation to our
theoretical light-ray as we choose. All we need do, according
to geometrical optics, is make the slits in the two screens
progressively narrower.

What do we find if we set up such an ap paratus? Up to a
point all goes as we expect. We erect a third screen (T) as a
target, and gradually make the slit in S 2 narrower and narrower;
and to begin with, the breadth of the bright line (b) where our
beam strikes the target decreases. But if we go on narrowing the
slit, then after a certain point we get no further advantage: the
only effect of doing so is to blur the line on the target, to spread
it out and make it fuzzier. We are up against the phenomenon
physicists speak of as diffraction.

What is the moral of this discovery? Is this the death-knell

LAWS OF NATURE 69

of geometrical optics must we conclude that its principles
have failed us, and must be given up? So must we abandon the
hope of extending to other fields the techniques which proved
so useful for explaining shadow-casting?

These reactions would be too drastic. For our discovery
need only remind us that, like all techniques, the inferring
techniques of geometrical optics have a limited scope. We
can rely on them to explain a great range of optical phenomena
with a high degree of accuracy, but beyond that point other
methods are needed. Further, it will remind us that when we
represent light by Euclidean straight lines we are setting up a
theoretical ideal: it remains to be discovered from experience
how far this theoretical ideal of a light-ray can be realized.
Just as it is too simple to regard the discovery that light travels
in straight lines as the discovery of an ordinary, but novel,
matter of fact, so the term 'light-ray' as it appears in theoretical
arguments must be understood as an ideal, introduced for the
interpretation of the inferences of geometrical optics: it should
not be thought of, so to speak, as the name of a new species
of object found in a hitherto-unexplored jungle, to which we
have to give a name, and whose habits it is for physicists to
study.

The actual practice of scientists in such a situation as this
is to recognize the existence of the limits set by diffraction,
and keep clear of them in all arguments and experiments in
geometrical optics. Diffraction effects will themselves be some-
thing to investigate in due course, but they are a subject for
physical optics, along with other problems connected with the
question "What is it that travels?" or, in physicist's language,
"What is the nature of light?" : the limitations we find ourselves
forced to place on the application of geometrical techniques are
themselves something to be explained though naturally
something which cannot be accounted for within geometrical
optics itself, but requires a richer and more refined mode of
representation for its explanation. With these allowances,
physicists can carry on as before. The discovery of diffraction
does not prove that it is untrue that light travels in straight
lines, for such a principle, as we shall see, cannot be spoken

70 THE PHILOSOPHY OF SCIENCE

of as true or untrue in any simple sense. No more did Einstein's
work prove that Newton's Laws of Motion were untrue. It
accounted for some limits, which had hitherto been unexplained,
to the accuracy with which Newton's mechanics can be used
to calculate the motions of the planets; but it superseded
Newton's mechanics only for the most refined theoretical
purposes, and could only whimsically be said to prove the older
laws of motion untrue.

3.3 Theoretical ideals and the world

It is worth while at this point considering a little more
carefully the status of theoretical ideals in physics, for it is by
using these ideals that the physical sciences become, as they are
sometimes called, exact sciences.

It is easy to misconceive the nature of this exactitude,
for two utterly different things have to be distinguished:
the mathematical exactitude with which inferences are drawn
in physics, and the practical exactness with which the conclusions
of these inferences can be applied to the systems physicists
study. It is the former which marks off the exact sciences from
other subjects, for this exactitude is characteristic of the in-
ferences we make in physics, genetics and the like, and is
commonly absent when we turn, say, to the study of ants'
eggs. The exactness of practical application, on the other hand
the degree of accuracy with which our theoretical conclusions
fit the facts is not something which marks off all the exact
sciences equally, being greater in some branches than in others.

Thus in geometrical optics, using the notion of a light-ray,
we can make all sorts of statements, such as Snell's Law, in
exact mathematically exact terms. Likewise we can draw
inferences, diagrammatically or trigonometrically, as exactly
as we please: so far as the mathematics of the subject is con-
cerned, we can compute the length of a wall's shadow from the
heights of the wall and the sun to as many places of decimals
as we choose. But all these statements and inferences will have
a physical meaning only up to a certain point. This is not only
because the sun itself has an appreciable width, so that the
shadows it casts cannot in practice have more than a certain

LAWS OF NATURE 71

sharpness: it arises also from the fact that the arguments of
physics are conducted in terms of ideals, and there is always
some limit to the extent to which we have found ways either
of realizing these ideals, or of recognizing bodies or systems
which can be accepted as realizing them as accurately as we
can measure.

Another example: if we do dynamical calculations in terms
of 'rigid rods', our conclusions will again be both unique and
indefinitely exact. But they will once more be about ideals:
before we can draw any conclusions about the actual rods
from which machines and houses are built, we must know
how far the rods with which we are concerned can be treated
theoretically as rigid rods, and the inferences will apply to them
only as accurately as they are rigid. And what goes for rigidity
goes also for other properties: there is a large family of words
in the physical sciences 'rigid', 'exact', 'straight' etc. whose
members lead this kind of double life. In each case, we may
contrast either the exactitude of mathematics with the in-
exactitude of experimental reports, the rigidity of the rods
we argue about with the flexibility of actual rods, the perfect
straightness of Euclid's lines with the imperfect straightness
of any line we draw, or the high degree of exactness with which
physical optics fits the facts with the comparative inexactness
of geometrical optics, the extreme rigidity of ferro-concrete
with the comparative flexibility of copper, the unusual straight-
ness of Roman roads with the comparative windingness of
most country lanes. Trouble begins in philosophy, and serious
trouble at that, when we use such words as these without being
clear which of the two contrasts we are intending to draw.

Furthermore, it is easy to overlook the ideal status of a term
like 'light ray', and to suppose that the phrase refers simply
to sunbeams and similar things. If we do this, we may be inclined
to regard the doctrine of rectilinear propagation as a way of
reporting such phenomena as the luminous streak which light
pouring through a window makes in the air. But this will not
do. For, to begin with, it is only because there are dust-motes
in the air, which scatter the incoming light, that one encounters
this phenomenon at all: the more visible the beam, the less

72 THE PHILOSOPHY OF SCIENCE

completely is the light actually travelling in a straight line,
In addition, the notion of a light-ray is tied to our optical
explanations in a way in which that of a sunbeam is not. A child
might learn to talk about sunbeams and yet have no conception
of geometrical optics; but a man cannot be said to know what
is meant by the term 'light-rays' if he does not understand the
diagrams which we use when explaining shadow-casting.
There is in fact no more direct a connexion between rays of
light in the everyday sense of the phrase, such as sunbeams,
and light-rays as physicists speak of them, than there is between
the light which, on an August afternoon, dapples the apples
and lies in great pools around the lawn and the physicist's
light, which could not meaningfully be said to lie around
anywhere.

This, of course, is not to deny that sunbeams are light-
rays, or composed of light-rays. Certainly we shall often be
able to apply to sunbeams the inferences that we draw in terms
of light-rays: we did this without hesitation in calculating the
depth of the wall's shadow. It is, rather, to mark the distinction
in logic between words like 'sunbeam' and phrases like light-
ray', i.e. to draw a distinction of logical type, like that between
the person and name of Winston Churchill and the title and
office of Prime Minister; and this can be done regardless of
whether or not in fact Winston Churchill at present holds the
office of, and so is describable as Prime Minister.

Similar distinctions are important when one examines the
use which is made in geometry and physics of the terms
'point', 'particle', etc. Old-fashioned text-books tend to start
off with mystifying definitions of these terms: Euclid's own
definition, "A point is that which has no part", is a good example.
After a perfunctory discussion of these, the author clears his
throat, begins a new chapter and gets going with some concrete
examples: the definitions are mercifully forgotten. And this
is as it should be. Definitions of these terms are not called for,
and the more self-conscious authors of text-books are at last
ceasing even to go through the motions of defining them.
For the questions to be asked about points, particles and the
rest arc not "What is a point?", "What is a particle?" etc.:

LAWS OF NATURE 73

they are "What can be regarded for physical purposes as a point,
particle, etc.?" Or rather, since we soon find out that under
some circumstances or other almost any region of space can be
treated as a point, and almost any body even the sun
as a particle, the sort of question to be asked is, "Under what
circumstances can the sun, say, be regarded as a particle?";
or, what comes to the same thing, "Under what circumstances
can the inferences we make in terms of particles in our dynamical
calculations be applied to the sun, and its dimensions be
neglected?" A particle in dynamics is not 'an indefinitely small
material object': if one insists on a definition, it is 'any material
object whose dimensions can, for the purposes of the present
calculation, be neglected'. 1

This brings us back to the notions of exactness and exacti-
tude. For in practice we shall always have to ask, not "Is an
aeroplane a particle, or a sunbeam a ray of light?", but "Under
what circumstances and with what degree of exactness, i.e.
accuracy, can one treat an aeroplane as a particle for dynamical
purposes, or a sunbeam as a ray of light for optical ones?"
The inferences of physical theory remain in every case exact:
it is the accuracy with which the conclusions are applied
that varies.

3.4 Establishing a law of nature (II)

So much for the first of our problems, that of realizing our
theoretical ideal of a light-ray. Let us suppose, then, that we
have assembled a pair of screens with narrow slits in them, a
bright lamp, and a target screen, and that the slits are set to
provide a beam which is as narrow as is practicable, bearing in
mind the limits that we have been discussing. Now we have
produced some light rays, or near enough, what about our
refracting medium?

At this point we encounter the second of the practical
problems facing us: how to ensure that we have steady and
reproducible phenomena to study. If we set up the same
apparatus two days running and go, to the best of our know-

l These remarks do not apply as they stand to the 'fundamental particles'
of atomic theory.

74 THE PHILOSOPHY OF SCIENCE

ledge and belief, through identical steps each time, and the
optical phenomena we observe on the two days are markedly
different, we are clearly in no position to make any worth-
while observations: still more so, if we set up the apparatus
and the phenomena fluctuate under our very eyes.

Any experimental set-up in a laboratory is inevitably a
highly artificial one. When it comes to studying refraction, say,
especially with such a specific end in view, one cannot hope
to find suitable specimens for one's experiments simply lying
around. Notice, incidentally, the contrast with natural history:
the naturalist must take his frogs as he finds them. Nor could
one be confident that one's apparatus was going to satisfy all
the required conditions if it consisted merely of a collection of
objets trouves. Such more or less transparent objects as one might
pick up would be of largely unknown physical and chemical
composition and of unsuitable shapes, whereas we require to
study only objects whose characteristics we know, and whose
shape in particular we can precisely control: hence the demand
for accurately ground prisms. Further, if we used any glass
a manufacturer happened to supply, we might still find that
the rays in the glass tended to waggle: we must therefore get
the manufacturers to supply specially homogeneous, so-called
'optical' glass, carefully mixed and slowly cooled for con-
sistency. It is worth noticing, by the way, that this recipe in-
volves a hypothesis that the optical properties of a material
depend on the constancy of its density and on its degree of
homogeneity: how far this is the case is something which requires
independent investigation. Again, if we are careless about
temperature variations, we shall find our results varying:
certain precautions will have to be taken keeping Bunsen
burners away from the apparatus and shading it from the sun
if our experiments are to be fruitful.

These, in the present example, are the most important
precautions. If others were needed in order to control relevant
factors and ensure steady and reproducible results, they could
no doubt be taken. But just what steps will be Heeded, just
what factors are relevant to any question and therefore have
to be controlled in an experiment, is something which will have

LAWS OF NATURE 75

to be found out: there can be no general recipe. In this respect,
the demand for homogeneous glass or the avoidance of tempera-
ture variations is on a different footing from the precautions
we take in order to obtain narrow beams of light with precisely
measurable directions: these last steps are essential, not for
practical reasons, but for theoretical ones, in order that the
apparatus shall be capable of helping us to the solution of our
theoretical problem.

The list of precautions may, in some experiments, be fairly
long, but it will always be finite and definite. If an experiment
gives an unexpected result, the conclusion that some relevant
factor must have been overlooked is normally acceptable only
when a possible factor can be suggested and investigated:
perhaps the test-tube was not clean. In a well-planned experi-
ment, this can be checked for all the factors which there is any
reason to consider as relevant. So, though in fact 'escape-
clauses' of this kind may sometimes be invoked, one cannot
do so arbitrarily, merely in order to preserve a particular
theory from discredit. We are not forced, accordingly, to
speak of physical theories as conventional: the discovery that
for some phenomenon to occur as it normally does, some
factor which is constant under normal experimental conditions,
magnetic field gradient, say, has to remain constant, may be a
major discovery. The Zeeman and Stark effects might be quoted
as examples of this sort of discovery: it would not at first occur
to one that the kind of light radiated by a body depended on
the strength of the magnetic and electric fields to which it
was exposed.

The apparatus with which we shall study refraction can
be thought of, then, as consisting of three things: a source of
light arranged to emit as narrow a beam as is practicable, a
parallel-sided specimen of the material being studied, carefully
made and mounted so that the direction at which the beam
strikes it can be accurately measured, and a target screen or
other device for observing how much the specimen deflects
the light which passes through it.

Two questions now need to be asked: what sort of obser-
vations shall we make with this apparatus, and how will they

76 THE PHILOSOPHY OF SCIENCE

be connected with the conclusion we are using them to establish,
viz. SnelFs Law?

There are various different sorts of observations we might
make: it is enough to consider a typical one. Supposing,
therefore, that we have arranged the specimen to swivel
through any angle we please, let us set it successively at angles
to the beam of 0, 5, 10, 15 ... and so on.

LIGHT

r

SPECIMEN

As we turn it progressively more and more askew, the bright
line on the target will be shifted more and more from its first
position. Let us make a note of the amount of the deflection
(x) corresponding to each angle (i) at which we set the specimen:
it will be a matter of simple geometry to compute from the
amount of the deflection the angle of refraction (r) of the light-
beam within the specimen. The results can then be tabulated
in three columns: letting of specimen, i\ 'reading of deflection,
x\ and 'corresponding angle of refraction, r\ The figures we
write down will in each case be subject to a 'probable error*,
to allow for inaccuracies in the measuring technique, the grind-
ing of the specimen and so on. It is always enough that the
predictions of theory should largely fall within the region
marked off by the probable error: one does not need to insist
that every reading made should tally exactly with the theory.

What, now, is the connexion between the figures we have
tabulated in our note-book and the law we are using them to
establish? Looking at the observations and the law from a
logician's point of view, what shall we say is the relation
between them? There is certainly no deductive connexion
either way between Snell's Law and the set of statements,

LAWS OF NATURE 77

"When the specimen was set at 5, the deflection of the beam
was 2 mms." Nor is the law to be thought of as a simple
generalization of the experimental results, despite the words
'whenever' and 'always' appearing in the formulation of it
quoted earlier on. These words are used misleadingly, for the
law is no more a universal generalization than the Rectilinear
Propagation Principle turned out to be: presented with any
of the situations and substances to which it ceases to be appli-
cable, a physicist will bring into the open his unstated qualify-
ing clauses 'anomalous refraction apart', 'the specimen being
homogeneous' and the rest. Leaving aside these clauses, which
are concerned with the application of the law, we are left with
the statement of the form of a regularity that the sines of
two angles are in a constant ratio and the value of the experi-
ment is to show just how accurately this form of regularity
fits the observed phenomena. "When the sun was at 30,
the shadow of a 6 ft. high wall was 10 ft. 6 in. deep . . . : ergo,
light travels in straight lines," "When the specimen was set
at 5, the deflection being 2 mms., the angle of refraction was
3 . . . : ergo the ratio of the sines of the angles of incidence and
refraction is constant": though these two steps are by no means
identical in type, they are at any rate alike in conforming tidily
to neither of the standard logic-book patterns of argument.

3.5 The structure of theories: laws, hypotheses and principles

In the last chapter, we remarked briefly on the special logical
character of the nature-statements we meet in physical theory,
and on the systematic character of scientific, as opposed to
everyday language. These are things which can be made more
intelligible with the help of the examples we have looked at
in the present chapter. By noticing how the different types of
statement we encounter in the theory of refraction are logically
related, we can see what people have in mind when they speak
of such theories as forming hierarchical or deductive systems.

Notice for a start, then, how the way in which physicists
handle their theoretical statements marks them off from the
familiar statements of everyday life, and from those of the
naturalist. First, since Snell's Law is stated in terms of 'light-

78 THE PHILOSOPHY OF SCIENCE

rays', it can be given a physical meaning only in circumstances
in which the term light-ray* is intelligible, i.e. within the scope
of geometrical optics. Where the optical phenomena are not
such as are explicable in terms of geometrical optics, Snell's
Law ceases even to be interpretable. Secondly, it is the practice
in the physical sciences to leave the application of a law to
be shown or stated separately: indeed, this itself is rather a
misleading thing to say, for that this should be done is not so
much a question of practice as the distinguishing mark of a law.
The statement, "Most transparent substances of uniform
density, excluding only certain crystalline materials, such as
Iceland Spar, have been found to refract light in such-and-
such a manner" is not what we call 'Snell's Law'. This state-
ment is a simple report of past fact, and its job is to tell us
about the circumstances in which Snell's Law has been found
to hold. To every law there corresponds a set of statements
of the form "X's law has been found to hold, or not to hold,
for such-and-such systems under such-and-such circum-
stances." Further, in order to discover how far this range of
substances and circumstances, i.e. the 'scope' of the law,
can be extended, a great deal of routine research is under-
taken, research which can in no way be said to call in question
the truth, or the acceptability, of the law itself.

This feature is one which is not shared in everyday speech
even by those statements which Ryle calls 'law-like statements',
such as "Glass is brittle". When a manufacturer produces
a new type of glass of exceptional toughness and resilience,
we say, "All glass except Tompkinson's Tuff glaze is brittle",
not " 'Glass is brittle' holds for all glass except Tompkinson's
TufFglaze". This invention certainly affects the truth of our
initial statement: after it, the law-like statement is said to be
'true on the whole but not true of Tompkinson's Tuffglaze',
whereas before the invention it had been 'true universally'.

Laws of nature, however, are different: to them the words
'true', 'probable' and the like seem to have no application. 1
To begin with, perhaps, we may suppose that light-rays are

*At any rate if we ask, "Is this law true?": on the other hand we can ask
the question, "Is this the true (form of the) law?'*

LAWS OF NATURE 79

always bent by transparent media in the way they are by the
glass specimen in our apparatus. We may, therefore, adopt
Snell's formula tentatively, hypothetically, as a guide to further
experiments, to see whether the phenomena always happen
so. On this level, we might ask "Is Snell's hypothesis true
or false?", meaning "Have any limitations been found to the
application of his formula?" But very soon indeed, as soon
as its fruitfulness has been established the formula in our
hypothesis comes to be treated as a law, i.e. as something of
which we ask not "Is it true?" but "When does it hold?"
When this happens, it becomes part of the framework of optical
theory, and is treated as a standard. Departures from the law
and limitations on its scope, such as double refraction and aniso-
tropic refraction, come to be spoken of as anomalies and thought
of as things in need of explanation in a way in which ordinary
refraction is not; and at the same time the statement of the
law comes to be separated from statements about the scope
and application of the law.

In this last respect, laws of nature resemble other kinds of
laws, rules and regulations. These are not themselves true or
false, though statements about their range of application can be.
Suppose there is a College rule against walking on the grass:
one can ask how widely it applies whether there is any class
of people, such as Fellows, who are not bound by it. Accord-
ingly, statements can be made about the rule which can be
true or false. If it is said that, despite the rule, Fellows are
allowed on the grass, one may reasonably ask, "Is that true?"
But one will not ask "Is the rule true?", nor will physicists ask
this of a law of nature.

This must not be misunderstood. Suppose one says that
laws of nature are not true, false, or probable ; that these terms
are indeed not even applicable to them; and that scientists
are accordingly not interested in the question of the 'truth'
of laws of nature all of which might fairly be said: one does
not thereby deny the obvious, namely, that scientists seek for
the truth. One points out, rather, that the abstract noun
'truth' is wider in its application than the adjective 'true',
that different types of statements need to be logically assessed

80 THE PHILOSOPHY OF SCIENCE

in different terms, and that not every class of statement in
which a scientist deals need be such as can be spoken of as
'true'/'false'/'probable'. This, of all things, is most often over-
looked in the logical discussion of the physical sciences: it
is therefore essential to insist on it. Saying a law holds univers-
ally is not the same as saying that it is true always and not only
on certain conditions. The logical opposition 'holds'/'does not
hold' is as fundamental as the opposition 'true'/'untrue', and
cannot be resolved into it.

Further, laws of nature are used to introduce new terms into
the language of physics the term 'refractive index', for
instance and such things as refractive index become in their
turn subjects for research. How, we may ask, does the re-
fractive index of a substance depend on its temperature? How,
to use onlooker's language, would we have to alter our ray-
diagrams in order to account for the way in which the optical
phenomena are affected by heating up the specimen, or for
such things as the shimmering in the air over a bonfire?
Notice one thing in particular: that questions about refractive
index will have a meaning only in so far as Snell's Law holds, so
that in talking about refractive index we have to take the appli-
cability of Snell's Law for granted the law is an essential part
of the theoretical background against which alone the notion
of refractive index can be discussed. This is something we
find generally in physical theory. Theoretical physics is
stratified: statements at one level have a meaning only within
the scope of these in the level below.

This fact must be borne in mind when we consider the
discussion between the hypothetical and established parts of
physics, for this is a distinction which has been widely mis-
conceived. It has been said by some philosophers, for example,
that all empirical statements are hypotheses, which can,
strictly speaking, never be called more than 'highly probable':
in support of this view they have pointed out that we could
always, by a sufficient stretch of imagination, "conceive the
possibility of experiences which would compel us to revise
them". Now it is important to recognize what violence this
sort of argument does to the terms 'hypothesis' and 'hy-

LAWS OF NATURE 81

pothetical'. For although all the statements we meet in science
are such that one can conceive of their being reconsidered in
the light of experience (i.e. empirical), only some of them can,
in the present sense, be called 'hypothetical'. We are now in
a position to see why this is.

One can distinguish, in any science, between the problems
which are currently under discussion, and those earlier prob-
lems whose solutions have to be taken for granted if we are even
to state our current problems. One cannot at the same time
question the adequacy of Snell's Law and go on talking
about refractive index. But the fact that, at any particular
stage, many of the propositions are taken without question
does not make the exact sciences any the less empirical: it
merely reflects their logical stratification. Certainly, every state-
ment in a science should conceivably be capable of being called
in question, and of being shown empirically to be unjustified;
for only so can the science be saved from dogmatism. But it
is equally important that in any particular investigation,
many of these propositions should not actually be called in
question, for by questioning some we deprive others of their
very meaning. It is in this sense that the propositions of an
exact science form a hierarchy, and are built one upon another;
and just as a bricklayer is only called upon at a given moment
to determine the positions of the bricks in a single course
which in their turn will become the foundation for the next
course so the scientist is only called upon at any one time to
investigate the acceptability of statements at one level. Now
and then there may have to be second thoughts about matters
which had been thought to be settled, but when this happens,
and the lower courses have to be altered, the superstructure
has to be knocked down, too, and a batch of concepts in terms
of which the scientist's working problems used to be stated
'phlogiston' and the like will be swept into the pages of the
history books. But for the time being it is only the top course
of bricks, the matters which are actively in question, which
the scientist has to deal with. From this we can see why the
discovery of phenomena which can be treated as standards
and of laws which can, to use a phrase of Wittgenstein's, be

82 THE PHILOSOPHY OF SCIENCE

put in the archives, is an essential step in building up a fruitful
body of theory.

The terms 'established' and 'hypothetical', as used in
science, need to be understood in terms of the distinction
between the parts of a science that are actually being called
in question, and those which we must take for granted in order
to state our working problems. It is the statements that figure
in the latter parts which are spoken of as established. Even a
few of these may eventually have to be reconsidered, but there
is no need nor are we in a position to anticipate the day
when this will happen. These parts provide the background
against which current problems are considered, and give a
meaning to the terminology in which they are stated. The
statements which we meet in them will be of two kinds: on
the one hand, laws of nature, and on the other, statements
about how far and in what circumstances these laws have been
found to hold. Neither of these classes of statement need, or
can, be spoken of as 'only highly probable': the experimental
reports are not unlimited generalizations, but simple statements
of past fact, while the laws of nature are not the sorts of thing
we can speak of as true, false or probable at all. Yet both can
reasonably be called empirical.

Contrasted with the established parts of a science, there are
those problems the solutions of which are not yet clear, and
about which we can at the moment say only tentative, hypo-
thetical things. These questions are indeed open, undecided,
matters for 'hypotheses'. But the statements in these hypothetical
parts of a science depend for their very meaning upon the accept-
ability of the lower levels of theoretical statement; so we are
debarred from speaking of the established propositions as
being hypothetical also, unless and until they themselves
become once again the subjects of active doubt. It could be
correct to speak of all empirical propositions as hypotheses
only in a language which was entirely devoid of logical stratifi-
cation the language of a people without any science. This
stratification is a feature of the theoretical sciences in particular,
as is borne out once more by the contrast with natural history.
We should not so much mind saying that the generalizations

LAWS OF NATURE 83

of natural history can be at most highly probable: next year
a pig might fly.

\ The distinction between laws and hypotheses is therefore
a logical matter, involving far more than the degree of con-
fidence with which we are prepared to advance them, or the
number of confirming instances we have observed. But what
about the distinction between laws and principles? Why is the
Rectilinear Propagation of Light called a 'principle' and Snell's
Law a 'law'?

This distinction turns upon something we noticed earlier:
namely, the role of the principle as the keystone of geo-
metrical optics. One can quite well imagine a geometrical
optics in which the law of refraction was different. The adopt-
ion of a different law in place of Snell's Law would, of course,
mean considerable changes our present notion of refractive
index would be one casualty. But geometrical optics could
still exist as a subject, and designers of optical instruments,
having learnt the new rule for tracing the passage of rays
through their assemblies of lenses, would soon accommodate
themselves to the change. By comparison, the principle that
light travels in straight lines seems to be almost indefeasible:
certainly it is hard to imagine physicists abandoning completely
the idea of light as something travelling in straight lines, for
to give up this principle would involve abandoning geometrical
optics as we know it. If we question the principle of rectilinear
propagation, the whole subject is at stake: that is why the
principle is not open to falsification in any straightforward
way.

It is not that, for physicists, the principle ceases to be
empirical and becomes tautologous or conventionally true.
They might, in circumstances sufficiently unlike the present
ones, decide to give it up entirely, but they would do so only
if they were ready to write off geometrical optics as a whole.
What the circumstances would have to be, in order for physicists
to decide that the methods of geometrical optics were no longer
any use, is something that is open to discussion, but this would
clearly require changes in the world far more drastic than those
which are needed to falsify any naive interpretation of "Light

84 THE PHILOSOPHY OF SCIENCE

travels in straight lines", e.g. as an empirical generalization.

It is the middle-level propositions in the hierarchy of
physics which alone are called 'laws', and they alone have an
ambivalent logical status. Such a proposition as Snell's Law
begins as an element in a hypothesis within geometrical
optics, something which cannot be explained without talking
about light-rays; but later it becomes an established part of
the theoretical background, while the foreground is occupied
by other propositions which have a meaning only where the
law holds. Since its place is within geometrical optics, to change
the form of the law is not to raze a whole subject to the ground.
There is, by contrast, no body of theory against which the
proposition that light travels in straight lines can be set. It
is as though this principle enshrined in itself the geometrical
mode of representation; and it can be discussed, accepted or
rejected on one level only.

One last point about the stratification of physical theory:
this is sometimes presented in a misleading way. It is suggested
that the relation between statements at one level and those
at the next is a deductive one, and the resulting hierarchy is
accordingly spoken of as a 'deductive system'. One is given the
idea that physical theories form a logical pyramid, with the
straightforward reports on our experimental observations at
ground level, and above them layer upon layer of progressively
more general generalizations. One can illustrate the sort of
thing envisaged by supposing it to be discovered that rodents
consume milk-products: this would be two layers up, since
from it we can deduce both "Mice eat cheese" and "Rats
drink milk", and from these again we can deduce, e.g., that
a mouse which we now have under observation will eat the
cheese we are about to offer it.

As here presented, the picture is open to several objections.
To begin with, the role of deduction in physics is not to take
us from the more abstract levels of theory to the more concrete:
as we have seen, these cannot, as Mach supposed, be thought
of as deductively related one to another. Where we make
strict, rule-guided inferences in physics is in working out,
for instance, where a planet will be next week from a knowledge

LAWS OF NATURE 85

of its present position, velocity and so on: this inference is
not deduced from the laws of motion, but drawn in accordance
with them, that is, as an application of them. Nor are state-
ments in terms of 'refractive index* deduced from SnelPs
Law. There is a logical connexion between them, certainly;
but this is because the term 'refractive index' is introduced by
reference to Snell's Law, and not because the two classes of
sentences can be deduced from one another. It is the terms
appearing in the statements at one level, not the statements
themselves, which are logically linked to the statements in
the level below.

One thing in particular would be especially mysterious
on the deductive system account; namely, the status of the
most abstract statements of all. For, if things were as suggested,
each of these would be the assertion of a tremendous coin-
cidence: if it were a coincidence that not only did mice eat cheese
but also rats drank milk, so that we could daringly generalize
that rodents consume milk-products, how much more of a
coincidence must it be, to cite one of Einstein's most abstract
principles, that Gp,v=\. Further, like all such coincidences,
the most abstract statements of all would be particularly open
to sudden upset; for surely some obscure South American
rodent might turn out to be entirely herbivorous, and, if so,
how much less likely still that no exception would ever be
found to < G/ui'=A'. But this, of course, is a caricature. It is clear
from a study of Einstein's work that he is concerned, not with
daringly wide generalizations from experiment, but rather
with conceptual matters: such an equation as 'G/-ii>=A' certainly
does not have the status the pyramid-model allots to it. Indeed,
it is no accident that one has to resort to habit-statements
about rats, mice and the like, in order to illustrate the point
of the pyramid-model; for, however it might do as a picture
of natural history, it misrepresents the logical structure of
theoretical physics.

3.6 Different kinds of laws and principles

In this chapter, as in the last, we must ask how many of the
things we have noticed about the example under detailed

86 THE PHILOSOPHY OF SCIENCE

examination apply more generally. How far, then, can we
regard Snell's Law as a typical law of nature?

Many of the things we have said about it would not be
true of all laws equally, for there is a wide range of things which
are spoken of in physics as laws of nature. At one extreme, one
finds statements of the sort which are sometimes called
'phenomenological laws'. These involve no theoretical terms at
all, not even such elementary ones as 'light-ray 1 : a good instance
is Boyle's Law, which states that the pressure and volume
of a gas vary inversely at a given temperature. At the other
extreme, one has such laws, or sets of laws, as Newton's three
Laws of Motion, or Maxwell's Laws or Principles of Elec-
tromagnetism: these are not used directly to express the form
of a regularity found in phenomena, as Boyle's Law is, but are
rather like the axioms of a calculus, which are accepted so
long as applications of them are found in practice to fit the facts.
It will be the test of such comparatively abstract laws, not so
much that they account directly for the observed phenomena,
as that they provide a framework into which can be fitted the
phenomenological laws which in their turn account for the
phenomena. Snell's Law is of an intermediate kind, though
one which is nearer to Boyle's Law than to Newton's three Laws ;
while the most abstract laws like Maxwell's Principles of
Electromagnetism and the Principles of Thermodynamics
come in time to have a position in their subjects almost like
that of the Rectilinear Propagation Principle in geometrical
optics, and are perhaps spoken of more naturally as principles
than as laws of nature.

Since the parts which different laws of nature play are
so very different, one cannot expect them to have many features
in common. But one such feature they do have; and it is one
which, in the case of Snell's Law, proved of the first import-
ance. They do not tell us anything about phenomena, if taken
by themselves, but rather express the form of a regularity
whose scope is stated elsewhere ; and accordingly, they are the
sorts of statements about which it is appropriate to ask, not
"Is this true or not?" but rather "To what systems can this
be applied?" or "Under what circumstances does this hold?"

LAWS OF NATURE 87

Boyle's Law, of all laws of nature, looks most as though one
could ask of it, "Is this true or not?" ; yet even it would nowadays
be treated in a way which rules this out. We know very well that
in some comparatively unusual circumstances gases can be shown
to behave in ways markedly at variance with Boyle's Law; and
that at all temperatures their behaviour deviates from it to
a minute but measurable extent. For theoretical reasons, as
well as reasons of convenience, however, it is preferable not
to regard this as a reason for scrapping Boyle's Law, but to
keep the law in circulation as a first, more-or-less approximate
expression of the way gases behave. The extent to which, in
different circumstances, the observed behaviour of gases con-
forms to or deviates from it is then recorded separately; and
accordingly the question whether the law is true or not no
longer arises.

There are indeed certain laws in physics that one might
take, at first sight, for exceptions to the rule that laws of nature
are not 'true or untrue' but rather 'hold or do not hold'; for
instance, Kepler's three Laws of jlanetary motion. These
laws tell usTamong other things, that the planets move round the
sun in ellipses, and they are unquestionably statements about
which one can ask, "Is this true or not?" if they correctly
represent the orbits of the planets, they are true: if not, they
are untrue. But along with this difference go others, which show
the force of our rule. For Kepler's Laws set out to tell us,
not about planets in general, but about the planets, viz. Mercury,
Venus, etc.; they summarize the observed behaviour of all
members of this class, and do not set out to explain it in terms
of the nature of things; they are thus even more completely
phenomenological than Boyle's Law; and correspondingly
no physicist would ever speak of them as laws of nature. One
could, no doubt, formulate three nature-statements, each of
which corresponded to one of Kepler's three Laws; but in
order to qualify as laws of nature these would have to be
expressed, not in terms of 'the planets', but in terms of 'bodies
moving under the influence of gravitation alone'. Such laws
would be the means, inter alia, of explaining Kepler's obser-
vational laws; but to identify them with Kepler's Laws would

88 THE PHILOSOPHY OF SCIENCE

be a mistake, since it would mean overlooking one logically
crucial step that of recognizing that 'the planets', viz.
Mercury, Venus etc., qualify for theoretical purposes as bodies
moving under gravitational attraction alone. As Wittgenstein
points out in the Tractatus, "The description of the world by
mechanics is always quite general. There is, for example, never
any mention of particular bodies in it, but always only of some
bodies or other."

The status of such sets of laws as Newton's three 'Axioms,
or Laws of Motion' is something which philosophers have
found perennially puzzling. Those students who take the
ordinary scientific training in dynamics find this question
passed over in text-books with a few embarrassed and incon-
sistent remarks. Experimental physicists like to talk as though
the laws were purely phenomenological; but this suggestion
is discredited by the discovery that three technical terms,
'mass', 'force' and 'momentum', are introduced into the subject
along with the three statements. After this, it is not surprising
if logicians who come to dynamics from a study of ordinary
discourse feel that the whole proceedings are tautological,
and the argument that the laws are thereby shown to be con-
ventional becomes attractive.

Each of these doctrines is in its way equally misleading,
for the true status of the laws of motion can be seen clearly
only if one examines in detail how they in fact enter into dy-
namical explanations. 1 When this is done, one finds that both
the everyday models with which one is tempted to compare
them are unsatisfactory. Newton's Laws of Motion are not
generalizations of the 'Rabbits are herbivorous' type; but
they are not for this reason any the more tautological (cf.
'Rabbits are animals'); and this is because they do not set out
by themselves to tell us anything about the actual motions of
particular bodies, but rather provide a form of description
to use in accounting for these motions. The heart of the matter
is put forcibly, and almost to the point of paradox r in a cele-
brated passage of Wittgenstein's: "The fact that it can be

1 Axiomatic theories really need a chapter to themselves: here there is
room only for the briefest of remarks tbout them.

LAWS OF NATURE 89

described by Newtonian mechanics tells us nothing about the
world; but this tells us something, namely, that it can be
described in that particular way in which as a matter of fact
it is described. " But we must notice that it is no denigration
of a system of mechanics to say that, by itself, it tells us nothing
about the world. This is not to say that it fails to do what it
sets out to do: it is to recognize its proper ambitions. As we saw
earlier, a description of the techniques of geometrical optics
by itself tells us nothing about shadows ; for this, we must find
out also how far and under what circumstances these techniques
can be employed. So also, laws of nature express only the forms
of regularities: the burden of our experimental observations
is borne, not by them, but by statements about when the laws
of optics hold, or how the laws of motion are to be used to
represent the actual motions of planets, projectiles, leaves,
ships and waves. There is, so to speak, a division of labour in
physics, between laws themselves and statements about the
ways in which, and the circumstances in which laws are to be
applied. It is by recognizing the nature of this division that
one comes to see how physicists steer their way between the
Scylla of fallible generalization and the Charybdis of empty
tautology.

If we are asked what the job of Newton's lawsjs, we may
not know at first whether to say thai they describe the way
things move, define such terms as 'force', 'mass' and 'momen-
tum', or tell us about the mode of measurement of force and
the rest. But there are very good reasons for this uncertainty.
The laws themselves do not do anything: it is we who do
things with them, and there are several different kinds of things
we can do with their help. In consequence, there is no need
for us to be puzzled by the question whether Newton's Laws
are descriptions, definitions, or assertions about methods of
measurement: rather, it is up to us to see how in some applica-
tions physicists use them to describe, say, the way a shell
moves, in others to define some such quantity as electromotive
force, and in others again to devise a mode of measurement
of, say, the mass of a new type of fundamental particle. It is
not that the laws have an ambiguous or hazy status: it is that

90 THE PHILOSOPHY OF SCIENCE

physicists are versatile in the applications to which they put the
laws.

3.7 Locke and Hume: Are laws of nature necessary or contingent!

In the light of this discussion of laws of nature, it will be
worth while examining the views philosophers have put
forward about them, to see how far these views truly reflect
the uses to which laws of nature are put in scientific practice,
and how far the disagreements that have arisen are a sign rather
of confusion or cross-purposes. But before we come to this,
it is important to do one thing: namely, to distinguish between
four different classes of sentence that one meets in books of
physics. When scientists use the word 'law', they do not always
trouble to show which class of statements they are referring to,
though when they do their usage is the one we have adopted:
only rarely, in fact, is there any strong reason for them to draw
these distinctions explicitly. As logicians, however, we cannot
afford not to distinguish between the various classes, since they
have markedly different logical characteristics; and in the
past philosophers have sometimes been less careful to do so
than they might have been.

The four classes of statement are the following:

(i) abstract, formal statements of a law or principle e.g.
Snell's Law, in the form quoted above;

(ii) historical reports about the discovered scope of a law
or principle e.g. the statement that Snell's Law has been
found to apply to most non-crystalline substances at normal
temperatures;

(iii) applications of a law or principle to particular cases
e.g. the statement that, in a particular prism now under
examination, the directions of the incident and refracted beams
vary in accordance with SnelPs Law; or the statement that the
sunlight getting over a certain wall is travelling to the ground
behind the wall in a straight line;

(iv) conclusions of inferences drawn in accordance with a
law or principle e.g. the conclusion that, the angle of incidence
and refractive index being what they are, the angle of refraction

LAWS OF NATURE 91

must be 36; or the conclusion that, with the sun at 30, the
shadow of a 6 ft. high wall must be 10 ft. 6 in. deep.

The main types of theory philosophers have put forward
about the logical character of laws of nature are also four in
number. This is no coincidence, for one finds exponents of
the four views citing, as support for their accounts, facts about
the appropriate one of our four types of statement. Accordingly,
these views may not really be the irreconcilable rivals they have
seemed. Perhaps their appearance of opposition reflects rather
a preoccupation with different aspects of laws of nature. How
far this is so, we must now consider.

On the one hand, then, one finds it suggested by Locke,
and more recently by Kneale, that laws of nature are principles
of natural necessitation, comparable with statements like
"Nothing can be both red and green all over" except in one
respect that where the necessity of the latter is something
we can 'see', the necessity of laws of nature is not immediately
visible, i.e. obvious, but is rather forced on us as a result of
our experiments. The metaphors 'transparent' and 'opaque to
the intellect' have been used by Kneale to mark the difference
between them. This view has been found objectionable by
such philosophers as Hume and Mach: they have felt that
nothing which a scientist can properly be said to discover
could be, in the logical sense, necessary, and they have accord-
ingly preferred to advance the theory that laws of nature are
statements of constant conjunction, which tell us that such-
and-such sets of characteristics have always been found to
go together. A third view, designed to circumvent the traditional
problems about induction, is that which Kneale attributes to
Whitehead: according to this, laws of nature should be regarded
as conjectures about uniformities holding over limited regions
of space, for limited periods of time, i.e. not as universal
generalizations, but rather as generalizations supposed to be
true throughout a vast but not infinite region and period of
time surrounding our own what may be called a 'cosmic
epoch'. Finally, Moritz Schlick and F. P. Ramsey have argued
that laws of nature are not "propositions which are true or false,
but rather set forth instructions for the formation of such

92 THE PHILOSOPHY OF SCIENCE

propositions . . . [being] directions, rules of behaviour, for the
investigator to find his way about in reality."

We must now notice, in turn, how each of these theories
reflects some aspect of the uses to which principles and laws
of nature are put in the physical sciences. Let us begin by look-
ing at the Lockean theory, that laws of nature are principles
of necessitation. To recognize the force of this view, recall the
way in which physicists use such words as 'must', 'necessarily'
and so on, especially in the conclusions of their arguments
cf. class (iv) above. In our first sample explanation, for instance,
we saw how a scientist will say that, the height of the sun being
30, and that of a wall being 6 ft., the shadow of the wall is
necessarily 10 ft., and indeed that it follows from, or in accord-
ance with, the Principle of the Rectilinear Propagation of Light
that it must have just that depth and none other. It is clear
from this that, in some sense or other, physicists do treat their
laws and principles as telling us, or enabling us to discover
how things necessarily are, and what in given circumstances
must happen; and the phrase 'principles of necessitation' is
presumably intended to reflect just this sort of fact about laws
of nature.

What needs to be made clear, however, is that the sense
in which one can speak of laws of nature as telling us how
things 'must' happen is not one that need be obnoxious to
Mach and Hume. So let us ask again: when the physicist says
that it follows from his principle that the shadow must have
just such a depth, what kind of inference is this, and what sort
of necessity? How can it be said to follow from any experi-
mentally established principle that the depth of the shadow
must be what it is?

To answer these questions correctly one must distinguish
between two pairs of things: first, between establishing a
theory and applying an established theory; and again, between
recognizing a situation as one in which a particular theory can
be employed, and employing the theory in that . situation on
the assumption that it has been correctly identified. It is part
of the art of the sciences, which has to be picked up in the course
of the scientist's training, to recognize exactly the situations

LAWS OF NATURE 93

in which any particular theory or principle can be appealed to,
and when it will cease to hold. Although a scientist can often
say what it is about one situation or another which makes a
particular theory applicable or inapplicable, there is always
a certain amount of room for the exercise of individual judge-
ment; and this makes it nearly as difficult to give rules for
deciding when a theory must be modified or abandoned as it
is to give rules for discovering fertile new theories. But pro-
vided that the scientist has correctly identified the situation
with which he is faced for what it is, and therefore knows
what principles and laws he can appeal to, it is the very business
of the theory to tell him what must happen, i.e. what he must
expect to happen, in such circumstances. If this is a field of
study which has been brought within the ambit of the exact
sciences at all, his theory will provide him, among other things,
with an inferring technique that is, with a way of arguing
from, e.g., the height of a wall and the angle of elevation of the
sun to the depth of a shadow. The actual technique of inference-
drawing may be a geometrical one, in which one draws infer-
ences by drawing lines, or it may be a more complicated,
mathematical one. But in either case it is essential, if the theory
is to be acceptable, that it shall license one to pass in one's
arguments from the conditions in which the particular
phenomenon takes place to the characteristics of the phenomenon
which are to be predicted or explained.

Now there is nothing that need worry Hume in the use
which, as a result, the physicist makes of words like 'must'
and 'necessarily'. For when he says, "In those circumstances
the shadow must be ten and a half feet deep", he does so always
with the tacit qualification, "If all the conditions are indeed
fulfilled for the application of this principle"; the depth of the
shadow is therefore not a necessary fact, but a necessary con-
sequence of applying the principle as it is meant to be applied.
And when we say that it follows from the principle that, in
such circumstances, the shadow must have that particular
depth, the principle finds its application, not as a major premise
in a syllogistic argument from generalization to particular
instance, but as the 'inference-ticket', to use a phrase of Kyle's,

94 THE PHILOSOPHY OF SCIENCE

which entitles us to argue from the circumstances of the
phenomenon to its characteristics. In the circumstances of
our example, it has been found that shadow-casting and the
like are explicable, representable or predictable in a way which
makes use of certain geometrical and trigonometrical relations:
arguing in accordance with the rules which express these
relations, one must in these particular circumstances expect the
shadow to be just the depth it is. It is because, and only because
a physical theory involves techniques of inference -drawing
that a 'must' enters in. Once we have been taught such a tech-
nique, a correctly performed computation of the depth of
the shadow must lead to the result it does, and any computation
which fails to lead to this result must be faulty.

Hume and Mach are, nevertheless, justified in insisting
on this: that the possibility of explaining particular phenomena
in a particular way is something which has to be found out.
One could not say that the techniques of geometrical optics
must be applicable in the ways in which they have been found
to be applicable, except in so far as this fact is, in its turn,
explicable by reference to a wider theory. One might, perhaps,
appeal to the wave-theory of light in order to show that ray-
diagrams must be applicable just when they are found to be;
but this simply shifts the burden. The important thing is not
to confuse the questions, what theory has been found reliable
in a given field, and what phenomena, according to this theory,
must occur in any given circumstances. When one is talking
about a theory whether establishing it, or identifying a
system as one to which it applies one is concerned with what
has been found to be the case, not with what must be ; but when
one is talking in terms of a theory applying it to explain or
foretell the phenomena occurring in such-and-such a situation
one is then concerned with what, according to that theory,
must happen in that situation. There are several mistakes into
which it is possible to be led if one fails to see just where it
makes sense to say 'must', 'necessarily' and 'cannot', and where
one has rather to say 'has been found' one such is the kind
of determinism which we shall have occasion to examine in
Chapter V. These mistakes are, one finds, made only easier

LAWS OF NATURE 95

by the scientist's customary idioms ("If the wall is 6 ft. high
and the sun is at 30, the shadow must be 10J ft. deep"), for
in these the currently accepted theories of optics are employed
without being explicitly mentioned. Logicians, for the sake of
clarity, can afford to say the same thing less compactly but more
explicitly, in onlooker's instead of participant's language: "If
the wall is 6 ft. high, and the sun is at 30, then a proper appli-
cation of the theories of optics which have been found reliable
in such circumstances as these will necessarily lead us to the
conclusion that the shadow will be 10| ft. deep."

What lies behind the Lockean view of laws of nature
seems, then, to be their use as principles of inference: the neces-
sity to which they point is the necessity with which conclusions
follow when one argues in accordance with these principles.
One may ask, then, why this necessity should seem 'opaque
to the intellect', when principles such as that nothing can be
both red and green all over are 'transparently necessary'.
The subject is too large for us to go into it fully here, but perhaps
a hint can be given. The difference seems to lie in this: we learn
words like 'red' and 'green' at an early age, at the same time
as we learn to sort, fetch, carry and label the things around us,
and our knowledge that nothing can be both red and green
all over is something which ordinarily shows itself in our ability
to give and obey orders, and to make and understand reports,
in which the words 'red' and 'green' appear. Only much later,
when both the use of these words and the activities in connexion
with which we have learnt to use them are second nature to
us, do we come to ask why such a principle holds ; and it then
seems to us, naturally enough, that anyone who has got the
hang of the words will recognize the force of the principle.
In the case of laws of nature, on the other hand, one has
neither the same strong association between the words appear-
ing in the laws and those particular inferring techniques
with which the laws belong, nor the same years of familiarity
with the use of these techniques. As often as not, in fact, terms
are taken over from outside physics and put to new jobs,
and in consequence it may well seem far from obvious that
'light' must travel in straight lines, or that 'action' and 'reaction*

96 THE PHILOSOPHY OF SCIENCE

must be equal and opposite. But perhaps, if dynamical cal-
culation were second nature to us, in the way colour-classifica-
tion is, and if we could all recognize, e.g., purely gravitational
systems by eye, in the way we can tell red from green, the
difference might not seem so great; and we might think the
Law of Gravitation quite as transparent as the more familiar
principles of colour-classification.

The point of Hume's 'constant conjunction' theory we
have seen in part already: it is to rebut the suggestion made
by advocates of the Lockean theory, that laws of nature some-
how provide us with information about 'necessary facts'.
(Recall also Mach's opposition to the idea that physics reveals
necessities in nature.) In consequence, one finds Hume and his
followers concentrating their attention, not on statements of
type (iv), but rather on those in class (iii). "The light getting
over this wall is travelling in a straight line/' "The beams of
light outside and inside this prism are oriented in such a way",
"The salt is dissolving in this water"; these statements may
constitute quite genuine applications of laws of nature, but
there is nothing necessarily true about them. They just represent
the sorts of thing that are in fact found to happen; and, by
contrast with statements in class (iv), there is not even any
'must' in them. Of course, if one has a satisfactory theory to
explain these facts, one will be able to show in any particular
case that things must happen just as they are found to do:
indeed, it would not be a satisfactory theory if one could not.
But, to repeat, this is not to say that the facts explained are
'necessary facts': rather it is to say that they are necessary
consequences of the theory. The distinction between necessary
consequences and necessary propositions is obvious enough
in elementary arithmetic: if a housewife argues, "I started with
twelve pounds of sugar, and I've used four, so I must have
eight pounds left," the formula on which she relies (12-4 = 8)
may be necessarily true or rather, unconditionally applicable
but the conclusion she reaches ("I have eight pounds left")
is to be accepted, not unconditionally, but rather as a necessary
consequence of her data. The same thing holds in physics:
when one applies a physical theory to a specific case, the con-

LAWS OF NATURE 97

elusions to which one is led may, in the circumstances, be
necessary ones, but it is a mistake to read this 'necessary'
in the logic-book sense, as 'necessarily true*.

If there is no need for the Lockean and Humean views to
be regarded as rivals, why then have they been so regarded?
This will be clearer if we ask the question, "Are laws of nature
necessary propositions or contingent ones?" For if we regard
this dichotomy as exhaustive, and try to fit laws of nature into
one category or the other, we shall find it hard to know what
to say. Are we to say that, despite their empirical origin,
laws of nature are necessary propositions? Or are we to say
that, despite their claim to tell us what 'must' happen, they are
only contingent propositions about constant conjunctions?
Or must we contradict ourselves, by saying that they are both
necessary and contingent? None of these alternatives is satis-
factory, and the moral of our earlier discussion is that we should
accept none of them. It is only because philosophers have
come to laws of nature from such everyday statements as
'Rabbits are animals' and 'Rabbits eat lettuce' that they have
supposed that laws of nature must be either necessary (like
'Rabbits are animals') or contingent (like 'Rabbits eat lettuce').
In fact, when they have attempted to establish their views
that laws of nature are the one or the other, they have talked
in either case, not about things which are properly called
'laws of nature', but rather about one or other of the types of
statement which we have distinguished from the laws themselves.

Advocates of the 'necessary' view have, as we saw, paid
special attention to those applications of laws of nature in
which one is led to conclude, e.g., that a particular shadow
must be 10 ft. 6 in. deep. But such a conclusion is not itself
a law or principle, or a deduction from any law or principle:
it is an inference drawn in accordance with the law or principle.
The appearance in this statement of the word 'must', reflecting
the use of a rule of inference, cannot therefore be taken as
evidence that laws of nature are necessary propositions in any
but a highly misleading sense.

Advocates of the 'contingent' view, on the other hand,
have concentrated their attention, not on the laws of nature

98 THE PHILOSOPHY OF SCIENCE

themselves, but upon the facts that they are used to explain
salt's dissolving in water, shadows being the depths they are,
light-beams having the directions they do all things which may
with some justice be spoken of as regularities or constant con-
junctions. But, once again, the statements they cite are not
laws of nature at all, and again nothing is proved about the
status of laws of nature by pointing to these facts.

In its way, to call laws of nature 'contingent' is as mislead-
ing as to call them 'necessary', for to do so is to focus too much
light on a set of questions which never arise with reference to
laws of nature, namely, questions about truth and falsity.
It may be clear enough what it would mean to deny, e.g.,
that the law of gravitation applied to electromagnetic radiation,
or again to deny that, the law being what it is, such-and-such
a configuration of bodies must move in such-and-such a way;
but it is quite unclear what it would mean to talk of denying
the law of gravitation itself. One might say "It needs recon-
sidering and reformulating to fit it into relativity theory",
but to say this is not to say that it is false: in such a case,
the word 'false' cannot get a grip. The facts which scientists
investigate experimentally have to do with the scope of their
laws, and with what, applying the laws in a particular context,
they must expect to happen. Physicists never have occasion
to speak of the laws themselves either as corresponding or as
failing to correspond to the facts. The logical relation between
the laws and the facts is indirect: by talking as though they were
connected any more closely than they are, one creates only
confusion and misunderstanding.

3.8 Whitehead and Schlick: Are laws of nature restricted general-
izations or maxims?

Where advocates of the first two views are preoccupied with
statements of types (iii) and (iv), the 'restricted generalization'
view seems to spring from a consideration of those in class
(ii): i.e. statements about the discovered scope of laws of nature.
As Kneale interprets him, Whitehead supposed that laws of
nature must be generalizations of some kind, either restricted or
unrestricted; and concluded, reasonably enough, that a few

LAWS OF NATURE 99

hundred years' experiments on this Earth could hardly justify
us in advancing generalizations of a completely unrestricted
kind. The natural consequence of this argument was the view
that laws of nature are generalizations of a kind that tacitly
refer to all places and times within a single, vast but bounded
cosmic epoch.

Now there is an important point behind this account, but
it needs re-stating. For, as stated, it assumes that the question
to be asked is "Are laws of nature true always and every whe re ?" ;
whereas the proper question is "Are laws of nature applicable
equally at all times and places?' ' And the answer to the question
is not "Yes, curiously and amazingly enough, they are found
to be universally true," but "Yes, they are formulated in such
a way as to be universally applicable: this is a feature which
marks off laws of nature from the other statements of physical
theory. " If laws were universal empirical generalizations, it
would indeed be a question whether they were always true;
but they are not, and the point at issue must be put otherwise.

The heart of it can perhaps be illustrated in this way: one dis-
tinguishes in physics between those expressions which are to
be labelled 'laws of nature/ and those expressions which are
not so much laws of nature as applications of laws to special
ranges of circumstances. Thus we can distinguish between
the Law of Gravitation, a genuine law of nature, and such a
statement as "Freely falling bodies accelerate by 32.2 feet/
second every second": this latter expression is not itself a law
of nature, but is an empirical law which can be accounted for
by applying the Law of Gravitation to the special conditions
of the Earth. Now it certainly makes sense to speak of our dis-
covering that what we now call 'the Law of Gravitation' should
itself be regarded as a law of this latter kind. This would happen
if it were found, e.g., that over the whole region to which we
had previously had access there was a constant 'field' of a
hitherto-unrecognized type; and if, on investigating the pro-
perties of this field, we found that the law of gravitation could
be expressed in its present form only for so long as this field
remained constant. One can imagine, say, the value of the
gravitational constant, *G', being found to depend on the strength

100 THE PHILOSOPHY OF SCIENCE

of this field. If this happened, we should have to reformulate
our law so as to take account of the new discovery, and the
present formula would be dethroned. The success of our present
law would then be spoken of as a local and temporary con-
sequence of the 'true law', in the way that the rate of gravitational
acceleration on the Earth is now regarded as a local and tempo-
rary consequence of our present law.

This, however, does not prove that laws of nature tacitly
apply only to limited regions of space and time, as it would if
our Law of Gravitation were a simple generalization. On the
contrary, the fact that such a discovery would be sufficient
reason for dethroning our present law shows just the reverse:
it shows that only those formulae we are ready to apply equally
at all places and times qualify for the title of 'laws of nature'.
But this, in its turn, does not imply that laws contain the words
'always and everywhere' in them either explicitly or tacitly.
These words would be out of place within a law, and belong
rather in statements of class (ii), about the circumstances in
which any particular law has been found to hold. So Whitehead's
suggestion, too, involves the confusion between laws and gene-
ralizations. Nor, for that matter, is the fact that it makes sense
to say, "Perhaps our so-called Law of Gravitation is only a
local affair", any reason for despondency: there is not the slight-
est reason at the moment to suppose the existence of the
undiscovered field which would force us to this conclusion.
Of course it makes sense to say, "Perhaps we have not got the
law right". Nevertheless, we shall need to have good reasons
before we abandon our present formulation of the law for
another.

Finally, let us consider the view about laws of nature put
forward by F. P. Ramsey, and quoted above in the words of
Moritz Schlick: the view that such laws are not so much
'statements', 'assertions' or 'propositions' as 'instructions for
the formation of propositions', 'rules of conduct', 'maxims' or
'directions for the investigator to find his way about in reality'.
Again we shall find that the theory draws attention to something
important about laws of nature, but once again this feature
is described in a needlessly paradoxical way.

LAWS OF NATURK 101

One can at any rate say in favour of this theory that its
advocates are genuinely concerned with laws of nature (i.e.
class i above), and not with those other, related classes of
statement (ii, iii and iv) which have so often been confused
with them. For the point which Schlick and Ramsey have
wanted to emphasize is the one we ourselves have recognized
as crucial: the fact that words like 'true', 'false' and 'probable'
are applicable, not to laws themselves, so much as to the
statements which constitute applications of laws ; and that any
abstract statement of a law or principle gives us only the form
of a regularity, telling us by itself nothing about the phenomena
it can be used to explain. As Schlick says, laws of nature "do
not have the character of propositions which are true or false",
and in some ways his alternative account of them is not at all
a bad one. If we consider the techniques of geometrical optics,
which give the Principle of Rectilinear Propagation its point,
we can indeed see grounds for speaking of the principle as a
means of finding one's way about in reality; and when we
remember how far laws of nature are used as principles of
inference, there is clearly some virtue in talking of them as
rules for the formulation of statements about the world.

There is, in fact, only one thing about Schlick and Ramsey's
account to which one can seriously object, and it is this same
thing that gives the account its paradoxical air the fact that
they use unduly imperatival words such as 'instructions',
'directions' and 'rules', instead of some rather less exciting
word such as 'principles'. If one makes this one amendment,
the objections brought against their view, e.g. by Kneale,
lose all their force. For Kneale argues that "if the sentence
which purports to formulate a law gives [as Schlick suggests] only
a general rule of conduct, what is derived from it can be no
more than a command or injunction": as he sees it, on this view
there would be no. possibility of using a law to derive genuine
propositions about the world one could get only a string of
particular injunctions. But Schlick and Ramsey are not claim-
ing that laws of nature are generalized commands; the point
of describing laws of nature in their way is to remind us of their
use as inference-licences entitling us to argue from known

102 THE PHILOSOPHY OF SCIENCE

facts about a situation to the phenomena we may expect in
that situation; and the weakness of Kneale's objection becomes
clear if one considers how his argument would affect other
principles of inference.

Consider, e.g., the Principle of the Syllogism. Lewis
Carroll showed in his paper, What the Tortoise said to Achilles,
what impossible conclusions one is led into if one treats the
Principle of the Syllogism as a super-major premise, instead
of as an inference-licence; yet it does not follow from his dis-
covery that the conclusions of all valid syllogisms, which may
loosely be spoken of as 'derived from 1 that Principle, must
therefore be commands or injunctions. This would be the case
only if one confused conclusions deduced from the Principle
with those inferred in accordance with the Principle: the
phrase 'derived from the principle* hides this distinction. It is
the same with laws of nature. The conclusions about the
world which scientists derive from laws of nature are not
deduced from these laws, but rather drawn in accordance with
them or inferred as applications of them, as our examples have
illustrated. It is only if one takes Schlick's phrase 'rules of
behaviour' too seriously that Kneale's objection carries weight.
Regarded as principles of inference though ones whose range
of application is empirically bounded laws of nature do indeed
have very much the sort of job that Schlick attributes to them.
Certainly they act hardly more as premises in physical argu-
ments than the Principle of the Syllogism does in syllogistic
ones.

What is it that makes Schlick's way of putting his thesis
especially paradoxical? It is perhaps this: that it snaps the link
between laws of nature and the world. Like the phrase 'laws
of our method of representation', Schlick's phrase 'directions
for the investigator' seems to sever laws of nature from the world
entirely, and makes it appear that they have to do solely with
physicists and their conduct. But to snap this link is, as we
saw earlier, an extremely misleading thing to do. "Through their
whole logical apparatus the laws of physics still speak about the
objects of the world"; and the fact that some inferences rather
than others come to be licensed usually tells us much more

LAWS OF NATURE 103

about the world than about the physicist and his methods.
(Though this is not equally so in every case, as will be seen
when we discuss Eddington's views on the subject in the next
chapter.)

How are we to account for Schlick's choice of this unhappy
form of words? The reason for it seems, strangely enough, to
be the same as that which distorts the Lockean and Humean
views the assumption that the only statements representing
genuine propositions' are those which are straightforwardly
classifiable either as necessary or as contingent. Where the
'principles of necessitation' view classes laws of nature as
opaquely necessary propositions, and the 'constant conjunction*
view classes them as contingent propositions of a somewhat
sophisticated kind, Schlick sees the unsuitability of putting
them in either category. But his reaction is too strong. For
his conclusion is that, if laws of nature are neither necessary
propositions nor contingent ones, they cannot properly be
spoken of as propositions at all: they must accordingly be found
a place with those other alleged quasi-propositions, the pre-
scriptions and recommendations of ethics and aesthetics.
Hence the imperatival words he chooses: 'instructions/
'directions' and 'rules of behaviour '. As so often in philosophy,
in objecting very properly to his opponents' conclusions, he
is betrayed into the same fallacy as they.

Schlick talks of the investigator finding his way about in
reality, Ryle of law-like statements as inference-tickets. Perhaps
these metaphors can be combined. For there is one variety of
railway ticket not unlike laws of nature the 'runabout ticket'.
Tickets of this kind do not have a single starting-point and
destination printed on them: they are valid, instead, for an
unlimited number of journeys within a given stretch of country.
The extent and limits of this region need not be, and usually
will not be stated on the ticket: they will be specified elsewhere
e.g. on posters and they can be varied by the railway
authorities without the ticket looking any different. Now one
might buy one of these tickets without knowing what its region
of validity was ; but one could find this out experimentally, by
seeing at what stations it was accepted. And one can do worse

104 THE PHILOSOPHY OF SCIENCE

than think of the physicist as a man who, in formulating laws
of nature, prints his own runabout tickets, and thereafter makes
it the goal of his experiments to discover where he can get
with their help. The formal statement of a law is like the
runabout ticket itself, which shows on it nothing as to its
scope: it is as a result of experience that the physicist comes to
know within what region it can be confidently employed.

By making the journeys (inferences) so licensed, the phy-
sicist finds his way around phenomena: by thinking of the
systems he studies in terms of appropriate models, he sees
his way around them and comes to understand them. But there
is one important preliminary first he must be able to identify
each system, classify it in theoretical terms, recognize its place
on the map. As we shall have reason to emphasize in Chapter V,
this is a logically vital step; and it is by no means as trivial as
one might think. Physical systems do not carry identification
labels, as railway stations do; nor is there any way in which
they can tell us themselves where on the theoretical map they
belong. Anyone who has studied chemistry will know what a
business identifying an anonymous specimen can be. What
still needs to be recognized is the logical burden which the
task of identification is made to bear.

CHAPTER IV

THEORIES AND MAPS

WE have seen how natural it is to speak of ourselves 'finding
our way around' a range of phenomena with the help of a
law of nature, or 'recognizing where on the map* a particular
object of study belongs. In doing so, we are employing a
cartographical analogy which is worth following up ; for whereas
to treat laws of nature on the pattern of generalizations is posi-
tively misleading, and to think of them as rules or licences
reflects only a part of their nature, the analogy between phy-
sical theories and maps extends for quite a long way and can be
used to illuminate some dark and dusty corners in the philosophy
of science. Of course, like any analogy, it will take us only a cer-
tain way, but after an overdose of arguments in which physics
is treated on the pattern of natural history, it can act as a healthy
purge. That this should be so is no accident, since the problems
of method facing the physicist and the cartographer are logically
similar in important respects, and so are the techniques of
representation they employ to deal with them.

4.1 Ray-diagrams and equations as maps of phenomena

Let us return, as a first application of this analogy, to a
question we considered in an earlier section. This is the question
the phenomenalists tried to answer: how we are to think of the
relation between a scientist's experimental observations, all
of which are expressible in everyday language, and the corre-
sponding theoretical statements in which the technical terms
of the science appear.

The difficulty to be overcome before we can answer this
question arises as follows. Mach wanted to insist, rightly, that
a scientific theory draws its life from the phenomena it can be
used to explain: furthermore, the idea that the scientist needed
insight into the causal connexion of things smacked to him of

105

106 THE PHILOSOPHY OF SCIENCE

metaphysics, and he tried to do without it. In view of this,
it was natural for him to suppose that, if a law of nature was
to contain no more than the phenomena it was used to explain,
it must be thought of as a summary of them, i.e. as an abridged
description or comprehensive and condensed report of the
experimental observations: "this," he concluded, "is really
all that laws of nature are." But such an account of the matter
may get us into difficulties. For to speak of laws as condensed
summaries, abridged descriptions, or comprehensive reports,
suggests that the connexion between any set of experimental
observations and the law they are used to establish is a deductive
one, so that it should be possible to give mechanical directions
for producing a theory from a set of observations, much as one
can produce a statement about the average schoolgirl from a
set of measurements of individual schoolgirls. And this, as
we saw, is a mistake: the relation between laws and phenomena
cannot be so described.

How then are we to restate this connexion without abandon-
ing the ground Mach gained? This is where the analogy
between theories and maps can help us, for a simple carto-
graphical example will show that no deductive connexion need
be looked for.

Consider, for instance, the imaginary motoring map
opposite, showing the town of Begborough and its environs.

We can ask about this section of map a question similar to
Mach's question: namely, what relation it bears to the set
of geographical statements that can be read off it, such as ,
"Potter's Bridge is 5 m. NE of Begborough on the road to
Little Fiddling", and "Great Fiddling is 3 m. due West of
Little Fiddling."

How are we to answer this question? Certainly the map
cannot be said to be deduced from the set of geographical
statements nor, in a logic-book sense of the phrase as opposed
to a Sherlock-Holmesian one, are the statements deduced
from the map. For in a deductive inference, such as "Fish are
vertebrates, mullet are fish, so mullet are vertebrates", the
same terms appear both in the premises and in the conclusion ;
whereas here the Conclusions' read off may be statements,

THEORIES AND MAPS

107

but the 'premise' is a map and contains no 'terms' at all. Only
where premises and conclusion are comparable in the way
that "Fish are vertebrates" and "Mullet are vertebrates" are
comparable, is there room for a deductive connexion, so the
relation between the map and the geographical statements
must be of a different, non-deductive kind. At the same time,
the map need not be said, in Mach's sense, to Contain' anything
which cannot be expressed as a geographical statement of the
kind included in our set: everything which one could read off
from the map is of this sort. Though the map and the geo-
graphical statements are not deductively related, one need not

LT. FIDDLING

conclude that the map goes beyond the surveyor's readings;
since it does not present us with additional information of a
novel kind, but represents the same information as the
statements in a different manner. This example shows that,
when we are presented with two logically incomparable forms
of expression, the question whether or no one form of expres-
sion contains more than the other is quite independent of the
question whether or no the one can be deduced from the other.
In fact, unless the expressions are of logically similar kinds,
there can be no question of such deduction.

108 THE PHILOSOPHY OF SCIENCE

The logical relation between, for instance, ray-diagrams
in geometrical optics and the phenomena they can be used to
represent, is a similar one. Here, too, neither can be spoken
of as being deduced from the other: yet a ray-diagram need not
be thought of as containing more than the phenomena. It is
rather that the diagrams present all that is contained in the set
of observational statements, but do so in a logically novel
manner: the aggregate of discrete observations is transformed
into a simple and connected picture, much as the collection
of readings in a surveyor's note-book is transformed into a clear
and orderly map.

The consequences of this analogy are worth noticing. For
if someone asks, "Doesn't the map tell us that Potter's Bridge
is 5 m. NE of Begborough, and a whole lot of similar things?",
we can only answer "Yes and No." Certainly, if you know how,
you can read off from the map a great range of geographical
information ; but the map on the one hand, and the geographical
statements on the other, tell us things in very different ways.
A man might own Ordance Survey maps of the whole country,
and yet, for lack of a training in map-reading, be quite unable
to tell us anything of a geographical kind: likewise, a man might
have memorized all the currently accepted laws of nature
and even know a vast amount about the calculative side of
mathematical physics, and yet not be equipped to explain or
predict any of the phenomena observed in the laboratory.
The most the first man could do would be to lend the appro-
priate map, on request, to a man capable of reading it: in physics,
too, the mathematician remains the servant of the man who
knows when and how the results of his computations can be
applied. Jeans and Eddington were both primarily mathe-
maticians, and in their popularizations of physics gave promin-
ence to the mathematical side of the subject, but the raults
were in certain respects misleading: the physics is not in the
formulae, as they suggested and as we are often inclined to
suppose, any more than being able to find your way about is
part of a map. The problem of applying the theoretical calculus
remains in physics the central problem, for a science is nothing
if its laws are never used to explain or predict anything.

THEORIES AND MAPS 109

To pursue our analogy yet further, we may ask: if the
map and the ray-diagram are counterparts, and the observations
of the surveyor and of the experimenter are also counterparts,
what exactly corresponds in cartography to laws of nature
in physics? Here the analogy begins to fail us, for interesting
reasons. For to press it at this point would mean saying that
laws of nature in physics were to be thought of as the counter-
parts of the laws of projection in accordance with which one
produces any specific type of map, such as Mercator's ; and this
leads to difficulties.

In certain respects the parallel holds: we have already seen
the parts the Rectilinear Propagation Principle and Snell's
Law play in the production of ray-diagrams, and the laws of
motion in dynamics play a similar part when one constructs
the equations of motion of a dynamical system. Up to a point,
therefore, the analogy with the laws of projection can be
illuminating. But the comparison is also an unhappy one.
The problems facing a cartographer have certain important
common features. In each case, it is his task to represent a
part of the surface of the Earth on a plane sheet of paper, so
as to preserve certain chosen features, such as equality of area;
and, the shape of the Earth being what it is, the rules of pro-
jection are calculable from his knowledge of the conditions
of his task. But in physics the situation is very different.
Though in some cases we may eventually come to be able to
work out what form laws of nature will take, as when one
derives the laws of geometrical optics from a knowledge of
physical optics, this knowledge is not like the prior knowledge
of the problem which we have in cartography.

In general, there seems to be no way of saying beforehand
what sort of techniques of explanation will be appropriate in
a given field of study. That is why laws of nature have always
to be discovered in a way in which the laws of projection do
not need to be. Our analogy could be preserved only by
imagining the figure of the earth to be both irregular and
discoverable only in the course of our cartographical survey:
if it were so, cartographers would be unable to pick on a method
of projection beforehand, and would have to find out empiric-

110 THE PHILOSOPHY OF SCIENCE

ally, as they moved from region to region, in what manner
each new area was to be mapped. Establishing a law by appeal
to the results of experiment would be like showing that a
satisfactory map of the new area could be produced using such-
and-such a method of projection as indeed we saw in the
case of Snell's Law. But even when so amended, the analogy
has its limitations: the problems to be tackled in physics differ
widely from one another in a way in which problems of map-
ping can never do.

4.2 The physicist as a surveyor of phenomena

In the traditional logical account of the sciences, one
encounters certain difficulties when explaining how it is that
experiments are used to establish theories. In the first place,
physicists seem to be satisfied with far fewer observations than
logicians would expect them to make: one finds in practice none
of that relentless accumulation of confirming instances which
one would expect from reading books on logic. This divergence
is partly to be accounted for by the logicians' confusion between
laws and generalizations one would hesitate to assert, say, that
all ravens were black if one had seen only half a dozen of the
species, whereas to establish the form of a regularity in physics
only a few careful observations are needed but this is not the
whole story. There is also a second, related difficulty to be over-
come: that of explaining how subsequent applications of a
theory are related to the observations by which the theory was
originally established.

To take the two difficulties together: it is worth noticing
that they arise for theories as much as, and no more than, for
maps. Not all the applications to which a theory is put need
have been specifically made in the course of the experimental
investigation by which it was established. But nor need all
the things which can be read off from a map have been specifi-
cally put in. A child might wonder how it was possible ever to
produce a map at all, since to tread every inch even of a small
area, and to measure all the distances and directions that one
can read off from a map, would take an unlimited length of
time. This, of course, is the marvel of cartography: the fact

THEORIES AND MAPS 111

that, from a limited number of highly precise and well-chosen
measurements and observations, one can produce a map from
which can be read off an unlimited number of geographical
facts of almost as great a precision. But it is not a marvel call-
ing for a general explanation, for only in some regions can the
techniques be implicitly relied on. In irregular country it is
always possible to be misled, and the number of observations
which have to be made per square mile will be much greater
in some areas than others just how many are needed being
something the practising cartographer must be able to judge.

Correspondingly, it is a fact that many physical systems
have been found whose behaviour can be similarly 'mapped'.
Having made a limited number of highly accurate observations
on these systems, one is in a position to formulate a theory
with the help of which one can draw, in appropriate circum-
stances, an unlimited number of inferences of comparable
accuracy. Thus it is always possible that the next time Boyle's
Law is applied, the particular combination of pressure and
volume concerned will be being observed for the first time.
But again, though this fact is in its way a marvel, it is not one
requiring a general explanation, any more than is the possibility
of mapping. For here, too, how far the behaviour of a given
system consists of phenomena which can be mapped in a
simple way, and just how many observations will need to be
made before we can be confident that our theory is a trust-
worthy one, are things which will vary very much from system
to system and which it is part of a physicist's training to
learn to judge.

The difficulties which logicians find in understanding the
role of experiments in physics arise, therefore, not only from
their thinking so much in terms of generalizations: to get clear
about it calls for quite a detailed study of the logic of physics.
To put our point succinctly only when a regularity has already
been recognized or suspected can the planning of an experi-
ment begin: until that time the mere multiplication of experi-
ments is comparatively fruitless. And when that time comes,
the problem for the physicist will not be like that for the
botanist or the naturalist, as it would be if his sole aim were to

112 THE PHILOSOPHY OF SCIENCE

generalize about 'all lumps of rock 1 or 'all flames' that is,
if physics were the natural history of the inanimate. His problem
will rather be like the surveyor's problem, and the accumula-
tion of observations in large numbers will be as much a waste
of energy in physics as in cartography. Faced with the demand
for more and more observations the surveyor and the physicist
might equally reply, "What's the point? We've been over that
ground already."

There is a further point about the sorts of observations
which need to be made in order to put a physical theory on
a satisfactory footing. Logicians have remarked, rightly, that
physicists prefer to make a limited number of observations
covering a wide range of circumstances, rather than a larger
number of observations covering a smaller range of circum-
stances. The point of this preference, they have concluded, must
be to show that the laws being established are true generally,
and not only true on certain conditions. From this point they
have gone on, first, to develop an elaborate theory of con-
firmation, analysing the way in which conditional clauses
might be eliminated from a hypothesis by reference to experi-
mental data; and secondly, to formalize the process of theory-
establishing in a way intended to fit in with the mathematical
theory of probability, the aim being to find a way of assessing
in numerical terms the probability of a given physical theory.

This account does not fit in with practice, nor does it
properly explain the preference for varied observations. For
physical theories are not spoken of in practice as true, false or
probable, nor is it clear what one could be expected to under-
stand by the statements, "The probability of the kinetic theory
of gases is r!" and "Five to one on Snell s Law". The point of
varying the conditions of observation is, in fact, otherwise: it
is to discover the scope of the theory, not its degree of truth
or the conditions on which it can be accepted as true. The 'logic
of confirmation' and the application of the probability-calculus
to theories have, therefore, hardly the slightest relevance to
the physical sciences. The mathematical theory of probability
has some place in the process of theory-establishing, certainly;
but it is a more restricted one than logicians have thought.

THEORIES AND MAPS 113

It has a central place only in limited branches of theory, such
as statistical mechanics and parts of quantum mechanics:
more generally, it has to do solely with questions of the form,
"Can such-and-such a specific set of experimental observations
be satisfactorily accounted for by applying a given theory in a
given manner?" i.e. the question whether the scatter in our
observations is significantly greater than the probable errors
in our measurements would lead us to expect. The application
of the calculus of probability in this sort of way raises no general
questions of a philosophical kind, but only particular questions
of statistical technique: questions to be answered in terms of
the theory of curve-fitting, significant deviations and so on.

4.3 Degrees of refinement in cartography and physics

There are many places in the physical sciences where one
finds a single field of phenomena covered by two or more
theories, in which techniques of different degrees of sophistica-
tion are employed. The optical phenomena with which we have
been concerned are a good example. We saw earlier how the
range of application of geometrical methods of representation in
optics is restricted by diffraction and the like, so that the limited
success of geometrical optics becomes itself something requiring
explanation. To explain the phenomena that cannot be accounted
for within geometrical optics, the wave-theory of light was
introduced, and this theory was particularly acceptable because
it could also be used to account for all the phenomena that
geometrical optics covers. It is true that what is simple in the
more elementary theory, explaining shadow-casting, for instance,
tends to become laborious in the more refined one ; but since the
wave theory can not only be used to explain a wider range of
phenomena, but does so. to a higher degree of accuracy, and
also explains just why the methods of geometrical optics break
down where they do, it is accepted as providing a more funda-
mental explanation than the simpler geometrical account
and reasonably enough.

Where there is such a multiplicity of theories, certain
things may appear mysterious to the outsider or to the beginner.
What is the relation of the two theories to one another, and how

114 THE PHILOSOPHY OF SCIENCE

does the development of the more refined theory affect the
status of the simpler one? Does the change-over mean that
the earlier theory has in some sense been falsified? If that is
so, surely it should be regarded as discredited; so how is it that
lens-designers, for instance, may prefer to go on using the
geometrical techniques of ray-tracing after the wave-theory
has been shown to be the true theory? Perhaps the most puzzling
thing is the way in which notions which were central in the
simpler theory that of a light-ray, for instance may dis-
appear almost completely in the more refined theory. So long
as we think in terms of the geometrical account, the term 'light-
ray' is indispensable: light-rays indeed seem to be the principal
actors on the optical stage. Yet in the wave-theory a light-ray
is an artificial construct as compared with, say, a 'wave-front' ;
and Snell's Law, which is stated, as we saw, in terms of rays
of light, has to be reformulated in quite a different way before
a niche can be found for it in the new theory. Yet the phenomena
are as they always were: lamps burn as they did, shadows fall
where they did, rainbows, reflections and all are as they were.
What then has happened to the light-rays?

The best answer can perhaps be given by pointing out
first the relation between different types of map. The imaginary
road map of the region between Begborough and the Fiddlings
which we discussed a few pages back, need not be the only
map of the region. There will also be some more elaborate
physical maps drawn to a larger scale and showing a great deal
more detail. In such maps as these, roads will perhaps be
drawn to scale, not represented by lines of purely conventional
widths, while towns and villages will be marked, not as mere
dots and blobs of standard sizes, but as having definite shapes
and made up of individual streets and blocks of houses.

Now a number of things should be noticed about the relation
between the road map and a physical map of the same region.
In the first place, many things can be mapped on the physical
map which there is no way of putting into the road map:
this is a consequence of the ways in which the two maps are
produced, and of the comparative poverty of the system of
signs used on the road map. On the other hand, given the

THEORIES AND MAPS 115

physical map, one could produce a satisfactory road map:
all that appears on the road map has its counterpart on the more
elaborate map, even though in a different form. But this does
not mean that the road map is not, of its kind, an unexception-
able map of the region. Providing that it is not thought of as
having irrelevant pretensions, there is nothing wrong with it:
indeed, for some applications one will be able to discover the
things one wants to know, e.g. distances by car, more easily
from the road map than from the physical one. Finally, it is
worth noticing what happens if we mix up the systems of signs
used on two different kinds of map. There are some motoring
maps in which one finds town-outlines and other features
sketched in on top of the simple road pattern: but since only
distances along roads can be given a satisfactory interpretation
on such maps, the result is usually confusing, and the simple
blob for a town is more consistent with the general scheme of
the map.

The relation between geometrical optics and the wave-
theory is not unlike that between a road map and a detailed
physical map. Thus the fact that one can explain on the wave-
theory, not only all the phenomena that can be accounted for
on the geometrical theory, but also why the geometrical
account holds and fails to hold where it does, is like the fact
that one can construct a road map from a physical map;
but again it is not a sign that the geometrical theory need be
superseded for all purposes. Road maps did not go out of
use when detailed physical maps were produced. It shows only
that, as one can produce a road map from a physical one but
not vice versa, so one could produce a ray-diagram from the
wave-theory picture of an optical system, but not vice versa.
The conceptual equipment of the geometrical theory, like the
system of signs on a road map, is too poor for one to do with
it all that can be done with the wave-theory. Indeed, the notion
of a light-ray is an artificial one in very much the way that the
conventional-width road is, and has to be abandoned in the
wave-theory because the accuracy with which one wants to
answer questions about optical phenomena is too great for the
conventional picture to be retained. No more can one, from a

116 THE PHILOSOPHY OF SCIENCE

simple motoring map, answer questions about the distance
from the northern verge of one road to the middle of another
these are things that a map of that type does not pretend
to show. Again, since there is no room within geometrical
optics for representing the phenomena of diffraction, a physicist
would hardly think it worth while to give any indication on
a ray-diagram of the shapes of any diffraction-fringes he
observed: they would be just as out of place there as town
shapes are on a bare motoring map.

If we look at the relation between different theories from
this angle, we can notice some points of importance about the
notion of a 'fundamental' or 'basic' theory. One finds that,
at a given stage in the history of physics, there is commonly
one theory, at any rate in a particular field, which is regarded
as the basic theory: this theory is thought of as capable of
accommodating all the phenomena to be observed in that field.
Now two questions need to be asked. Since it will never be
the case that all the phenomena have in fact been explained,
all that need be claimed is that the basic theory can in principle
explain them all: the first question is, what are we to under-
stand by this claim? Secondly, when physicists talk about
explaining everything, what are the criteria by which they
would judge that everything had in fact been explained?

It is helpful to compare the basic theory with the funda-
mental map on which the Ordance Survey might record all
the things which it is their ambition to record. This would, of
course, be a map drawn on the very largest scale, but it would
not be the only true map of the country: rather it would be
the one which most fully and precisely represented the region
mapped, and the one from which by appropriate selection and
simplification all others could be produced. For many purposes
it will be too elaborate to be of practical use, but for some
purposes none else will do, and the lover of cartography for
its own sake must have a special place for it in his heart.

The value of the comparison lies in this: it suggests that
the standards of what constitutes a complete theory in physics
may change. For we could say that the fundamental map was
complete only if it showed all the things which in that region it

THEORIES AND MAPS 117

was the cartographer's ambition to record. Now it is always
possible for cartographers to develop fresh ambitions: the
criteria of the completeness of a map are, accordingly, at the
mercy of history. So are they with the theories of physics.
One is at first inclined to suppose that the physical sciences
have a definite goal, the same for Aristotle, Newton, Laplace,
Maxwell and Einstein, but a closer look at the history of the
subject will show the mistakenness of this idea. Rather there is
at any given stage a standard of what sorts of things require
explaining: this is something with which scientists grow
familiar in the course of their training, but which is hardly
ever stated. The standard accepted at any time determines
the horizon of physicists' ambitions at that time, the goal
which for them would have been reached if 'everything'
i.e. everything thought of as requiring explanation had been
found a place in the theories of physics.

In physics, as in travelling, the horizon shifts as we go
along. With the development of new theories new problems
are thrown into prominence, ways are seen of fitting into physical
theory things which before had hardly been regarded as
matters requiring a place at all: the horizon accordingly ex-
pands. Classical physics, for instance, was thought of as
potentially exhaustive. Yet, looking back at it, we must feel
that nineteenth-century standards of exhaustiveness were
strangely unexacting. The existence of ninety-two elementary
kinds of matter, their relative abundance, and the colour of
the light emitted by each element: these things, to mention
only a few, were hardly even asked about. They were not things
to be explained but things to which, in a phrase of Dr. Wais-
mann's, 'one had to take off one's hat'. Perhaps this is why
the claim of some classical physicists, that they had the ex-
planations of everything in principle in their grasp, was
peculiarly distasteful. For what was repugnant was not just
the fact that the theories advanced were so bare and mechanical
but, quite as much, the fact that their idea of what it would be
to have explained everything was so much smaller than life.

On the whole, then, the horizon of physics expands. From
time to time, however, the ideal changes in a way which cannot

118 THE PHILOSOPHY OF SCIENCE

be described so simply, and these are occasions when disputes
of a philosophical kind frequently arise. In the change-over
from Aristotelian to Newtonian dynamics, for example, cer-
tain phenomena which were previously regarded as 'natural*
and taken for granted, such as carts stopping when the horses
ceased to pull and heavy bodies falling to the ground, came to
be thought of as complex phenomena needing explanation:
in these respects the horizon expanded. But at the same time
certain other phenomena, which had until then been regarded
as complex and in need of explanation, were reclassified as
simple, natural and to be taken for granted; notably, the con-
tinued flight of an arrow after it had left the bow, and the un-
faltering motion of the planets along their tracks. The need for
this second kind of reclassification was the great obstacle to
the development of the new dynamics: it was easy enough to
recognize as complex something previously accepted as simple,
but the reverse change was a bitterly hard one to make. And
so it has been elsewhere. One finds the same thing happening
around 1700, in the dispute between Leibniz and the New-
tonians over the mechanism of gravity and action at a distance;
and the same thing again in the late nineteenth-century dis-
putes over the luminiferous ether.

One of the most instructive disputes of this kind is in
progress at the moment, and concerns the adequacy of quantum
mechanics as a basic theory. Einstein, on the one hand, refuses
to accept the changes in our standards of what needs explaining
which have to be made when one introduces quantum mechanics:
in his view, these changes require one to restrict the horizon
of scientific endeavour in an unjustifiable way. His opponents,
on the other hand, claim that his objections show only that he
has not properly understood the theory. This is not the place
to deal with the substance of the dispute. But for our purposes the
language in which the dispute is carried on must be noticed;
for it is couched in terms of the question, "Is a quantum-
mechanical description of a physical system complete or not?"
This way of posing the problem confuses the issue, giving to
it too sharp an appearance of opposition. For a complete
description of a physical system is one from which one can,

THEORIES AND MAPS 119

using the currently accepted laws of nature, infer all the pro-
perties of the system for which it is a physicist's ambition to
account: where two physicists do not share a common standard
of what does and does not need to be explained, there is no
hope of their agreeing that the corresponding description can
be called complete. This is the case in Einstein v. the Rest:
the use of the word 'complete', with its implicit reference to
particular criteria of completeness, conceals rather than reveals
the point at issue between the parties. A similar moral holds
more generally: by using the words 'exhaustive', 'all', 'every-
thing' and 'complete' in stating the goal of their investigations,
physicists have hidden from themselves as well as from others
the changes in the horizon towards which they work.

4.4 Causes are the concern of the applied sciences

A subject which receives a good deal of attention in tradi-
tional treatments of 'induction and scientific method' is that of
causes. It is a proposition often taken as obvious that the task
of the sciences is the discovery of causes: Mill's four methods
arid similar formal analyses can, indeed, be regarded as relevant
to the physical sciences only in so far as this is so. Some
logicians go further: the existence of causal chains is said by
them to be a condition of the possibility of science, and certain
features of quantum theory are accordingly interpreted as a
breakdown of the causal principle or an abandonment of
causality. Causes, causation, causality: these are the staple of
much philosophical and logical writing about the sciences.

If one turns from the logic-books and the spare-time
philosophical works of scientists, to the professional journals
in which the sciences really progress, one is in for a surprise.
For in the papers there printed the word 'cause* and its
derivatives hardly ever appear. In works on engineering,
perhaps ; in medical journals, certainly ; wherever the sciences
are applied to practical purposes, there one finds talk of causes
and effects. But in the physical sciences themselves, the word
'cause' is as notable an absentee as the word 'true'. Why is this?

To recognize the reason, consider first the sorts of every-
day situation in which we have occasion to ask questions

120 THE PHILOSOPHY OF SCIENCE

about causes. A wireless set, instead of giving out a Haydn
symphony, howls dismally; an invalid's temperature, instead
of staying at 98.4F., soars to 105F. ; a stretch of railway
embankment crumbles and falls away, leaving the lines in a
dangerous state; a field of barley grows unevenly, sturdy and
thick in some parts, sparse and weak in others; and in each
case we ask about the cause why the wireless does not work
properly, what is wrong with the invalid, what has happened
to make the embankment collapse, in what respect the fertile
parts of the field differ from the infertile ones. Developments
which we are interested in producing, preventing or counter-
acting these are the typical sorts of thing about whose causes
we ask. Correspondingly, to discover the cause of one of these
developments is to find out what it is that needs to be
altered, if we are to produce, prevent or counteract it. To
discover the cause of the howling is to discover, say, that
a particular valve is faulty and needs replacing; the patient, it
may be, has an infected sinus ; the foundations of the embank-
ment have been sapped by an underground stream; the fer-
tility of the different parts of the field depends upon their
nitrogen content. In each case, we speak of that as the cause
which in tJie context would have to be, or have had to be other-
wise, for the development on which our attention is focused
to go differently. Where there is no one thing in the antecedents
rather than anything else which could reasonably have been
wished different, we may accordingly find no use for the term:
"Nothing particular caused it", we sometimes say, "Things
just worked out that way".

Now these everyday cases are all anthropocentric: the things
for whose causes we seek are those we human beings want to
produce, prevent or counteract. Our examples of typical
everyday uses of the term 'cause' are, that is, all concerned with
people getting somewhere. That they should be anthropocentric
is not essential. One can ask about the cause of the explosion
of a distant star as well as of an invalid's temperature: things
which, humanly speaking, are indifferent can have causes
just as well as things we care about. But one feature of our
examples is essential. Wherever questions are asked about

THEORIES AND MAPS 121

causes, some event, which may matter to us or may not, has a
spotlight turned on it: the investigation of its causes is a scrutiny
of its antecedents in order to discover what would have to be
different for this sort of thing to happen otherwise what in the
antecedents God or man would need to manipulate in order
to alter the spot-lighted event. It is not essential that the search
for causes should be anthropocentric ; but that it should be
diagnostic, i.e. focused on the antecedents in some specific
situation of some particular event, is essential. People sometimes
mystify one by asking what would happen if the order of all
physical events were reversed, suggesting that as a result
effects would then precede causes. This suggestion misses
the point of the notion of cause in particular, its dependence
on context. If one puts a steam-engine into reverse, one has
to apply the brake at quite a different point in the cycle in
order to achieve a given result, e.g. to stop it at top dead centre:
in the new context the same pairs of happenings no longer
belong together as causes and effects. But the causes are still,
necessarily, among the antecedents of the effects.

It is, then, still in cases where our interest is in how one
might 'get somewhere', i.e. produce or counteract some spot-
lighted development, that we talk about causes though the
destination need not be one that we care about either way.
From this we can see why the term 'cause* is at home in the
diagnostic and applied sciences, such as medicine and engineer-
ing, rather than in the physical sciences. For the theories of the
physical sciences differ from those of the diagnostic and applied
sciences much as maps differ from itineraries. If the term
'cause' is absent from the physical sciences, so also a map of
South Lancashire does not specifically tell us how to get to
Liverpool. To a man making a map, all routes are as good as
each other. The users of the map will not all be going the same
way, so a satisfactory map is route-neutral: it represents the
region mapped in a way which is indifferent as between
starting-points, destinations and the like. An itinerary, however,
is specifically concerned with particular routes, starting-points
and destinations, and the form it takes is correspondingly
unlike that of a map. Often enough, of course, a map may be

122 THE PHILOSOPHY OF SCIENCE

used to work out the itinerary for a particular journey, and
from one map an indefinite number of routes may be read off,
as occasion requires. But, from its form, there is nothing about
a map to show that it is to be used for this, rather than any other
of a wide range of purposes.

In the physical sciences, likewise, the regularities we find
in any particular field of phenomena are represented in a way
which is application-neutral. The theories which are produced
to explain optical phenomena, for instance, do not specifically
tell us how to bring about this or that optical effect how to
produce a shadow a hundred feet deep, or how to create a
mirage. Rather they provide us with a picture of the sorts of
phenomena to be expected in any given circumstances, which
can then be used in any of a number of ways. The study of
the causes of this or that event is, therefore, always an applica-
tion of physics. It is not of direct importance to the physicist,
and can at best suggest to him something which may turn out
to be of theoretical importance. In the case of theories, as of
maps, there will be an indefinite number of applications to
be made, say, in engineering. But the way in which the theory
is formulated will not show that it is to be applied in this or
that particular kind of way, for the production or prevention
of this or that particular kind of development. Problems of
application and questions about causes arise with reference
to particular contexts, but physical theories are formulated
in a manner indifferent to particular contexts: it is when we
come to apply theories that we read off from them the causes
of this and that, but there is no call for the term 'cause' to
figure within the theories themselves.

This analogy shows us something about the relation between
the fundamental and applied sciences, and about such phrases
as 'applied physics'. For in many fields of science practical
skills preceded theoretical understanding, and even provided
the first data for systematic study. Sundials were in use for
centuries before their operation was properly understood, and
there are still plenty of familiar processes, in cooking for
instance, about whose physico-chemical nature we have only
the sketchiest of ideas. There is therefore only a part of engin-

THEORIES AND MAPS 123

eering which can be called 'applied physics', even though this
part may be continually growing and may in some divisions,
such as atomic energy, be all but exhaustive. This state of
affairs also has its natural counterpart in cartography. For a
long time, travellers relied on itineraries rather than on maps;
Greek seamen and Roman legionaries as often as not followed
set routes for which itineraries had been written out; there
must still be today a few more remote parts of the world which
are totally unmapped, but around which a guide could take
one; and even in our own well-mapped country we all know
some short cuts and refinements that are shown on no map.
So though the preparation of itineraries may in fact often be
applied cartography, it need not be. Itineraries preceded
maps. The development of cartography has given us a way of
understanding the relations between different routes, and at
the same time a source of new itineraries whose possibility had
not previously been recognized. And there may be some parts
of the world so remote, so mountainous, that one could hardly
hope to work out itineraries for them except by first mapping
them from the air.

The absence of the term 'cause' from the professional
writings of physicists can therefore be explained. But this
explanation in its turn creates a fresh problem: for if the prime
aim of the physical sciences is not the discovery of causes
or causal chains, what are we to make of the elaborate dis-
cussions of causality and indeterminacy provoked, e.g., by
quantum mechanics? The subject is too complex to go into
in detail here. But one thing may be worth saying: the idea of
causality reigning unchallenged seems to be accepted by philoso-
phical scientists so long as the basic theories of the time appear
capable, in principle, of explaining all the things it is hoped
eventually to explain. It is no surprise, accordingly, to find
Einstein, whose horizon stretches further than quantum
mechanics can reach, calling for a re-establishment of causality,
and saying reproachfully that Born and his colleagues 'believe
in a dice-playing God'. Restated in our terms, the question of
causality becomes the question whether all physical phenomena
are completely mappable ; and this, like other general philoso-

124 THE PHILOSOPHY OF SCIENCE

phical questions containing the words 'everything', 'air and
'complete', depends very much on one's standards of complete-
ness. The determinate, correspondingly, is that for which a
place can be found on the map; so that the very name 'Indeter-
minacy Principle' for Heisenberg's relation seems to rest on a
misunderstanding.

The notion of causal chains and causal contiguity, which
Russell for one regards as central for the justification of scientific
method, must wait for a proper discussion till we consider
determinism and the 'causal nexus' in Chapter V, but again
a word is in place here. The idea that events form chains, each
drawing the other inevitably after it, originates in what we have
called the diagnostic field rather than in the physical sciences.
It is catastrophes of which we most want to know the causes,
and the discovering of such a cause is spoken of as 'laying
bare the chain of circumstances which led to the disaster'.
Two things must now be noticed. First, the idea of a chain
of circumstances tends to be taken too seriously on such
occasions just because it is a disaster whose causes we are
concerned to diagnose, i.e. the sort of thing we tend also to
think of, as often as not mistakenly, as fated or destined to
happen: apart from this association, there is no reason to under-
stand the 'chain' metaphor as any more than a metaphor.
Secondly, this tendency is reinforced by special features of
the diagnostic, as opposed to the physical sciences. To under-
stand the causes of something is the first step towards being
able to cause it to happen. Success in the applied sciences
may therefore lead us to think of events as at the ends of chains ;
all we need is to know which chain to pull and the required
result will follow. But simple chain-like prescriptions can be
given only in restricted sets of circumstances: we can confidently
match causes and effects only in a given context. So once we
shift from the diagnostic to the physical sciences the idea of a
causal chain is of as little use as the term 'cause' itself.

4.5 Eddington and the fish-net

A perplexing question about the theories of physics was
raised by Sir Arthur Eddington, and has been widely discussed

THEORIES AND MAPS 125

in the last few years. "How much," he asked, "of the structure
of our theories really tells us about things in Nature, and how
much do we contribute ourselves?" This question was of
importance to him because of his own professional activities,
for it was his aim to 'work out from first principles*, and treat as
a conceptual matter, quantities which many of his fellow-
physicists regarded as matters of brute fact. One instance is the
ratio of the mass of the proton to that of the electron, a quantity
which many physicists regard as something to be discovered
only by looking and seeing, like the ratio of the populations of
London and Liverpool: another is the number of protons and
electrons in the Universe, which Eddington regarded as a
conceptual matter but his critics as a pure matter of fact, like
the aggregate population of the Earth.

Now there is an important philosophical question here,
which is worth a more careful examination than it has so far
received. Much of the discussion that it has been given has
been needlessly mystifying, and some of it is completely
misconceived. The conclusion has even been drawn from
Eddington's suggestions that the theories of physics are
essentially subjective imposed on the facts, even to the extent
of falsifying them, rather than built up so as to give a true picture
of them. One is reminded of Bergson's thesis, that we falsify
by abstraction.

Eddington has certainly been in part to blame for this, for
he himself called his doctrines 'Selective Subjectivism', and
introduced the two analogies which have dominated and con-
fused later discussions. Suppose, he says, that an ichthyologist
trawls the seas using a fish-net of two-inch mesh: then fish
less than two inches in length will escape him, and he will
find when he pulls up the net only fishes two inches long or
more. This, Eddington suggests, may tempt him to conclude
that the world contains no fish of smaller size ; he may generalize
and announce, "All fish are two inches long or more" ; and until
he has the sense to examine his own methods of fish-catching,
he may fail to realize that these methods, not the ichthyological
facts, are what have led him to the conclusion. This, Eddington
argues, is what happens in physics: the theorist trawls the

126 THE PHILOSOPHY OF SCIENCE

results of the experimenters' work through his net and announces
as discoveries about the world things that he himself forces
on the facts by his methods of trawling. Eddington also recalls
the old story of Procrustes, the giant who obliged unfortunate
travellers to sleep in his bed and always trimmed them to fit,
stretching the shorter ones on the rack and lopping pieces off
the longer ones until their corpses were exactly the right length.
The theorist for him is Procrustes: the experimental observa-
tions are the travellers, and are adjusted willy-nilly until they
fit exactly into the theoretical bed. "Let us therefore/'
Eddington implies, "be more self-conscious about our methods
of theorizing, recognize that it is to subjectively selected data
that the generalizations of physics the so-called laws of nature
apply, and see what surprising things may not be discovered
from a careful examination of our explanatory techniques."

One thing about Eddington's fish-net analogy must be
pointed out at once. The conclusion which the incautious
ichthyologist announces is one of natural history, an empirical
generalization of the purest kind, "^4//fish share such-and-such
a property". Elsewhere we have seen the disastrous effect the
use of this model can have on our understanding of the physical
sciences, and we must take care not to be misled here also.
So let us pose Eddington's question in a way which is truer to
life, and see how much of the problem remains. For these
purposes the cartographical analogy is a useful guide, and can
make Eddington's professional activities look less disreputable
than they have tended to look to some of his colleagues.

We saw earlier how some features, even of the simplest
theories, must be understood in terms of the method of
representation we employ as much as of the phenomena
represented. The central notion of geometrical optics, that
of a light-ray, holds the centre of the theoretical stage only
so long as the geometrical method of representation (ray-
tracing) remains our basic technique of inference-drawing:
as soon as the wave-theory displaces the simpler picture as
the basic theory, the notion of a ray of light loses its. theoretical
importance. Nor is there anything mysterious about this, any-
thing in particular which can be regarded as falsification of

THEORIES AND MAPS 127

the facts. In cartography, too, there is a good deal which has
to be contributed by us before there can be a map at all, and
this contribution is again of an unmysterious kind. Carto-
graphers and surveyors have to choose a base-line, orientation,
scale, method of projection and system of signs, before they
can even begin to map an area. They may make these choices
in a variety of ways, and so produce maps of different types.
But the fact that they make a choice of some kind does not
imply in any way that they falsify their results. For the alter-
native to a map of which the method of projection, scale and
so on were chosen in this way, is not a truer map a map undis-
torted by abstraction: the only alternative is no map at all.
To draw an analogy between a cartographer's method of
projection and the ichthyologist's fish-net would accordingly
be misleading. There is no question of falsification here.
Quite the reverse: it is only after all these decisions have been
taken and a map has been produced, that the question can
even be raised, how far the product of the cartographer's work
is true to the facts, for only then will there be anything which
can be true to or falsify them.

If physicists are to be spoken of as in any way responsible
for the structure of physical theory, the reasons are similar.
For in physics, as much as in cartography, some decisions
have to be taken, consciously or no, before a theory can be
produced at all. If Eddington's remarks appear mysterious,
this is probably because these decisions are so obvious,
elementary and easily made that one is liable to overlook them,
forget that they have ever been taken, and even take them
without recognizing them for what they are. In geometrical
optics, for instance, it is easy to forget that we have decided
to represent optical phenomena by the use of lines drawn on
paper or on the blackboard; and perhaps no one has come
to understand the logic of physics who has not at some time
been amazed that there should be any connexion between such
things as shadows, lamps and patches of light on the one
hand, and graphite streaks on paper on the other. The lines in
our ray-diagrams are not, so to speak, thrown in with the pheno-
mena: they have to be put into relation with the phenomena

128 THE PHILOSOPHY OF SCIENCE

by our adoption of a particular theory, view of light and tech-
nique of representation. Wherever in physics we introduce
numerical concepts, such as temperature, or employ mathemat-
ical techniques of inference of a geometrical or of a more
sophisticated kind, decisions of this sort must have been taken.

Once again, this does not imply that the statements which
the theoretical physicist advances for our acceptance are in
fact falsehoods, which he is able to misrepresent as true as
a result of his methods of theorizing. Here, too, the fish-net
analogy is quite misleading. For the alternative to a theory
which has been built up with the help of decisions of this kind
is not a truer theory, 'free from the distorting effects of abstrac-
tion': the only alternative is no theory at all. Some contribution
on our part to the structure of theoretical physics is needed
if the statements within the theory are to be capable of having
any application to the world; and only when this connexion
has been established will there be anything to be spoken of
either as 'true to the facts' or as 'falsifying the facts'.

The air of mystery and the suggestion of subjectivity,
which have marked the discussion of Eddington's problem,
are both therefore unnecessary. There is no need to feel that
the physicist's contribution to his own theories is either
personal, or necessarily unstateable: it is something as public,
and as open to inspection and description, as a cartographer's
methods of projection and representation. Reading about this
subject, as when reading Kant, one gets the impression that
to try to say where to draw the line between our own contribution
and that of the facts is in some curious way an impossibility
rather like trying to chew your own teeth. But this is a mistake.
It is not that the physicist has a mysterious predilection for
some theoretical mould, into which he thrusts all the experi-
mental results he meets, nor is it a deep necessity of experi-
ence that he should handle these results in the way he does. His
part is no more than that played by anyone who introduces a
language, symbolism, method of representation or system of
signs.

Perhaps if Kant's arguments were stripped of their unhappy
air of psychological discovery and re-expressed in similar

THEORIES AND MAPS 129

terms, they too would cease to be so obscure. For if the
decisions on which our physical theories rest are easy to for-
get, those which have gone to the making of everyday speech
are yet more easily forgotten; and the philosophical effects of
forgetting them, as Wittgenstein saw, are yet more pervasive.
To talk, in the philosophy of science, of theoretical physics
falsifying by abstraction, and to ask for the facts and nothing
but the facts, is to demand the impossible, like asking for a
map drawn to no particular projection and having no particular
scale. In epistemology, too, to argue that our everyday concepts
falsify by abstraction or are necessary conditions of experience,
with the suggestion that one thereby points to a defect in our
conceptual equipment or to an unfortunate limitation on our
capacity for experiencing, is to evince a similar misconception.
If we are to say anything, we must be prepared to abide by the
rules and conventions that govern the terms in which we speak:
to adopt these is no submission, nor are they shackles. Only
if we are so prepared can we hope to say anything true or
anything untrue. It is unreasonable to complain, as philosophers
have so often done, because we cannot tell the truth without
talking.

4.6 Facts and Concepts: the Absolute Zero

In order to indicate what sort of thing the physicist's
contribution to his theories consists of, let us look at a simple
example. For it is possible to show, with a very little technical
explanation, how the acceptibility of statements which at first
glance seem to be pure matters of fact may depend, rather, on
the technique of representation employed in a physical theory.

A suitable example is at hand if one considers the physicist's
notion of temperature. When one first learns about tempera-
ture and about thermal phenomena, the existence of the Absolute
Zero of temperature may appear to one as a strange and
ineluctable fact about the Universe. The world of thermal
phenomena, it seems, has a curious and unforeseen feature.
As we work our way down lower and lower, we cannot go on
for ever, but after a time come up against an adamantine layer,
against which even our best drills are blunted: all attempts to

130 THE PHILOSOPHY OF SCIENCE

penetrate it are in vain. The existence of the Absolute Zero
may thus present itself to us as the brutest of brute facts ; and
the natural geological analogy, between up and down in tempera-
ture and up and down from ground-level, reinforces this
impression. Of course the Absolute Zero is not something which
one comes up against with a bang: rather, as one produces
lower and lower temperatures, all further reductions get harder
to make, so that at 270C. it may be more difficult to
cool things by i 3 bC. than, at ordinary temperatures, it is to cool
them by 10C. But the geological picture will accommodate
this additional feature easily enough: it is as though, as our
drills went down, we came up against progressively more
impenetrable strata, the Absolute Zero being the limit beyond
which, it seems, there will never be any hope of piercing, how-
ever much we improve our drills.

This geological picture is totally misleading. The existence,
at some point, of an Absolute Zero of temperature is not a brute
fact at all, but a conceptual matter i.e. a consequence of the
way in which we give a meaning to the notion of temperature,
and put degrees of warmth and cold into relation with the
number-series. We who grow up familiar with thermometers
tend to overlook the fact that this has had to be done. Yet there
is no more connexion between numbers and the notions of
heat and cold, until we create one, than there is between pencil-
marks on paper and optical phenomena. In either case, some-
one had the genius to see what a help it would be to introduce
a new concept ('light-ray* or 'temperature'), and so the crucial
steps were taken. When Galileo invented the notion of tempera-
ture and designed the first thermometer, he knew very well
what he was doing. He saw that to produce a thermometer
would not just be to find a way of measuring something which
before we had been able to estimate only roughly: rather, it
would be to alter the whole status of our thermal notions. He
did what he did as part of a deliberate campaign, the first
stage in his programme of making physics mathematical, and
'turning secondary qualities into primary ones'. Likewise, the
physicists who helped to extend our scale of temperature
were not just developing fresh instrumental techniques, but

THEORIES AND MAPS 131

helping to fix the meaning of the term 'temperature' in respects
in which it had previously been indeterminate. This shows why
the title 'theory of measurement', which has often been used
for our present field of discussion, may be misleading. Tech-
niques of measurement and conceptual refinements do often
proceed pari passu, but for logical purposes we must keep
conceptual matters distinct from questions of experimental
technique.

If one wants to understand about the Absolute Zero, the
crucial thing to examine is the introduction of the ideal gas
scale of temperature as the basic theoretical scale. This scale
is introduced by three steps. First, it is remarked that the
behaviour of all gases tends to conform the more nearly to a
single law, the more we heat them up and the lower we make
their pressures. This law is Charles' Law, according to which
each degree through which we heat or cool a closed container
of gas, as measured, for instance, on a mercury thermometer,
should produce the same change of pressure whatever the gas.
The more we cool different gases down, on the other hand, and
the more we increase their pressures, the more markedly their
behaviours diverge from each other: they liquefy and solidify
at quite different temperatures from one another, and their
compressibilities vary more and more as they approach the
temperature of condensation.

Next, the common behaviour of all gases at high tempera-
tures and low pressures is taken as a theoretical standard,
deviations from which require to be explained. To mark the
adoption of this standard, physicists proceed to introduce the
notion of an ideal gas, which is defined as one behaving at all
temperatures in the manner in which actual gases tend the
more nearly to behave, the higher the temperature and the
lower the pressure. This notion is, of course, even more of a
theoretical ideal than that of a light-ray. Finally, temperature on
the ideal gas scale is introduced by reference to the properties
of this ideal gas: equal changes in temperature, on this scale,
are defined as those which produce equal changes of pressure
in a closed container of ideal gas. To measure temperature on
this scale thermometers containing simple gases, such as

132 THE PHILOSOPHY OF SCIENCE

hydrogen, are used, their readings being corrected wnere
necessary to allow for deviations from the theoretical scale.

Now notice one thing about the ideal gas scale: it cannot
help having an Absolute Zero. For, whatever may be the pres-
sure of a given mass of ideal gas when it occupies one cubic
centimetre at the freezing point of water, it will not make sense
to talk of cooling it down by more degrees of temperature
below 0C. than will reduce this pressure to zero. The precise
numerical value of the Absolute Zero, in degrees Centigrade,
is a brute fact which has to be found out by investigating the
properties of actual gases at high temperatures. But that there is
an Absolute Zero at all is something which does not have to be
found out by experiment, being ensured by our way of introduc-
ing the ideal gas scale. It turns out in fact to be 273.1 6 C.
This figure was, of course, known very precisely long before
physicists had any means of approaching it in practice. It is a
conceptual matter, a fact about our notion of temperature, not
as one might at first suppose, a fact about thermal phenomena at
very low temperatures.

The statement, "Nothing can be cooled below the Absolute
Zero" or, to put the same thing less misleadingly, "The ideal
gas scale has a lower bound", is accordingly one of those
theoretical statements which may look at first like a fact about
actual phenomena ; but which turns out on closer inspection to
be a consequence of the technique of representation adopted
in this case, of the particular manner in which the notion of
temperature is fitted into our theories. The existence of the
Absolute Zero can be compared with the existence of the
boundary in a map of the World drawn to a stereographic or
orthographic projection. On these projections, the surface of
the Earth does not cover the whole of any sheet of paper you
use, as a Mercator's map is capable of doing, but fills only two
circles. If there is blank space round the circles, that is not
because the cartographer has chosen to cut off the map half-
way up Greenland, say, but because, the nature of the
projection being what it is, no point on the Earth can be
mapped outside the circles. One can, of course, decide to
make the circles as large as one chooses; but, however large

THEORIES AND MAPS 133

one decides to have them, there will still be a boundary,
whereas a map drawn to Mercator's projection is capable of
going on indefinitely.

If we prefer, it is open to us to stop using a map of one
kind and start using one of the other kind; and to abolish the
boundary in this way shows nothing about the area we are map-
ping. The presence or absence of such a boundary tells us
nothing about the surface of the Earth. The same is true in
physics. One can, if one chooses, change over from the ordinary
ideal gas scale to a logarithmic scale, which extends without
limit in both directions; and to make this change implies
nothing about actual thermal phenomena. In neither case does
one, by changing the method of representation, burke any
facts about the World.

Here the defects of the geological analogy become clear.
For so long as we think in terms of this picture, the inaccessible
strata below the adamantine layer seem as authentic as those
above it: that is why it seems a simple question of fact that we
cannot break through to the 'inaccessible' temperatures below
the Absolute Zero. But the truth is quite otherwise. The way
we line up degrees of warmth and cold with numbers in the
ideal gas scale is such that numbers below 273.16 are given
no interpretation as temperatures: all the thermal phenomena
that are conceived of in the current theories are mapped on to
the range of numbers from 273.16 upwards. So the inacces-
sible temperatures below the Absolute Zero are a myth. On
our standard theoretical scale, figures like ' 300' no more
represent inaccessible temperatures than do the blank spaces
round a stereographic map represent inaccessible places: all
genuinely inaccessible places, such as the top of Mt. Everest,
have a place within the circles, quite as much as Leicester
Square. It is true that our theories may perhaps come to be
altered some day, and a fresh temperature-scale introduced
along with new theories, but there is no reason to anticipate this ;
and in any case, if it happens, it will not mean that a new,
sharper drill has been built which has torn a way through the
adamantine layer, but rather that we, who put the layer there to
begin with, have moved it elsewhere.

134 THE PHILOSOPHY OF SCIENCE

4.7 Do sub-microscopic entities exist?

Non-scientists are often puzzled to know whether the
electrons, genes and other entities scientists talk about are to be
thought of as really existing or not. Scientists themselves also
have some difficulty in saying exactly where they stand on this
issue. Some are inclined to insist that all these things are just as
real, and exist in the same sense as tables and chairs and omni-
buses. But others feel a certain embarrassment about them, and
hesitate to go so far; they notice the differences between
establishing the existence of electrons from a study of electrical
phenomena, inferring the existence of savages from depressions
in the sand, and inferring the existence of an inflamed appendix
from a patient's signs and symptoms; and it may even occur to
them that to talk about an electromagnet in terms of 'electrons' is
a bit like talking of Pyrexia of Unknown Origin when the patient
has an unaccountable temperature. Yet the theory of electrons
does explain electrical phenomena in a way in which no mere
translation into jargon, like 'pyrexia', can explain a sick man's
temperature; and how, we may ask, could the electron theory
work at all if, after all, electrons did not really exist?

Stated in this way, the problem is confused: let us there-
fore scrutinize the question itself a little more carefully. For
when we compare Robinson Crusoe's discovery with the
physicist's one, it is not only the sorts of discovery which are
different in the two cases. To talk of existence in both cases
involves quite as much of a shift, and by passing too swiftly
from one use of the word to the other we may make the problem
unnecessarily hard for ourselves.

Notice, therefore, what different ideas we may have in mind
when we talk about things 'existing'. If we ask whether dodos
exist or not, i.e. whether there are any dodos left nowadays, wfe
are asking whether the species has survived or is extinct. But
when we ask whether electrons exist or not, we certainly do not
have in mind the possibility that they may have become extinct:
in whatever sense we ask this question, it is not one in which
'exists' is opposed to 'does not exist any more'. Again, if we ask
whether Ruritania exists, i.e. whether there is such a country as

THEORIES AND MAPS 135

Ruritania, we are asking whether there really is such a country
as Ruritania or whether it is an imaginary, and so a non-existent
country. But we are not interested in asking of electrons whether
they are genuine instances of a familiar sort of thing or non-
existent ones: the way in which we are using the term 'exist 1 is
not one in which it is opposed to 'are non-existent 1 . In each
case, the word 'exist' is used to make a slightly different point,
and to mark a slightly different distinction. As one moves from
Man Friday to dodos, and on from them to Ruritania, and again
to electrons, the change in the nature of the cases brings other
changes with it: notably in the way one has to understand
sentences containing the word 'exist*.

What, then, of the question, "Do electrons exist?" How is
this to be understood? A more revealing analogy than dodos or
Ruritania is to be found in the question, " Do contours exist ?"
A child who had read that the equator was 'an imaginary line
drawn round the centre of the earth' might be struck by the
contours, parallels of latitude and the rest, which appear on
maps along with the towns, mountains and rivers, and ask of
them whether they existed. How should we reply? If he asked
his question in the bare words, "Do contours exist?", one could
hardly answer him immediately: clearly the only answer one
can give to this question is "Yes and No." They 'exist* all
right, but do they exist? It all depends on your manner of
speaking. So he might be persuaded to restate his question,
asking now, "Is there really a line on the ground whose height
is constant?" ; and again the answer would have to be "Yes and
No", for there is (so to say) a 'line', but then again not what you
might call a line. . . . And so the cross-purposes would con-
tinue until it was made clear that the real question was: "Is
there anything to show for contours anything visible on the
terrain, like the white lines on a tennis court? Or are they only
cartographical devices, having no geographical counterparts?"
Only then would the question be posed in anything like an
unambiguous manner. The sense of 'exists' in which a child
might naturally ask whether contours existed is accordingly
one in which 'exists* is opposed not to 'does not exist any more*
or to 'is non-existent*, but to 'is only a (cartographical) fiction*.

136 THE PHILOSOPHY OF SCIENCE

This is very much the sense in which the term 'exists* is used
of atoms, genes, electrons, fields and other theoretical entities
in the physical sciences. There, too, the question "Do they
exist?" has in practice the force of "Is there anything to show
for them, or are they only theoretical fictions?" To a working
physicist, the question "Do neutrinos exist?" acts as an invita-
tion to 'produce a neutrino', preferably by making it visible.
If one could do this one would indeed have something to show
for the term 'neutrino', and the difficulty of doing it is what
explains the peculiar difficulty of the problem. For the problem
arises acutely only when we start asking about the existence of
sub-microscopic entities, i.e. things which by all normal stan-
dards are invisible. In the nature of the case, to produce a
neutrino must be a more sophisticated business than producing
a dodo or a nine-foot man. Our problem is accordingly compli-
cated by the need to decide what is to count as 'producing* a
neutrino, a field or a gene. It is not obvious what sorts of thing
ought to count: certain things are, however, generally regarded
by scientists as acceptable for instance, cloud-chamber
pictures of a-ray tracks, electron microscope photographs or, as
a second-best, audible clicks from a Geiger counter. They would
regard such striking demonstrations as these as sufficiently like
being shown a live dodo on the lawn to qualify as evidence of
the existence of the entities concerned. And certainly, if we
reject these as insufficient, it is hard to see what more we can
reasonably ask for: if the term 'exists* is to have any application
to such things, must not this be it?

What if no such demonstration were possible? If one could
not show, visibly, that neutrinos existed, would that necessarily
be the end of them? Not at all; and it is worth noticing what
happens when a demonstration of the preferred type is not
possible, for then the difference between talking about the
existence of electrons or genes, and talking about the existence
of dodos, unicorns or nine-foot men becomes all-important.
If, for instance, I talk plausibly about unicorns or nine-foot men
and have nothing to show for them, so that I am utterly unable
to say, when challenged, under what circumstances a specimen
might be, or might have been seen, the conclusion may reason-

THEORIES AND MAPS 137

ably be drawn that my nine-foot men are imaginary and my
unicorns a myth. In either case, the things I am talking about
may be presumed to be non-existent, i.e. are discredited and
can be written off. But in the case of atoms, genes and the like,
things are different: the failure to bring about or describe cir-
cumstances in which one might point and say, "There's one!",
need not, as with unicorns, be taken as discrediting them.

Not all those theoretical entities which cannot be shown to
exist need be held to be non-existent: there is for them a middle
way. Certainly we should hesitate to assert that any theoretical
entity really existed until a photograph or other demonstration
had been given. But, even if we had reason to believe that
no such demonstration ever could be given, it would be too
much to conclude that the entity was non-existent; for this
conclusion would give the impression of discrediting something
that, as a fertile explanatory concept, did not necessarily deserve
to be discredited. To do so would be like refusing to take any
notice of contour lines because there were no visible marks
corresponding to them for us to point to on the ground. The
conclusion that the notion must be dropped would be justified
only if, like 'phlogiston', 'caloric fluid' and the 'ether', it had
also lost all explanatory fertility. No doubt scientists would be
happy if they could refer in their explanations only to entities
which could be shown to exist, but at many stages in the
development of science it would have been crippling to have
insisted on this condition too rigorously. A scientific theory is
often accepted and in circulation for a long time, and may
have to advance for quite a long way, before the question of the
real existence of the entities appearing in it can even be posed.

The history of science provides one particularly striking
example of this. The whole of theoretical physics and chemistry
in the nineteenth century was developed round the notions of
atoms and molecules: both the kinetic theory of matter, whose
contribution to physics was spectacular, and the theory of
chemical combinations and reactions, which turned chemistry
into an exact science, made use of these notions, and could
hardly have been expounded except in terms of them. Yet not
until 1905 was it definitively shown by Einstein that the

138 THE PHILOSOPHY OF SCIENCE

phenomenon of Brownian motion could be regarded as a
demonstration that atoms and molecules really existed. Until
that time, no such demonstration had ever been recognized,
and even a Nobel prize-winner like Ostwald, for whose work as
a chemist the concepts 'atom' and 'molecule' must have been
indispensable, could be sceptical until then about the reality of
atoms. Moreover by 1905 the atomic theory had ceased to be
the last word in physics: some of its foundations were being
severely attacked, and the work of Niels Bohr and J. J. Thom-
son was beginning to alter the physicist's whole picture of the
constitution of matter. So, paradoxically, one finds that the
major triumphs of the atomic theory were achieved at a time
when even the greatest scientists could regard the idea of atoms
as hardly more than a useful fiction, and that atoms were
definitely shown to exist only at a time when the classical atomic
theory was beginning to lose its position as the basic picture of
the constitution of matter.

Evidently, then, it is a mistake to put questions about the
reality or existence of theoretical entities too much in the centre
of the picture. In accepting a theory scientists need not, to
begin with, answer these questions either way: certainly they
do not, as Kneale suggests, commit themselves thereby to a
belief in the existence of all the things in terms of which the
theory is expressed. To suppose this is a variant of the Man
Friday fallacy. In fact, the question whether the entities spoken
of in a theory exist or not is one to which we may not even be
able to give a meaning until the theory has some accepted
position. The situation is rather like that we encountered earlier
in connexion with the notion of light travelling. It may seem
natural to suppose that a physicist who talks of light as travelling
must make some assumptions about what it is that is travelling:
on investigation, however, this turns out not to be so, for the
question, what it is that is travelling, is one which cannot even
be asked without going beyond the phenomena which the
notion is originally used to explain. Likewise, when a scientist
adopts a new theory, in which novel concepts are introduced
(waves, electrons or genes), it may seem natural to suppose
that he is committed to a belief in the existence of the things in

THEORIES AND MAPS 139

terms of which his explanations are expressed. But again, the
question whether genes, say, really exist takes us beyond the
original phenomena explained in terms of 'genes'. To the
scientist, the real existence of his theoretical entities is con-
trasted with their being only useful theoretical fictions: the fact
of an initial explanatory success may therefore leave the
question of existence open.

There is a converse to this form of the Man Friday fallacy.
Having noticed that a theory may be accepted long before visual
demonstrations can be produced of the existence of the entities
involved, we may be tempted to conclude that such things as
cloud-chamber photographs are rather overrated: in fact, that
they only seem to bring us nearer to the things of which the
physicist speaks as a result of mere illusion. This is a conclusion
which Kneale has advanced, on the ground that physical
theories do not stand or fall by the results obtained from cloud-
chambers and the like rather than by the results of any other
physical experiments. But this is still to confuse two different
questions, which may be totally independent: the question of the
acceptability of the theories and the question of the reality of
the theoretical entities. To regard cloud-chamber photographs
as showing us that electrons and a-particles really exist need
not mean giving the cloud-chamber a preferential status among
our grounds for accepting current theories of atomic structure.
These theories were developed and accepted before the cloud-
chamber was, or indeed could have been invented. Nevertheless,
it was the cloud-chamber which first showed in a really striking
manner just how far nuclei, electrons, a-particles and the rest
could safely be thought of as real things; that is to say, as more
than explanatory fictions.

CHAPTER V

UNIFORMITY AND DETERMINISM

IT is often said in philosophical discussions about the sciences
that either they, or scientists, or scientific arguments pre-
suppose (or take for granted, or assume) some fact (or general
principle, or major premise) which is spoken of as 'the Uniform-
ity of Nature'. It is time to re-examine this notion, and see what
light is thrown on it by the results of our discussion.

This extremely vague way of introducing the subject has
been chosen deliberately. Different writers present the Uni-
formity of Nature in different guises, feel bound to invoke the
idea for different reasons, and formulate it very differently.
Some see in it the solution of the problem of induction, the
logical bridge spanning the 'gulf ' between observations made in
the past and predictions made about the future; others see it
as an article of scientific faith, the expression of the scientist's
confidence in the possibility of solving his problems; others
again look backwards at the achievements of science in past
centuries, and see in them evidence of uniformity already
revealed. The arguments advanced and the points made differ
correspondingly, and each requires a separate examination.
All that we can hope to do in this chapter is to put the doctrine
in a form which bears directly on the physical sciences, and see
what light is thrown on it by our examination of the types of
arguments physicists have occasion to employ.

5.1 Are laws of nature universally applicable?

The Principle of the Uniformity of Nature, then, takes many
forms, and is asked to do many jobs. Let us begin by considering
one of the more extreme suggestions in connexion with which it
appears: namely, the doctrine that, taken by themselves, the
arguments employed in the physical sciences are logically un-
sound, and that the holes in them can be plugged only by

140

UNIFORMITY AND DETERMINISM 141

introducing as a major premise in all such arguments some
statement about the uniformity of things-in-general. Have
philosophers good reasons for thinking that any such extra
premise is called for? Is it in fact needed? And, if anything is, in
some sense or other, assumed in scientific arguments, will a
general principle about uniformity help to justify the assump-
tions made? From Mill's System of Logic down to Russell's
Human Knowledge, a long and important series of writers
has answered "Yes" to all these questions.

One thing springs to the eye as soon as one begins to watch
scientists at work, which strongly suggests that assumptions
are being made. When, for instance, physicists calculate the
manner in which falling apples, the moon, the satellites of
Jupiter and double-stars many light-years away may each be
expected to move, they employ the same law of gravitation in
each case: it does not seem to occur to them that the form of
the equation might need to be modified in passing from one
system to the next. Yet surely it should: surely, one may feel, it
is a question whether the law of gravitation is the same on Mars
and a fortiori in the distant nebulae and surely a million
years ago its form might not have been the same. Does not this
fact alone show that pre-suppositions are being made about the
Uniformity of Nature? To be specific: is it not being assumed
that the laws of nature take, will take and have always taken one
form everywhere the same in distant parts of the Universe and
at remote epochs as here and now?

Certainly if laws of nature are put in the same pigeon-hole
as empirical generalizations, this conclusion seems irresistible ;
and it is easy to see how, having this analogy in mind, one might
come to accept it without question. Suppose, for instance, that
there turned out to be cats and rabbits on Mars as well as on the
Earth. Then it is quite on the cards that their diets would be
completely unlike those of their terrestrial fellows. It might be
the case, for instance, that the rabbits on Mars ate nothing but
mice, while Martian cats turned up their noses at flesh and
lived on lettuces. At any rate, the generalizations that cats are
carnivores and that rabbits live on lettuces both of which are
what we have called habit-statements will require further

142 THE PHILOSOPHY OF SCIENCE

checking as soon as the Martian species are discovered ; and it
will be dangerous for natural historians to jump to conclusions.
To say, e.g. "So there are rabbits on Mars, are there? Well,
then, there must be vegetables there, too, for them to live on,"
would very definitely be to take it for granted that the habits of
rabbits were the same there as here. Granted that this is so in
the case of rabbits, what about gravity? Might not gravity too
work differently on Mars, and the law of gravitation itself be
different there from here? Are not physicists running as much
of a risk in taking this for granted as natural historians would be
if they assumed too much about Martian rabbits?

To get the first indication of the right answer to this ques-
tion, compare for a moment, not the things which are allegedly
assumed to be the case, but those things which are allegedly
assumed not to be the case. This is worth doing because, for an
assumption to amount to anything, it must rule out some
possibility: an assumption which could be invalidated by no
describable happening is only misleadingly so called. Now the
two assumptions are, respectively, that rabbits eat the same
sorts of food everywhere, and that the law of gravitation takes
the same form everywhere. So what is said to be ruled out is, in
the one case, that rabbits eat elsewhere kinds of food they do
not eat on the Earth ; and in the other, that the law of gravita-
tion takes a different form elsewhere from that which it takes
on the Earth. The first of these two possibilities is, clearly, a
perfectly meaningful one: one knows well enough what it would
be like to discover rabbits eating mice rather than lettuces.
But what about the second one? This requires a closer examina-
tion; first there is an ambiguity to be resolved; and when this
has been done it will be questionable whether after all, in the
required sense, the suggestion means anything.

To deal with the ambiguity first what are we to under-
stand by the phrase 'the law of gravitation'? As in Chapter III,
we must distinguish between Newton's inverse-square law or
Einstein's refinement of it, these being the sorts of thing one
might call 'laws of nature* ; and such a statement as that freely
falling bodies accelerate by 32.2 ft./sec. every second, which
might loosely be called 'the law of gravitational acceleration on

UNIFORMITY AND DETERMINISM 143

the Earth*, but is not the sort of thing physicists would call a
law of nature. What we have to say about the alleged assump-
tion depends entirely on which of these we consider. It would
certainly be dangerous to assume that freely falling bodies
accelerated at 32.2 ft./sec./sec. on Mars, as they do on the
Earth. But this is not an assumption of a sort that physicists
would dream of making. The rate of acceleration of bodies
moving under gravitational attraction alone, expressed in ft./
sec. /sec., and other such specific gravitational effects, will
without doubt be different on Mars from what they are on the
Earth: in this sense, gravity unquestionably works differently,
and the law of gravitational acceleration will be different, on
Mars. In this sense, the assumption that physical laws operate
in the same way everywhere may make perfectly good sense;
but it is also quite unfounded, and physicists would never
make it. This example, however, goes no way towards establish-
ing the need for a general Principle of the Uniformity of Laws
of Nature, since the so-called 'law' considered is not a law of
nature at all, but rather an empirical discovery which is to be
accounted for by applying the law of gravitation to the special
circumstances of the Earth.

If we turn to a real law of nature, the situation changes at
once, since there is now no room to say what it is that is being
ruled out. For if, as one might conceive happening, the study of
gravitational phenomena on Mars obliged us to amend the law
of gravitation, we could not let things rest there: we could not
cheerfully say "The law takes a different form on Mars", as we
might say "Rabbits eat different food on Mars". Our concep-
tion of a law of nature requires that, if the law has to be so
amended, the modified law must continue to explain the
terrestrial and other phenomena previously accounted for in
terms of the unmodified law. Any discovery which forced us to
amend the law of gravitation itself would therefore be regarded
as revealing not a gravitational non-uniformity, but rather an
inadequacy in our present ideas about gravity, a defect in the
theory of gravitation with implications as much for the Earth
and the distant nebulae as for Mars. If there were, say, un-
commonly strong gravitational fields on Mars, such an in-

144 THE PHILOSOPHY OF SCIENCE

adequacy might be shown up first by a study of Martian gravity;
but once discovered it could not be put on one side with a mere
"They order these things differently on Mars' '.

The statement that the law of gravitation might be different
on Mars is, therefore, of doubtful meaning. This being so, the
suggested assumption that it is not the case that the law is
different on Mars is also of doubtful meaning. So after all, it is
not as clear as it at first seemed to be that physicists are assuming
anything when they apply the same law of gravitation to gravita-
tional phenomena in different places or at different times.

5.2 Physicists work on presumptions, not assumptions

The point is worth expanding. All we have shown so far is
this: that, by itself, a physicist's expressing his Law of Gravi-
tation in an identical form on all occasions proves nothing about
the Uniformity of Nature. To add the conclusion, that employ-
ing the same law to account for all gravitational phenomena
does not entail making any assumptions, may be misleading.
For what are we to count as gravitational phenomena? Once we
have a theory of gravitation of any standing, it will be just those
phenomena which can be, or which there is reason to suppose
might be explained in terms of that theory that will be called
'gravitational' phenomena, so our conclusion is at first sight an
empty one.

But while the conclusion may rest at bottom on a tautology,
it is none the less an important one. Once again, the thing that
matters is the difference between laws and generalizations. In
physics, if we start by taking a phenomenon to be purely
gravitational, but it turns out not to be properly explicable on
the current theories, then one of two things may happen: either
we must conclude that the phenomenon is not after all a purely
gravitational one, and look elsewhere for an explanation of the
deviations from the expected behaviour; or we must call the
current theory of gravitation in question. In the former case,
other laws must be brought in to account for the unforeseen
features of the phenomenon: in the latter, the laws and theories
must be reconsidered and revised. But if reconsideration is
forced on us, the revised law (e.g. Einstein's) will no more be

UNIFORMITY AND DETERMINISM 145

expressed in a way referring to a particular place or occasion
than was the unmodified law (e.g. Newton's): it will accordingly
be applicable, if at all, to every appropriate system of bodies,
regardless of the place and time in question. In neither case will
there be any room to talk of a law of nature having a different
form at different places or times; nor, except misleadingly, of
its having the same form at different places and times either,
for only one expression can be entitled to such a name as 'the
Law of Gravitation'.

Generalizations, such as we meet in natural history, are
treated in quite a different way. The statement "All rabbits eat
lettuces" is liable to sudden upset, and might well be rewritten
"All known species of rabbit eat lettuces". Here there is an
unspoken reservation "On Mars perhaps, who knows?" The
generalization requires to be covered by a guarding clause of a
kind which would be totally out of place if added to a law of
nature.

This difference between laws and generalizations is con-
nected with something we noticed earlier, the fact that natural
historians are committed for the most part to the everyday
classification of their subject-matter, whereas it is open to
physical scientists to reclassify theirs as they go along: "What
is or is not a cow is for the public to decide" but how different
it is with cadmium, a diffraction pattern, an electron or a meson
field. Each term in the generalization "Rabbits eat lettuces" is
accordingly given a meaning before the generalization is, or
indeed could be formulated. Supposing a zoologist were faced
with lettuce-loathing rabbits brought from Mars by inter-
planetary travellers, it would not be open to him to say, "Let us
call these 'tibbars', to distinguish them from rabbits which, as
everyone knows, eat lettuce". He might, if he chose, give the
Martian rabbits a different Latin name and a special taxonomi-
cal status, but he would not be entitled to resort to ad hoc
reclassification simply to save the everyday generalization that
"Rabbits eat lettuces". For him to do so would be like a man's
saying "No Briton would lay violent hands on a woman", and
then trying to save his claim from falsification in the light of the
Law Reports by amendments of the form "No true Briton

146 THE PHILOSOPHY OF SCIENCE

would . . ." a patriotic move, perhaps, but a logically un-
systematic one, of a kind for which there can be no room in the
sciences.

In formalized sciences such as physics, by contrast, the
terminology is not fixed beforehand, least of all by the public.
Theories, techniques of representation and terminologies are
introduced together, at one swoop. It is thereafter a technical
question, in what specifiable circumstances a given metallic
strip can be accepted as cadmium, some apparatus regarded as
a neutron source, or a particular body treated for theoretical
purposes as moving under gravitational influences alone. The
application of a chemical substance-name such as 'cadmium'
involves much more than the use of words like 'wooden' or
'stone J , for it not only labels the specimen by origin and every-
day characteristics, but places it on the physico-chemical map.
We shall have to look more carefully at this point later. It is the
same with a phrase like 'purely gravitational phenomenon': to
use this phrase for the motion of some system such as a double-
star is likewise to place it on the physical map to commit
oneself to the belief that the phenomenon can be explained by
the application of some one particular theory alone. The
phenomena which form the physicist's field of study are
classified in a systematic way, which reflects the terms in which
and the methods by which he sets about explaining them; and it
is the systematic nature of this reclassification which dis-
tinguishes it from ad hoc unsystematic distortions of an existing
classification, like the "No true Briton" move.

The fact that physicists always speak of one and the same
thing as their 'law of gravitation', regardless of the place and
time referred to, involves them, in consequence, in no par-
ticular assumptions: it would not be a law of nature if they did
otherwise. They would be making assumptions only if, e.g.,
they were to suppose that all the systems they studied would
turn out to be purely gravitational, and so did not bother to
consider the possibility that other kinds of theory besides
gravitation theory would be required to account for their be-
haviour. But this sort of assumption, again, is one they would
never make. They never assume that all the systems they study

UNIFORMITY AND DETERMINISM 147

are of one type: the most that they do is to presume (a) that the
existing theories will, between them, suffice to explain the
behaviour of each fresh system of bodies which they choose to
study and (b) that any fresh system of bodies they examine will
resemble most closely in behaviour those systems which it most
closely resembles in structure. This can be illustrated with the
help of our original example. The fact that an astrophysicist
uses the same law, when explaining the motion of the parts of a
double-star, as has already been used to account for the motion
of falling apples, the moon and the satellites of Jupiter, re-
presents a uniformity in his techniques for dealing with the four
systems. This uniformity in technique reflects the presumption
that the four phenomena can be regarded as similar in type,
viz. gravitational: it would not be found if we looked at physic-
ists working in different fields gravitation and magnetism,
say.

Further, a physicist's presumptions are only initial pre-
sumptions. Our astrophysicist, for instance, must be on the
look-out for deviations; and if he finds things working out
otherwise than he was led by gravitation theory to expect, he
will have to ask himself why this happens. If he were not on the
look-out for such deviations, and did not even bother to' ask
whether the theory of gravitation would explain the star's
motions by itself, or whether other forces were involved, then
one might indeed say that he was presupposing or taking for
granted something which was in need of justification. But, in
fact, he will always be ready to reconsider the initial pre-
sumption that a double-star and the solar system are, theoreti-
cally speaking, strictly comparable as soon as there is any
reason to do so. As soon as he does begin to look elsewhere for
an explanation of observed deviations, he will have abandoned
the initial presumption that the motion of the parts of the
double-star can be regarded as purely gravitational: presump-
tion (b) above. But this is not all. If and when there is an
adequate reason for doing so, he will abandon also the deeper
presumption (a), that there is a place for the novel system on the
map as it is, i.e. that the existing theories between them are
capable of explaining the behaviour of the new double-star, and

148 THE PHILOSOPHY OF SCIENCE

will try to discover how the existing theories can be modified
or supplemented in order to account for its behaviour.

What to begin with looked like an assumption turns out
therefore to be hardly more than a piece of common sense. If
physicists use the same form of law in widely differing cases,
that is the mark, not of a daring presupposition about the
Uniformity of Nature, but of a decently methodical procedure.
And if we try to express in words what it is that physicists
thereby presume, it will take the form, not of a grandiose
principle about things-in-general, but rather of some such trite
expression as this: that, unless there is some reason to suppose
that a novel phenomenon cannot be explained in terms of the
theory which it is natural to turn to first, there is every reason
to turn first to that theory. This is not a very dangerous pre-
sumption, in need of a reasoned defence. In any case, it is one
which carries its own shield with it: if in practice a physicist is
mistaken in his first attempts at an explanation, taking for a
purely magnetic phenomenon, say, what on further investiga-
tion turns out to be partly an electrical one, this is something
which will soon show up. Once he has found out his mistake,
he will be warned, and will know what to expect next time he
encounters a similar system. So it is not Nature that is Uniform,
but scientific procedure ; and it is uniform only in this, that it is
methodical and self-correcting.

5.3 Criteria of similarity within and outside science

One last attempt might be made to 'make honest generaliza-
tions' out of the statements of theoretical physics, and so to
find a place in the arguments of physical theory for a Principle of
the Uniformity of Nature. For on the previous page we spoke of
systems being 'similar in structure*: might one not accordingly
say, at the very least, that physicists assume similar phenomena
to occur always when structurally similar systems are placed in
similar situations?

This suggestion is an attractive one only so long as we leave
the physicists' criteria of similarity unexamined. For where do
these criteria come from? Suppose that phenomena and
situations were to be classified as similar within the physical

UNIFORMITY AND DETERMINISM 149

sciences on the same grounds as outside: if this were so, then
we might yet have material for the kind of overriding general-
ization which could be spoken of as a Trinciple of the Uniform-
ity of Nature', without being a complete truism. But is it so?

At first sight the everyday criteria of similarity, with which
we are familiar outside physics, seem to fit well enough. One
might reasonably claim to see a resemblance between a punt-
pole sticking out of the river and a walking-stick half-immersed
in a water-butt; and one might suppose that this resemblance
accounted for the similarity in the explanations which a
physicist would give of the way the two things looked. So, one
might conclude, the resemblance between physically similar
situations is something which can be seen; and the Uniformity
Principle can be put in the form, " Where two systems can be
seen to resemble one another, the explanations of their be-
haviour are similar/' But one has only to look at a punt-pole
sticking out of the river and another punt-pole lying broken
on the river-bank in order to see a resemblance between them,
too, yet in this case the explanations a physicist would give of
the way they looked would be quite unlike one another: what-
ever resemblance we may see between the objects is, for physical
purposes, irrelevant. This kind of thing happens very often.
Even where there are resemblances to be seen, these may be of
no interest to the physicist. The reason is that the criteria of
physical similarity between phenomena, objects and situations
are fixed by our experience within physics, and not beforehand.
So the statement, "Similar phenomena occur always when
similar systems are placed in similar situations", is true only if
one counts as similarities those resemblances, and those alone,
which turn out to be physically significant ; and then it is not so
much a generalization as a truism.

To know what phenomena, systems or situations to speak of
within physics as 'similar' requires not merely an eye for re-
semblances, but a knowledge of what resemblances matter; and
this knowledge comes only when one has some acquaintance,
however rudimentary, with the theories physicists have come to
accept. If you point out as similar phenomena the ways in which
a punt-pole looks in the river and a walking-stick looks in a

150 THE PHILOSOPHY OF SCIENCE

water-butt, you thereby show your familiarity with elementary
optics: had you no such familiarity, you could not know that
the seen resemblance was physically significant, i.e. that the
explanations would be sufficiently alike to justify your remark.
Consider a contrasted example: if your wireless howls and the
man comes from the radio shop to have a look at it, he may
invoke all sorts of resemblances in the course of his diagnosis
which are far from obvious to a layman. Perhaps the aerial is
festooned around the valves, and the mechanic points to it,
saying, "If you do that sort of thing with your set, you can't
expect good reception: why, it's like trying to hold a public
meeting in the Dome of St. Paul's." No doubt you will find this
remark mystifying: the resemblance between your coiled aerial
and the Dome of St. Paul's may not be striking. Yet there is
nothing fundamentally different about the example: only here it
is manifest that the criteria of physical likeness depend entirely
on the formulation of a satisfactory theory. No doubt the mecha-
nic will use some technical term such as 'resonance' to mark the
likeness; but this acts simply as a sign-post pointing towards
the theory which justifies the comparison. This function of
technical terms seems to be overlooked by Mach, who tends to
use words like 'refraction' and 'diffraction' in a misleading way;
as though the layman could tell diffraction from refraction as
surely as he can tell a cow from a pig. Whereas it is only with
the development of optical theories that the need for fresh
terms and fresh criteria of similarity becomes clear.

If we try to formulate the Principle of the Uniformity of
Nature in terms of similarities between different phenomena
and different situations, the result will be either vacuous or
untrue. One can say, of any particular phenomenon, "Things
always happen in that way under such circumstances", and to
do so shows that you have a grasp of the particular factors
required if an explanation of this phenomenon is to be
given. But that is all; and there is no place for any general state-
ment, any common-sense generalization, which. will in all cases
say what factors are physically significant. This is something
which has to be discovered afresh in each part of the subject.

There seems, then, no hope of finding a place for the pro-

UNIFORMITY AND DETERMINISM 151

posed Principle of Uniformity as a premise in the arguments of
physicists. This conclusion is borne out if we look at an argu-
ment in which one might genuinely talk of a premise being
assumed. For instance, there has been developed recently a
method of dating archaeological finds, known as the 'radio-
carbon method*. Wherever a find includes organic remains
bones of animals or men, or ashes, or relics of wooden struc-
tures or implements the date of death can be computed from
the amount of radio-active carbon present in the remains. Now
this calculation can be made only on the assumption that, during
the lifetime of the animals or men or trees concerned, the radio-
carbon content of the atmosphere was effectively the same as it
is now; for it is the decrease since that time in the proportion of
radio-carbon present in the remains from which is computed
the lapse of time. Here we have a very genuine assumption, and
one which in fact there is every reason to suppose reliable. If,
however, any reason were shown for modifying the assumption
if, for instance, evidence were found that the radio-carbon
content of the atmosphere had been greater 10,000 years ago
than it is now then all our calculations would have to be re-
viewed, and the dates inferred from radio-carbon measurements
would have to be altered. This is the mark of a genuine assump-
tion: modify the assumption, and the conclusions will change.

What would change if we gave up the Principle of the Uni-
formity of Nature? How would it alter our scientific conclusions
if we modified this assumption? This is never explained; and it
is not easy to see how scientists could be led, without it, to con-
clusions other than those they reach anyhow. This being so,
it is better to avoid calling the Principle an assumption at all:
when compared with any specific, concrete assumption such as
is involved in the radio-carbon method of dating, it hardly
seems to qualify for the name.

5.4 Uniformity as a principle of method

Perhaps we can look at the Principle in a different light.
There is a weaker claim, according to which we can speak of the
very success of the sciences as showing the Uniformity of
Nature. On this account of the matter, any general statement

152 THE PHILOSOPHY OF SCIENCE

about the uniformity must remain very vague ; but it will at any
rate not pretend to be a presupposition, or assumption, or
indeed anything which scientists could be described as making
'blindly'. In this sense of the phrase, there will be no room to
deny that there is some uniformity in Nature: the fact that
physical theories have been developed which have some
application to the world, is all the evidence of this uniformity we
need require the Uniformity of Nature has been discovered,
once for all. Even if the particular theories now accepted prove
to have their weaknesses, that will not wipe out the successes
already achieved: the existence of some degree of uniformity, in
this sense, will be a fact beyond dispute. Thus interpreted, the
Uniformity Principle is very unexciting, and we may prefer to
pitch our standards of uniformity higher as we go along; but
then, as the limitations of our present theories are discovered,
that will be reason not so much for abandoning the physical
sciences in despair as for developing more and better theories.
At times, indeed, the Uniformity Principle has been treated
almost as a manifesto, or as the statement of a programme: as
if one said, "There are always uniformities which remain to be
discovered/' So understood, to say that physicists believe in the
Uniformity of Nature will be to say, not that they have had
some success in the past, nor that their present procedures are
methodical; but rather that they are optimistic, and have hopes
of getting somewhere in the future.

But, in whatever sense we understand the Uniformity
Principle, whether as assumption, as discovery or as manifesto,
it has one special weakness: that of irremediable vagueness.
A principle stated in such general terms can be of no practical
significance. For to talk of Nature as uniform without saying in
what respect or to what degree it is uniform, is to say hardly
anything: no one either assumes, or has discovered, or expects
to discover an unlimited degree of uniformity in an unlimited
number of respects.

The astrophysicist studying a new double-star, for instance,
presumes not a general Uniformity but a particular and explicit
similarity namely, that this one double-star now under obser-
vation is comparable as a dynamical system with the sun and

UNIFORMITY AND DETERMINISM 153

planets, in just such respects and to just such a degree as will
entitle him to calculate the motions of its members by using
Newton's inverse-square law. He does not need to presume
anything more general than this: he does not assume, e.g., that
all double-stars will turn out to be strictly comparable with the
solar system whether or no they are remains to be seen. A
chronologist using the radio-carbon technique likewise assumes,
not a general Uniformity, but a highly specific constancy in the
atmospheric conditions since the time at which his specimen
was formed. Whatever scientific problem one considers, one
finds fresh assumptions and presumptions, all highly specific,
and differing from one case to another. Nor indeed is it neces-
sarily uniformities and correlations which are specially interest-
ing. Non-uniformities and non-correlations, independencies
and disconnections are quite as important, for instance, in dis-
crediting old wives' tales and quack remedies.

As a result, it is impossible to state in any but completely
formal terms a Principle of Uniformity common to all the
sciences alike: different scientists working in different fields
start off with different initial presumptions, and nothing more
general will be of any use to them. There is nothing to prevent
one's saying, "Scientists presume that, or have discovered that,
or believe that Nature is Uniform", leaving what it is exactly
that they presume, have discovered or believe entirely vague;
but to say this is to make the very weakest of claims, which
does no more than to indicate the form taken by scientists'
presumptions, discoveries and ambitions. So if, in practice,
scientists never seem to worry about the trustworthiness of
their Uniformity Principle, that need be regarded neither as
surprising nor as a sign of blindness on their part.

To conclude, then the Principle of the Uniformity of
Nature will not do the job designed for it by philosophers from
Mill to Russell: being at best purely formal, it can serve as a
premise in no physical arguments. But need it be any the less
important for that? Might one not hold that the principle must
be treated as a principle; that the inadequacies we have dis-
covered come from forgetting this, and mistakenly treating it as
a premise instead? Recognized for what it is, cannot a place in

154 THE PHILOSOPHY OF SCIENCE

fact be found for it like that which we have seen is allotted to
principles within physics?

The Rectilinear Propagation Principle, as we saw, has a
place in physics for so long as the methods and arguments of
geometrical optics are found of use: the abandonment of this
principle would mean the end of geometrical optics as we know
it. In a similar way, one can perhaps speak of all science as rest-
ing on certain formal principles, provided that these are
recognized as being principles of policy, of method, or of
'reason', and not premises: abandoning these principles means
the end, not of a single subject, but of science as we know it.
Scientists certainly do, on occasion, invoke principles of this
kind. If one reads the recent disputes over genetics, one finds
Lysenko criticized not merely for failing to explain the observed
facts and going against established theories, but even more for
proceeding in an unscientific, unmethodical, if not a positively
irrational manner. What repels scientists educated in the
European tradition is the way in which he resorts to invective
and ideological dogma to bolster his case: he seems to them to
be attacking not just the particular theories they accept, but the
very practice of rational scientific investigation.

If we interpret the idea of 'the uniformity of nature' in this
particular way, the only question is, whether we should not
replace it entirely by the idea of the uniformity of scientific
procedures. Perhaps we ought. But it is worth recalling how
"through all their logical apparatus'* the principles of physics
do "still speak about the world". The same may hold here: it is,
after all, as a result of experience that we find out what are the
rational ways of studying the world and its contents.

5.5 Determinism: stuffs and substances

We have remarked in several places on the differences
between our everyday classification of stuffs, as 'wood', 'stone',
'water', etc., and the chemical classification of substances, as
'cadmium', 'sodium hydroxide' and the like. This difference
becomes important if we consider one particular kind of
experiment in which the claims of the Uniformity Principle are
especially appealing. Suppose, for instance, that we take a

UNIFORMITY AND DETERMINISM 155

cadmium vapour discharge lamp and pass an electric current
through it: it will then emit its characteristic red spectrum line.
So confident are we that this will happen that we feel that the
lamp is, as it were, obliged to glow just so, and that its con-
forming to expectation is evidence of a uniformity in the
properties of chemical substances. Furthermore, though
it seems quite meaningful to suggest that, when we
switched on again, a cadmium lamp might perfectly well
glow differently, we believe this to be in the last degree unlikely,
and our confidence that it will not do so seems again to be
evidence that we are making some genuine assumption about
chemical uniformity.

This example also brings to the fore another vexed question,
that of determinism. It is easy to suppose that, when we talk of
systems obeying the laws of physics and chemistry, the meta-
phor of obedience can be taken seriously, i.e. that the systems
are in some way compelled by these laws to behave as they do.
For instance, we may have the idea that, when the cadmium
lamp is switched on, it cannot help but emit just such kinds of
light as it does; and the same idea is attractive in the case of
other phenomena it seems, e.g., that the planets are constrained
by the laws of dynamics, as by tramlines, to follow the elliptical
paths they do. On this view, the more science advances, the
more the Universe must be thought of as resembling a vast
machine. Colour is given to this type of determinism by the
physicist's use of the word 'must', and by the characteristic
logic of nature-statements. For, using the methods of quantum
mechanics, it is possible to infer from the atomic specification
of cadmium that cadmium vapour must emit radiation of just
such-and-such wavelengths when a current passes through it.
So it seems that the lamp in our apparatus has no choice, poor
thing: it must glow just as it does.

Two points can be made, which will help us to see how
these conclusions are to be avoided. To begin with, we must
not overlook all that goes on in the geological survey, and in the
process of getting and refining, which take place before the
lamp is ever constructed. As we saw before, if we forget what
an astrophysicist presumes when he begins to study a double-

156 THE PHILOSOPHY OF SCIENCE

star, we may be led to suppose, mistakenly, that he is making
assumptions of a general kind about the constancy of laws of
nature: so here, if we forget what geologists presume about the
particular lumps of rock they unearth, we may be led to sup-
pose, equally mistakenly, that chemists have to make general
assumptions about the uniformity of the properties of chemical
substances. For the question can always be raised, whether the
manufacturers who delivered the metal from which our lamp
was constructed were not mistaken in supplying it as pure
cadmium; and this question is on a par with the question,
whether the astrophysicist was right in presuming that the
motion of the parts of his double-star was a purely gravitational
phenomenon. It is no accident that we apply the adjective 'pure'
both to kinds of chemical substance and to types of physical
phenomenon. In their turn, the manufacturers rely on the sur-
veying geologist and on their own testing procedures: they
presume that ore from a correctly identified vein will, after
a given process of refining and testing, yield an end-product
which they will be entitled to sell as pure cadmium. If, how-
ever, the surveyor has made a mistake, or the vein was impure,
or their tests gave deceptive results, it is always possible that the
stuff they send out will contain other substances than cadmium,
and even, though improbably, no cadmium at all.

The geologist responsible for identifying the vein of
cadmium ore has, of course, his techniques for deciding when
a stratum is of the composition required. Having identified a
vein from tests on a sample, he will then expect the mass of
rock from which the sample was taken to go on yielding the
same substances, for as long as there is no reason to suspect
changes in the composition of the rock. Our use of the word
Vein' helps to conceal this point. It is often left indeterminate
whether a vein is to be identified by its texture, colour, etc,
or by its chemical composition; and that is natural, for the first
are taken as reliable signs of the second. Once again, nothing of
a general nature can be said about what should lead him to sus-
pect such changes: this is something which will depend entirely
on the circumstances of any particular case. Furthermore, the
geologist's presumptions will again be only initial ones. If any-

UNIFORMITY AND DETERMINISM 157

thing goes wrong, he will reconsider them; and the first sign
that something has gone wrong may be that a lamp, on being
switched on, shines in a quite unexpected way. As with any
initial presumption,the geologist will certainly be surprised if
this happens, but he will not be desperate. He will take it, not as
evidence of the breakdown of a Uniformity Principle, but
rather as evidence of an undetected variation in the ore, and so
of an unforeseen failure on the part of his surveying procedure:
his presumptions, like the astrophysicist's presumptions, are
both highly specific and open to rebuttal.

Our confidence that the lamp will glow in the way we have
been led to expect accordingly reflects, not a general assump-
tion about the uniformity of chemical substances, but rather a
specific confidence that the chemists who supplied the material
for the lamp sent what we ordered, and this in its turn depends
on the geologist's highly specific presumption, that the next foot
of ore will have effectively the same composition as the last
hundred feet. It points also towards an important difference
between everyday stuff-words and chemical substance-words, a
difference worth comparing with that between generalizations
and laws. For while a statement like "Wooden objects float"
resembles in its logic the habit-statements of natural history,
the statements of chemical theory, such as "Two molecules of
hydrogen combine with one of oxygen to form two molecules of
water", ('ZHg+Oa ^2H 2 O'), are as much nature-statements
as the law of gravitation.

The consequences of this fact are crucial. First, and above
all, it means that a chemical expression such as '2H 2 +O 2 >
2H 2 O' will be connected with the experimental results in-
directly: like a law of nature, it will tell us about the world only
if read in conjunction with other statements in this case,
directions for identifying such-a-stuff as qualifying for the
chemical symbol 'H 2 ' and other stuffs as qualifying for the
symbols *O 2 ' and 'HgO'. Accordingly, no experimental state-
ments can be deduced from the chemical formula: rather, if we
are given the chemical specification of the system under investi-
gation, we can infer experimental conclusions by arguing in
accordance with the formula.

158 THE PHILOSOPHY OF SCIENCE

This point is easily overlooked, since words which figure
both within chemistry and outside it, such as 'water', 'iron' and
'salt 1 act as logical bridges: they are used sometimes as everyday
stuff-words, sometimes as chemical substance-words, and often
enough in a way which has about it something of both uses.
Distinctions are not made in the sciences until they need to be ;
so, where the origin of a stuff carries with it a presumption as to
its chemical nature, as is the case with water, one word may
well be used to mark both origin and presumed nature. From the
logician's point of view, however, there is one drawback: this
double function helps to conceal the transition from the state-
ments of theory to those of the laboratory, so that one may not
notice that a distinction can be made between them.

Only from occasional remarks does the need to distinguish
between the two uses of such words become clear, but these
remarks are significant. The section on 'Water' in one well-
known text-book of Inorganic Chemistry opens with the words,
"Water is found in large quantities in the sea, rivers, etc." To
the non-scientist this sentence is incurably comic: to the
chemist, it is deadly serious. For the non-scientist reads it as he
would the sentences, "Trout are found in large numbers in the
streams of Dartmoor" and "There's gold in them thar hills."
Read this way, it looks like a joke, since 'water' is what we call
the stuff of the rivers and sea; so that to say this is as unhelpful
as to say "I'll tell you what I've got in my pocket ... its con-
tents." In each case, what starts promisingly ends in bathos.
The chemist, on the other hand, thinks of water more as a
chemical substance than as an everyday stuff, and accordingly
the trite-looking everyday sentence is transformed, for him,
into the significant chemical statement, "Much of the stuff of
which the sea and rivers consist can be counted as 'H 2 O'."
This sentence is far from tautologous: it is, indeed, a very
necessary and practical piece of information, for only with this
assurance can we confidently apply to the liquid we get from the
sea the statements about 'H 2 O' in books of chemical theory
such as that 'H 2 O can be decomposed by electrolysis into H 2
and O 2 '.

With this in mind, we can reconsider the case of the cad-

UNIFORMITY AND DETERMINISM 159

mium lamp. The source of our problem was this: one can infer
from atomic theory that cadmium vapour must emit light of
such-a-wavelength when a current passes through it, and it
seemed that our cadmium lamp could hardly do anything else
but conform to the general rule. Apparently, then, a chemist
was justified in asserting, quite baldly, that our particular
specimen must radiate just such wavelengths that it was
chemically compelled so to do. But now we can see part of the
way out: the argument has both a major premise, 'Cadmium
vapour when excited must emit such-and-such radiation', and a
minor premise, 'This lamp contains cadmium vapour', and this
minor premise is evidently not as trifling as it at first appeared.
For, in so far as the stuff of which the lamp is made was identi-
fied by geological tests alone, there i ; no necessity that it shall
satisfy also the chemical criteria, but only a presumption. This
is the distinction which is hidden by our word Vein*. The
minor premise is, therefore, to be understood as saying, 'The
stuff from which this lamp was made can be counted as cad-
mium', and the conclusion of the argument will accordingly be,
not the bald 'This specimen must emit light of such-a-wave-
length', but either 'In so far as the stuff from which this lamp
was made has been correctly identified as cadmium, it must
emit light of such-a-wavelength', or else 'This specimen will
emit such light.' Only in so far as the stuff can properly be
counted as cadmium or rather, to use the chemical symbol,
as Cd must it emit just that light. The minor premise is more
than a simple class-membership statement: it is the essential
identification-statement, without which there can be no bridge
between a theoretical doctrine in chemistry and any experi-
mental conclusion.

5.6 Determinis?n: theoretical necessities are not constraints

There is a second point to be considered, which also helps to
remove the force of the deterministic doctrine, and connects
with the things we noticed in an earlier chapter about the
physicist's use of the word 'must'. When we use the word
'must', it is not always the thing which we say 'must do or be
so-and-so* which is subject to compulsion, obligation or con-

160 THE PHILOSOPHY OF SCIENCE

straint. This is especially the case when the word 'must' is used
in connection with inferences, and with the application of rules.
You may, for instance, be struck by something in a girl's
features and say, "Why, she must be Jack's sister"; but to say
this is not to say that the girl is subject to any obligation or
constraint to be Jack's sister. Rather, it is to show that some-
thing forces you to the conclusion that she is Jack's sister: it is
you, not she, who is 'constrained'. Again, you may read in the
newspaper that "32.000 people visited the Zoo on Sunday" and
say, "They must mean 32,000 people, not 32": in this case also
it is you, not they, who are 'driven' driven, that is, to the con-
clusion that '32,000' was meant. The conclusions of arguments
are very commonly expressed in such terms; and wherever this
is done, the word 'must' marks the fact that the inferred con-
clusion has been drawn in a manner which could be justified
by appeal to a rule of inference, law of nature, or generally
accepted principle. Where there is such a rule of inference and
a suitable set of premises, there can be only one conclusion:
this, we say, 'must' be the conclusion i.e. must be the proper
conclusion to draw.

Inferences in the physical sciences, chemistry included, are
no exception. When, for instance, we read off from a ray-dia-
gram the depth which a shadow may in given circumstances be
expected to have, we put our conclusion in the words, "So the
shadow must be 10 ft. 6 in. deep." In saying this we are saying,
not that there is any compulsion on the shadow to have just
this depth if indeed it means anything to speak of a shadow
being under compulsion but rather that, when one applies
the methods of inference-drawing found reliable in such cases,
there can be only one conclusion as to the depth the shadow
can be expected to have. It is, accordingly, not the systems
which physicists study which are forced by the Laws of Nature
to be or do this or that, so much as physicists themselves: by
accepting particular laws of nature as applicable in particular
types of situation, they are required to draw just those con-
clusions about physical phenomena to which the laws lead,
rather than others.

Perhaps the manner in which mathematical physics origin-

UNIFORMITY AND DETERMINISM 161

ally developed was, from a philosopher's point of view, un-
fortunate. The solar system, which provided the testing-ground
for the first coherent theories of dynamics, was too good an
example: the parallel between theory and fact was there so
close that the long-term prediction of actual developments came
to seem a more reasonable and practical aim than we are
entitled to expect. If there had been no such isolated, better-
than-laboratory prototype to study, we might have been less
inclined to overlook the steps involved in applying physical
theories. For, when one checks the motions of the planets
against the astronomer's dynamical calculations, it almost seems
that Newton's Laws are plain statements of fact about the
planets themselves: for a moment the logical gulf between
Kepler's Laws and Newton's seems to vanish. The tramlines of
our dynamical calculations are projected into the sky, and the
planets are seen to be running along them. In this mood, we
tend to think of the logical articulation of our mathematical
theories as having a physical counterpart in the Celestial Tram-
way along which the planets are constrained to move the
Celestial Tramway being, so to speak, only the Inner Circle of
the Causal Nexus: determinism then seems an inescapable con-
sequence of the success of our theory of planetary dynamics.
Alternatively, since there is manifestly no Tramway there in
fact, it is as though the 'laws of nature' had as counterparts
Divine Regulations, which the planets, being obedient crea-
tures, conscientiously observed.

The Causal' Nexus is, nevertheless, a myth. The necessities
of dynamics, and of all theories in the exact sciences, are of
another kind. It is not that the physicist believes the world to be
a machine, and that this premise is essential to the success of his
theories. Rather, the physicist develops as the central parts of
his theories techniques of inference-drawing and ways of
representing physical systems which can be used inter alia to
make exact predictions ; and, if his inferences are to be regarded
as correct, they must be drawn by the use of the appropriate
techniques. Since only those theories are accepted which can
be made to fit accurately a considerable range of observed
phenomena, the results of correctly performed calculations can

162 THE PHILOSOPHY OF SCIENCE

thereafter be expected to fit the behaviour of appropriately
chosen systems. But the 'must' which appears in any physical
argument remains the 'must' of a correctly drawn inference, and
we can read it into the conclusions we draw about actual collec-
tions of bodies or pieces of apparatus only by overlooking the
fact that these conclusions depend on two presumptions: the
overriding presumption that the theories employed are not in
need of correction, and the particular presumption, that the
system or apparatus being studied has been correctly identified
('placed') as falling within the scope of these theories.

The physicist's use of the word 'must' provides no warrant,
therefore, for the idea that physics has proved that the Universe
is a machine. On closer examination, in fact, one feature of this
idea appears decidedly peculiar. For the machine of the
determinist's picture is no ordinary machine rather, it is the
machine of an engineer's dreams. Likewise with all the para-
phernalia of determinist metaphysics: causal chains, billiard
balls and the rest.

All actual machines wear out. Their behaviour can be fore-
told from the design specification only for a limited time after
manufacture; and further, it departs from the specified be-
haviour progressively, and in a more-or-less unpredictable
manner, up to the moment of breakdown. What happens after
that moment is quite unpredictable from the specification: it
depends entirely on what the engineer decides to do with the
broken-down machine. The determinisms machine, however, is
unlike actual machines in this most characteristic respect: it did
not occur to the nineteenth-century mechanist that the world-
machine was liable to wear and tear. No wonder, for the
machine he had in mind is the ideal machine, which will by
definition behave for all time in the way laid down in the design
specification. In this it betrays its origin. The determinist's
machine, churning on to eternity with mathematical precision,
bears the marks of its maker it is the ghostly counterpart of
our own mathematics: its mathematical precision reflects the
rule-guided exactitude of the steps in our calculations.

Wittgenstein illustrated this point by considering the
diagram, shown opposite, of a piston moving in a cylinder.

UNIFORMITY AND DETERMINISM

163

We are inclined to say that, in the machine here represented, as
A rotates, B must move first one way, then the other. The
piston, like the planets, seems compelled. But notice one thing:
to say only this is to say nothing about any actual machine in
the world. The diagram could be part of the specification of a
number of possible machines, but it is not itself a machine:
the 'must' of our conclusion can be read into statements about

actual machines only by overlooking the vital minor premise,
that the present state of some particular machine can be
accurately represented by this diagram. All that we are entitled
to infer from the diagram is this: that the more nearly the state
of an actual machine can be so represented, the more closely
can it be expected to move in the manner stated. Actual
machines, being subject to wear and tear, do not conform to
specification for an indefinitely long time: no single diagram
such as this can, therefore, be accepted for ever as a reliable
guide to their performance.

As with the determinist's world-machine, so with 'causal
chains'. The chain manufacturer who made chains having the
properties of causal chains would soon be a millionaire. For, as
machines wear out and break down, so also do chains wear and
snap. We do riot find unbreakable chains in Nature, nor do we
know how to make them. The unbreakable chain is an ideal,
towards which our manufacturers work: actual chains are all
liable to break and wear out, but the better the manufacturer
succeeds in making them, the greater will be the strain which
they will stand without breaking and the longer they will last.
If causal chains seem so particularly tough and long-lasting,

164 THE PHILOSOPHY OF SCIENCE

that is again a mark of their origin: the unbreakable causal
chains in the determinisms picture of the world are unbreak-
able because they are the shadows cast by the logical chains
of inferences in scientific arguments. If we do not find any
unbreakable chains in Nature, this is a reminder that only
exceptionally well-made artefacts behave according to specifica-
tion for more than a limited time; and likewise it is only
exceptional systems of bodies, like the solar system, whose
behaviour continues for more than a limited time to be explic-
able in terms of a single, simple theory.

There is one last point to be made about causal chains,
which connects with what was said earlier about the notion of a
cause. The causal chains which are the metaphysical shadows of
the arguments we employ in the physical sciences the chains
by which a shadow, for instance, is bound to be the depth it is
are not to be confused with the chains of circumstances which
are features of the diagnostic sciences. They are distinct in two
respects. First, the chains of circumstances which we speak of
as leading, e.g., to a railway accident have nothing in the way of
necessity about them: when we say "The chain of circumstances
was as follows . . .", our aim is to tell how it was that the
accident came to happen, not to show that it must inevitably
have happened. It is a further question whether or no, under
the circumstances, the accident was in any sense bound to
happen. And secondly, such chains as these do not correspond
one for one with causal chains. They could be fitted into the
determinisms picture only by including in them links from many
different causal chains as many, indeed, as the different
branches of scientific theory which would have to be invoked,
if we were to produce an exhaustive picture of the physical
processes involved in the accident. The ways in which things
happen outside the laboratory conform with unlimited exact-
ness neither to the decrees of a machine-like Destiny nor to the
pattern of any one simple argument.

5.7 'Believing that . . .' and 'Regarding as . . .'

To leave the question of determinism at this point
would, nevertheless, be unsatisfactory: one other thing

UNIFORMITY AND DETERMINISM 165

urgently needs saying. Even supposing we grant that it is a
mistake, for philosophical purposes, to project the logical
necessities of our calculations into the sky, and to think
that the planets are in any everyday sense constrained by
them; nevertheless, it is a physicist's business to do
something very like this. To the physicist, understanding
why the planets move as they do means not simply having a
mathematical theory with the help of which their orbits can be
computed, but also being able to think of them in a way which
makes sense of that theory. The scientist must be able, that is,
to look at the systems of bodies he studies with a professional
eye, and 'see' their behaviour in the way his theories require:
this, as we have seen all along, is the purpose of using models
in the physical sciences.

But though the use of models may at first look very like the
determinist's philosophical mistake, the two things are in fact
very different.To think that A is B is one thing, to think of A as
B is another; and the Man Friday fallacy is one consequence of
overlooking the difference between them. In some ways, indeed,
the two things seem to be mutually exclusive, There is no room,
e.g., to think of a cylinder of gas as a box full of fast-moving
billiard balls, unless one knows very well that it is not in fact
such a box. One cannot use the model of a box full of fast-
moving billiard balls to explain the behaviour of a box full of
fast-moving billiard balls: a model can only be used to explain
the behaviour of things which are in fact distinct from it. 1 Nor
is the physicist, whose explanation of the behaviour of gases
requires us to think of a cylinder of gis as a box full of balls, in
any danger of mistaking the one for the other he knows very
well the difference between them. Coming to understand the
kinetic theory of gases does not involve coming to believe, in any
everyday sense, that a cylinder of gas is such a box; and yet
we do need, in learning the theory, to be able to look at the one
as though it were the other, for only so shall we be able to use

x This is a logical point which crops up in connexion with many phrases
containing the word 'as'. One can paint a picture of Mrs. Siddons as Ariadne,
but a portrait of Mrs. Siddons is not 'a picture of Mrs. Siddons as Mrs.
Siddons'.

166 THE PHILOSOPHY OF SCIENCE

the theory to understand the observed behaviour of hydrogen,
oxygen, carbon dioxide and the rest.

What goes for the particular models of physics applies also
to the physicist's general method of approach, of regarding his
objects of study as articulated structures. In the physical sense
of the phrase, one can hardly 'regard' the bones of the hand or
the parts of an Anglepoise lamp 'as' articulated, for they are
just that; and to explain the action of each it is sufficient to
describe its method of articulation. But few natural objects are
of this kind, and these few are mainly the concern of biologists;
so that it is in accounting for the behaviour of systems which
are in fact not articulated that the physicist has to look for
as-it-were connexions, as-it-were structure, and as-it-were
mechanism. Systems which are not in fact articulated struc-
tures are just those that he has to regard as articulated struc-
tures.

This point is sometimes hinted at by a distinction between
the methodological determinist and the metaphysical deter-
minist. It is suggested that the physicist does not need to assert
that the world is a machine, i.e. that the behaviour of any
system he chooses to study will prove to be as mechanical as
the movements of, say, a steam-engine; but that he needs
only to assume, for professional purposes, that everything in
the particular field he is studying is determined and mechanical.
This latter, tentative assumption is spoken of as methodo-
logical determinism, and contrasted with the more general and
dogmatic, metaphysical determinism.

To put the point in this way is, however, still likely to
mislead, for it conceals one essential feature of theoretical
models. Remember: the physicist who uses the idea of light as a
substance travelling does not assume for purposes of geometrical
optics that light is literally travelling. To state his method
in this way is to commit the Man Friday fallacy, for one can be
said to assume only those things which could have been stated
beforehand; whereas the point of this view of optical pheno-
mena is, inter alia, that it brings with it a new way of talking and
thinking about them, and only in this new way of talking can the
so-called assumption even be stated. What the physicist does is,

UNIFORMITY AND DETERMINISM 167

rather, to think of the old phenomena in this new way: shadows
and the rest are now, for him, the consequences of something as-
it-were travelling from the lamp to the illuminated object,
though by all everyday criteria nothing need be travelling at all.
Nor does an astronomer assume that the planets are, literally,
constrained by the inverse-square law to follow elliptical orbits:
the Celestial Tramway is rather as-it-were a tramway, the Con-
nexions, Structure, Mechanism and Articulation of the Uni-
verse as-it-were connexions, structure, mechanism and articula-
tion. If one speaks of the physicist's idea of mechanism as a
provisional and professional assumption, people will be entitled
to suppose that the advance of science may eventually prove the
assumption justified. But there is no question of this: scientists
will never be entitled to say to the public, "At last we are in a
position, not merely to assume, but to announce definitely that
the universe is a machine/' any more than they will ever be able
to say, "At last we have proved definitely that a hydrogen
cylinder is a box full of fast-moving billiard balls." Models
remain models, however far-reaching and fruitful their applica-
tions may become.

5.8 Why popular physics misleads the layman

One last point: the models of the theoretical sciences
have parts to play not only on paper, but also in scientists'
minds. And here one thing must be noticed about the
way in which this book has been written: throughout, a
great deal has deliberately been made explicit which in
practice might often go unstated. Frequently one can see how a
shadow comes to be the depth it is, without going to the
length of drawing a ray-diagram: given the model of light as
something which travels in straight lines, one can understand
the phenomenon well enough and, having learnt to think of
light in this way, one will often be able to dispense with all
formulae and diagrams. Yet this fact does not mean that the
formulae and diagrams are, logically speaking, any less central: if
one were required to set out one's argument in full, or to ex-
plain the subject to a novice, it would be essential to use them.
For the logician, therefore, such things as these, which in prac-

168 THE PHILOSOPHY OF SCIENCE

tice may sometimes be left unmentioned, are as important as the
things which are always stated or worked out on paper; for our
purposes, it has been necessary in every case to bring them into
the open.

With this point in mind, we can return to a difficulty which
we encountered at the very beginning of the book, and see the
reason for it. There we noticed how easily misunderstandings
arise when professional physicists set out to explain their
theories to outsiders. The physicist says, "Heat is a form of
motion", or "The universe is the three-dimensional surface of a
four-dimensional balloon", or "A gas is a collection of minute
particles moving with high velocities in all directions"; and in
each case the onlooker either does not know what to understand
by the pronouncement, or overlooks the unspoken, qualifying
'as it were* and so draws the wrong conclusions.

This sort of cross-purposes will perhaps be less surprising in
the light of our subsequent discussion. For the physicist learns,
as part of his training, to think and speak in terms of his theoreti-
cal models, and when he is required to popularize his subject he
naturally turns to these for help. But to the outsider these
theoretical models, however vivid, are neither familiar nor
immediately intelligible, and their role is itself something which
he needs to have explained. Inside physics, speaking within a
theory and in terms of it, the scientist can do without the quali-
fying phrase 'as it were': he will perhaps see the implications of
the kinetic theory for the gases he is studying in the laboratory
the more clearly, the more vividly he can visualize gases as com-
posed of minute billiard balls. In the laboratory, therefore, there
will be every reason to say, "A gas is composed of . . ." instead of
"A gas is, as it were, composed of. . . ." But when the scientist
turns to speak to the outsider, the qualifying phrase becomes
vitally necessary. After all, the gas is not in fact composed of
minute billiard balls: the thing he has to explain is how physics
is advanced by using billiard balls as a model in terms of which
to think about gases.

There need be no mystery about this contrast. Often enough,
a remark which is immediately intelligible in one situation will
be either misleading or unintelligible in another. Thus, in the

UNIFORMITY AND DETERMINISM 169

theatre, a member of the audience can whisper to his neighbour,
"Here comes Cleopatra," as an actress comes on to the stage;
and, provided the neighbour understands what is going on, he is
in no danger of being misled. But the same words, used when
passing the actress in the street next day, would be open to
serious objection: it is in fact Edith Evans, not Cleopatra, we
have met. Whether we may safely speak of 'Cleopatra', or must
say rather 'as it-were Cleopatra' ('the actress playing Cleopatra')
depends entirely on the situation in which we are placed.

To explain the theories of physics in a manner which would
be both genuinely intelligible to the outsider and free from risk
of misunderstanding, a scientist must therefore reverse com-
pletely the language-shift to which he becomes accustomed in
the course of his training, and use all the terms affected by the
shift (such as 'force', 'energy', 'surface', 'billiard-ball', 'light',
'travel', 'structure', 'mechanism') in their everyday senses once
again. Anything less than this will leave room for cross-purposes
and misunderstandings of the old, deplorable kind.

One result of this reversal will be to increase the length of
any account though this is a small price to pay for under-
standing. We saw in an earlier chapter, for instance, how much
longer a statement of Snell's Law must be if the technical
vocabulary of 'light-rays' is eschewed, and the whole thing put
in explicit terms. Where a physicist, among his colleagues,
would describe the investigation leading up to the discovery of
the law as 'an investigation of the optical properties of refracting
media', the onlooker needs to think of it as 'seeing if a way can
be found of extending the techniques of geometrical optics
(ray-tracing, etc.) so as to be applicable when such things as
shadows are formed under water, or the far side of a sheet of
glass'. And whereas a physicist would state Snell's Law in the
form, "The angle which the incident ray makes with the normal
to the surface of the refracting medium (i) is related to the angle
between the refracted ray and the normal (r) by the equation

_ =/Lt", we laymen have to precede this statement by the
sin T

preamble, "The techniques can be extended by altering the
directions of the straight lines in our ray-diagram where

170 THE PHILOSOPHY OF SCIENCE

they cross the surface, and thinking of light-rays as bending
where they pass from one transparent medium to another, in
such a way that . . ." ; while the physicist's brief 'The refractive
index of water is 1.33" becomes "The constant in the equation
governing the amount by which the lines in our ray-diagram
are to be deflected when passing from one transparent medium
to another is 1.33, for the transition from air to water."

This increase in length should have been foreseen. Physical
scientists do not adopt their models and terminology for
nothing, and greater conciseness of expression (what Mach
calls 'economy') is one of the important advantages they aim at.
But it imposes on the popularizer a duty which he is often
tempted to ignore to remember that theories draw their life
from the phenomena they are used to explain, and to make sure
that, in squeezing his account into a nutshell, he does not sacri-
fice first what he should retain till the very last: an adequate
account of the physical phenomena in question, and of the
manner in which the models used in the theory help the
physicist to make sense of them.

SUGGESTED READING

INTRODUCTORY LEVEL

Philosophy of Science
Campbell, Norman, What is Science? (1921).

Mathematics and its Applications
Sawyer, W. W., Mathematician's Delight (Penguin ed., 1943).

Inductive Logic
Black, Max, Critical Thinking (1946), part III.

CLASSICAL DISCUSSIONS OF THE PHILOSOPHY OF SCIENCE

Galilei, Galileo, Dialogue concerning the Two Principal Systems

of the World (1632, tr. 1661).
Newton, Isaac, Mathematical Principles of Natural Philosophy

(1687, modern tr. Cajori, 1934).

Locke, John, An Essay on Human Understanding (1690).
Hume, David, A Treatise of Human Nature (1739).
Kant, Immanuel, Critique of Pure Reason (1781, 1787, modern tr.

Kemp Smith, 1929).

All of these contain sections dealing with problems in the philo-
sophy of science, and have greatly influenced the course of all
subsequent discussion.

MODERN CLASSICS IN THE PHILOSOPHY OF SCIENCE

Whewell, William, The Philosophy of the Inductive Sciences (1840).

Mill, J. S., A System of Logic (1843), esp. Bk. III.

Mach, Ernst, The Science of Mechanics (1883, tr. 1907); Mach's

essays on 'Economy* and 'Comparison' in Popular Scientific

Lectures (1895) may also be recommended.
Hertz, Heinrich, The Principles of Mechanics (1894, tr. 1899),

Introduction.

Poincare*, Henri, Science and Hypothesis (1902, tr. 1905).
Bridgman, P. W., The Logic of Modern Physics (1927).

OTHER GOOD GENERAL DISCUSSIONS

Born, Max, Experiment and Theory in Physics (1943).

Clifford, W. K., The Common-sense of the Exact Sciences (1885).

171

172 SUGGESTED READING

Clifford's essay on The Aims and Instruments of Scientific

Thought' reprinted in The Ethics of Belief (1947) is excellent.
Dingle, Herbert, Through Science to Philosophy (1937).
Eddington, A. S., The Nature of the Physical World (1928,

repr. Everyman ed.).

Einstein, A. and Infeld, L., The Evolution of Physics (1938).
Frank, Philipp, Between Science and Philosophy (1941).
Pearson, Karl, The Grammar of Science (1892, repr, Everyman

ed.).

Planck, Max, A Scientific Autobiography (1948, tr. 1950).
Stebbing, L. S., Philosophy and the Physicists (1937, repr.

Penguin ed.).

MORE ADVANCED DISCUSSIONS AND WORKS OF IMPORTANCE FOR
PARTICULAR TOPICS

Campbell, Norman, Physics, the Elements (1920).

Dingier, Hugo, Die Methode der Physik (1938).

Eddington, A. S., The Philosophy of Physical Science (1939).

(ed. Schilpp), Albert Einstein, Philosopher- Scientist (1949).

Kneale, William, Probability and Induction (1949).

Popper, K. R., Logik der Forschung (1935).

Ramsey, F. P., The Foundations of Mathematics (1931).

Rylc, Gilbert, The Concept of Mind (1949).

Schlick, Moritz, Gesammelte Aufsatze (1938).

Watson, W. IL, On Understanding Physics (1938).

Whitrow, G. J., The Structure of the Universe (Hutchinson's

University Library, 1949).
Wittgenstein, L., Tractatus Logico-Philosophicus (1922), esp.

section 6.3 ff.
Woodgcr, J. H., Biological Principles (1929).

The British Journal for the Philosophy of Science, published
quarterly, contains important articles on a variety of subjects: those
in early issues by Prof. H. Dingle can be particularly recommended.
The Penguin series Science News also contains worth-while articles
on the philosophy of science from time to time. From more out-of-
the-way periodicals, two papers are worth special mention, Prof.
G. G. Simpson's article on classification in taxonomy (Bulletin of the
American Museum for Natural History, 1945) and Prof. K. R.
Poppers article on the part played by tradition in science (Rationalist
Annual, 1949).

INDEX

ABSOLUTE Zero of Temperature,

129 ff
Accuracy, degrees of, 29, 70 ff,

111, 113

Action at a distance, 118
Aristotle, 46, 117, 118
Atomic model, 12, 39, 40, 137-9

BACON, Roger, 64

Bent stick phenomenon, 61-2,

149-50.

Bergson, H., 125
Black, Max, 171
Bohr, N., 138
Born, Max, 13, 123, 171
Boyle's law, 86-8, 111
Bridgmaii, P. W., 171
Brownian motion, 137-8

CALORIC fluid, 39, 137
Campbell, Norman, 171, 172
Carroll, Lewis, 102
Causal chains, 119, 124, 162,

163-4

connexions, 41, 54

nexus, 124, 161
Causality, 9, 119, 123
Cause, notion of, 10, 119 ff.
Charles' law, 131
Chemical substances, 53

and stuffs, 51 n., 146

uniformity of, 154 ff.
Chemistry, 46, 137
Churchill, Winston, 72
Classification, taxonomic, 50-2,
145-6

Cleopatra, 169
Clifford, W. K., 171-2
Cloud chamber, 136, 139
Complete description, 118-9, 124
Confirmation, 110

theory of, 112-3
Constant conjunction, 91, 96, 98,

103

Contours, existence of, 135, 137
Conventionalism, 75, 83, 88
Cosmic epoch, 91, 99
Crusoe, Robinson, 19-20, 134

DALTON, J., 46

Deduction, role of, 41, 84, 106 ff.
Deductive systems, 62, 77 ff., 84
Description and explanation,

53 ff.
Determinism, 94, 154 ff.

metaphysical and methodo-
logical, 166

Diagnostic sciences, 121 ff.
Diffraction, 29, 68, 69
Dingle, H., 172
Dingier, H., 172
Discovery and inference, 19,
24-5, 42, 43 n., 76-7

accidental, 44

Dynamics, Newtonian, 33, 46,
70, 118

Aristotelian, 46, 118

EDDINGTON, Sir A., 11-12, 103,

108, 124 ff., 172
Einstein, Albert, 13, 15, 16, 38,

43, 70, 85, 117-19, 123-4,

137, 144, 172

173

174

INDEX

Empirical character of science,

80-1

Ether, 38, 118, 137
Exactitude, mathematical, 70 ff.
Exactness, practical, 70 ff.
Euclidean definition of point, 72

straight lines, 71
Evans, Dame Edith, 169
Experiments, 49, 57, 65 ff., 73 ff.,

110, 111

FRANK, P., 172

GALILEO, 130, 171

Geiger-counter, 136

HABIT statements, 50, 85, 157

Heisenberg, W., 124

Heraclitus, 22

Hertz, H., 171

Horizon of science, 117-19, 123-4

Hume, D., 91-6, 103, 171

Hypotheses, 11, 49, 79, 80-3

IDEAL Gas, notion of, 131
Ideals, theoretical, 70 ff.
Identification in chemistry, 104,

147, 155-9, 162
Imagination, 43
Induction, 9, 43, 140
Inference, syllogistic, 33, 49, 102
Inference tickets, 93-4, 103
Inferring techniques, 23 ff., 30,

33,58,61,64,93,95, 128, 160,

161-2

JEANS, Sir J., 11, 12-13, 108

Kinetic theory of matter, 39,

165-6, 168
Kneale, W., 91, 98, 101-2, 138-9,

172

LANGUAGE everyday, 18-19
onlookers and participants,

13, 58, 169-70
scientific, 13, 21

conciseness of, 15, 170
and everyday, 35-6, 47,
50 ff., 105, 145, 169
Language shift, 13, 169
Laplace, 117
Law-like statements, 78
Laws, scope of, 31, 63
and principles, 83-4
phenomenological, 86-8
Laws of Nature, 11, 49, 52, 57 ff.
and generalizations, 10, 34,
77, 99, 105, 110, 126,
141-2, 144-6, 151
as maxims, 100 ff.
cf. laws of projection, 109-10
logical character of, 11, 78 ff.,

90 ff.

not 'true', 78, 101
Leibniz, 38, 55, 118
Light, everyday view of, 21-3, 26
Greek view of, 23, 26, 30,

39

rectilinear propagation of,
17 ff., 23 ff., 29-30, 57,
71, 83, 86, 109, 138, 154
wave theory of, 94, 113 ff.
Light-ray, idea of, 26-9, 60-1, 65,
69, 70-2, 77-8, 114, 126,
130-1

Locke, 91-2, 103, 171
Lysenko, 154

KANT, I., 128-9, 171

Kepler, 64

Kepler's Laws, 87-8, 161

MACH, E., 40-3, 54, 84, 91-6,
105 ff., 150, 170, 171

INDEX

175

Man Friday, 23, 135

PARTICLE, idea of, 72

fallacy, 20, 40, 138-9, 165, Pearson, Karl, 172

166
Maps and itineraries, 121-3

Phenomenalism, 40-1, 105 ff.
Phlogiston, 81, 137

and methods of projection, Physics, popularization of, 11 ff.,

108, 167 ff.

127, 132-3

Mathematics, role of, 11, 26, 31-2, Planck, M., 38, 172
70, 108, 128, 130, 162, 165 ff. Poincare, H., 10, 171

world of, 33
Maxwell, J. C., 117

Point, idea of, 72
Popper, K., 54, 172

principles of electromagnet- Principles and laws, 83-4, 86

ism, 86
Mill, J. S., 141, 153, 171

Methods, 9, 119

Models, 11, 12, 29, 30, 34-5, 39,
165-6, 167-9

fertility of, 37, 50
Motion, equations of, 33, 109

NATURAL HISTORY, 34

and physics, 44 ff., 50 ff., 55,
67, 74, 82-3, 85, 111-2,
141-2, 145
Nature-statements, 50, 87, 155,

157

Necessity and laws of nature, 91,
92, 96, 103

in physics, 159 ff.
Newton, Isaac, 46, 117, 171

Laws of Motion, 86, 88, 90,
161

Probability calculus, 10, 49, 112-3

Procrustes, 126

Protons & electrons, mass-ratio

and number of, 125
Ptolemy, 64

QUANTUM mechanics, 35, 113,
118-19, 123-4

RADIO-CARBON dating, 151, 153
Ramsey, F. P., 91-2, 100 ff., 172
Ray-diagrams, 25-6, 108, 127-8,

167
Refraction, 27, 29, 55, 58 ff.,

63 ff, 73 ff., 169

anomalous, 60, 64, 77, 79
Refractive index, 60, 63, 80, 85,

170
Regularities, form of, 44-5, 64,

77

Law of Gravitation, 99-100, Relativity, general theory of,

145, 153

12, 85
Representation, methods of, 32,

115 ff., 122, 126 ff.,

of phenomena, 27, 29, 41-2
Romer, 37

OBSERVATIONS, 54

Optical homogeneity, 63, 74, 77 R u f sel /> B -> 12 Q 4 >

Optics, geometrical, 17 ff., 23 ff., R y le > G -> 78 > 93 ~*> 1

36, 57 ff., 65-6, 69, 83, 108,

113 ff., 127, 154, 166

physical, 36-8, 69, 89, 113 ff. SAWYER, W. W., 171

Ostwald, W., 138 Scattering, 71

*, 172

176

INDEX

Schlick, M., 91-2, 100 ff., 172
Scope, 31, 63, 69, 78

of theories, 112-13
Simpson, G. G., 172
Snell's law, 58, 63 ff., 70, 73 ff.,

77-8,85,86, 109, 114, 169
Stark effect, 75
Statistical mechanics, 113
Stebbing, L. S., 172
Stuff and chemical substance,

51 n., 146, 154 ff.
Syllogism, 33, 49, 102
System, 47, 77, 146

Truth and laws of Nature, 78-9,
86-7, 98, 99, 101, 112
and theories, 114 ff., 128

UNIFORMITY of chemical sub-
stances, 154 ff.

of Nature, 9, 140 ff.
Universe as a machine, 155,
162-4, 167

spherical model of, 1 2, 1 5-16,
40, 168

TEMPERATURE, 128, 129 ff.

ideal-gas scale of, 131-3
logarithmic scale of, 132-3

Thermodynamics, principles of,
86

Theories, basic, 113 ff., 123
Pyramid model for, 84
Stratification of, 80-1

Theoretical entities, 11

existence of, 38, 134 ff.

Theoretical ideals, 70 ff.

Thomson, J. J., 138

Time and causation, 121

WAISMANN, F. 117
Watson, W. H., 172
Whewell, W., 171
White, Gilbert, 54
Whitehead, A., 91, 98 ff.
Whitman, W., 22
Whitrow, G. J., 172
Wittgenstein, L,, 13-14, 51, 81,

88-9, 129, 162-3, 172
Woodger, J. H., 172

ZEEMAN effect, 75