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EDITED BY J.
H.
MUIRHEAD,
LL.D.
INTRODUCTION TO MATHEMATICAL PHILOSOPHY
the
same j4uthor.
PRINCIPLES OF SOCIAL RECONSTRUC TION. yd Impression. Demy 8vo. 73. 6d. net.
"Mr
Russell has written a big
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book."
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Nation.
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attempt to extract the essence of these three doctrines, then as guidance for the coming recon
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London
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&**
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INTRODUCTION TO
MATHEMATICAL
PHILOSOPHY BY
BERTRAND RUSSELL
LONDON
:
GEORGE ALLEN & UNWIN, LTD.
NEW YORK: THE MACMILLAN
CO,
May 1919 Second Edition April 1920
First published
[All rights reserved}
PREFACE " intended essentially as an Introduction," and does not aim at giving an exhaustive discussion of the problems
THIS book
with which
is
it
deals.
It
seemed desirable to
set forth certain
results, hitherto only available to those who have mastered logical symbolism, in a form offering the minimum of difficulty
to
the beginner.
The utmost endeavour has been made
to
avoid dogmatism on such questions as are still open to serious doubt, and this endeavour has to some extent dominated the choice of topics considered. The beginnings of mathematical logic are less definitely known than its later portions, but are of Much of what is set forth at least equal philosophical interest.
not properly to be called " philosophy," though the matters concerned were included in philosophy so long as no satisfactory science of them existed. The nature of infinity and continuity, for example, belonged in former days
in the following chapters
is
to philosophy, but belongs now to mathematics. Mathematical philosophy, in the strict sense, cannot, perhaps, be held to include
such definite
scientific
results
as
have been obtained
in
this
region ; the philosophy of mathematics will naturally be ex pected to deal with questions on the frontier of knowledge, as
which comparative certainty is not yet attained. But speculation on such questions is hardly likely to be fruitful unless the more scientific parts of the principles of mathematics to
A
known. book dealing with those parts may, therefore, claim to be an introduction to mathematical philosophy, though
are
can hardly claim, except where it steps outside its province, to be actually dealing with a part of philosophy. It does deal,
it
vi
Introduction
to
Mathematical Philosophy
however, with a body of knowledge which, to those
who
accept appears to invalidate much traditional philosophy, and even a good deal of what is current in the present day. In this way, as well as by its bearing on still unsolved problems, mathematical it,
logic is relevant to philosophy.
For
this reason, as well as
on
account of the intrinsic importance of the subject, some purpose may be served by a succinct account of the main results of
mathematical logic in a form requiring neither a knowledge of mathematics nor an aptitude for mathematical symbolism. Here, however, as elsewhere, the method is more important than the results, from the point of view of further research ; and the
method cannot
well be explained within the
framework
of such
be hoped that some readers following. be interested to advance to a study of the sufficiently may method by which mathematical logic can be made helpful in a
book as the
It is to
But that investigating the traditional problems of philosophy. is a topic with which the following pages have not attempted to deal.
BERTRAND RUSSELL.
NOTE
EDITOR'S
Mathematical relying on the distinction between this Philosophy and the Philosophy of Mathematics, think that book is out of place in the present Library, may be referred to what the author himself says on this head in the Preface. It is
THOSE who,
not necessary to agree with what he there suggests as to the readjustment of the field of philosophy by the transference from it to mathematics of such problems as those of class, continuity, infinity, in order to perceive the
discussions that follow on the
bearing of the definitions and
work
of
" traditional philosophy."
philosophers cannot consent to relegate the criticism of these at any categories to any of the special sciences, it is essential, If
know the precise meaning that the science these concepts play so large a part, which mathematics, other hand, there be mathematicians on the to them. If, assigns to whom these definitions and discussions seem to be an elabora rate, that
they should in
of
and complication of the simple, it may be well to remind them from the side of philosophy that here, as elsewhere, apparent tion
conceal a complexity which it is the business of whether philosopher or mathematician, or, like the somebody, author of this volume, both in one, to unravel. simplicity
may
vii
CONTENTS
........ ....... ....
CHAP.
PREFACE
EDITOR'S NOTE
PAGE
V vii
1.
THE SERIES OF NATURAL NUMBERS
2.
DEFINITION OF
3.
FINITUDE AND MATHEMATICAL INDUCTION
4.
THE DEFINITION OF ORDER
29
5.
KINDS OF RELATIONS
42
6.
SIMILARITY OF RELATIONS
7. 8.
9.
NUMBER
12.
.
.
.
.
..... ...... ... ..... .... ...... .
.
.
AND CONTINUITY LIMITS AND CONTINUITY OF FUNCTIONS SELECTIONS AND THE MULTIPLICATIVE AXIOM .
.
.
I
,11
RATIONAL, REAL, AND COMPLEX NUMBERS INFINITE CARDINAL NUMBERS INFINITE SERIES AND ORDINALS
10. LIMITS 11.
.
.
2O
S2
63
77 89
97 107 IJ 7
.
14.
THE AXIOM OF INFINITY AND LOGICAL TYPES INCOMPATIBILITY AND THE THEORY OF DEDUCTION
15.
PROPOSITIONAL FUNCTIONS
155
16.
DESCRIPTIONS
167
17.
CLASSES
18.
MATHEMATICS AND LOGIC
194
INDEX
207
13.
.
.
..... ........ .... ...... .
Viii
.
131
144
l8l
Introduction to
Mathematical Philosophy CHAPTER
I
THE SERIES OF NATURAL NUMBERS MATHEMATICS
is
when we
a study which,
start
from
its
most
familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards
gradually increasing complexity real
:
numbers, complex numbers
plication to differentiation
mathematics.
The other
and
;
from integers to fractions, from addition and multi
and on to higher
integration,
direction,
which
is
less
familiar,
proceeds, by analysing, greater and greater abstractness and logical simplicity instead of asking what can be defined to
;
is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced.
and deduced from what
It is the fact of
pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics.
But
it
should be understood that the distinction
the subject matter, but in the state of
mind
is
one, not in
of the investigator.
Early Greek geometers, passing from the empirical rules of
Egyptian land-surveying to the general propositions by which those rules were found to be justifiable, and thence to Euclid's axioms and postulates, were engaged in mathematical philos ophy, according to the above definition ; but when once the axioms and postulates had been reached, their deductive employ ment, as we find it in Euclid, belonged to mathematics in the I
2
Introduction sense.
ordinary
The
Mathematical Philosophy
to
between
distinction
mathematics
and
one which depends upon the interest and the research, upon the stage which the research inspiring has reached ; not upon the propositions with which the research
mathematical philosophy
is
is
concerned.
We may state the same distinction in another way. The most obvious and easy things in mathematics are not those that come logically at the beginning they are things that, from ;
the point of view of logical deduction, come somewhere in the middle. Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are
"
"
neither very complex nor very simple (using in a simple And as we need two sorts of instruments, the logical sense). telescope
and the microscope, for the enlargement of our visual we need two sorts of instruments for the enlargement
powers, so
of our logical powers,
one to take us forward to the higher
mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics. We shall find that by analysing our ordinary
mathematical notions we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects
by adopting It is the
fresh lines of
purpose of this
advance
after our
backward journey.
book to explain mathematical philos
ophy simply and untechnically, without enlarging upon those portions which are so doubtful or difficult that an elementary
treatment
is
in Principia
A
full treatment will be found scarcely possible. * the treatment in the present volume ;
Mathematica
intended merely as an introduction. To the average educated person of the present day, the obvious starting-point of mathematics would be the series of
is
whole numbers, i, 1
2,
3, 4,
Cambridge University Press,
By Whitehead and
Russell.
vol.
... i.,
1910
etc.
;
vol.
ii.,
1911
;
vol.
iii.,
1913.
The
of Natural Numbers
Series
3
Probably only a person with some mathematical knowledge would think of beginning with o instead of with i, but we will this degree of knowledge the series point
presume
;
we
take as our starting-
will
:
o,
and "
it
i,
2,
3,
this series that
is
series of natural
.
we
.
.
n,
shall
n+ 1,
.
.
.
mean when we speak
of the
numbers."
high stage of civilisation that we could take our starting-point. It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2 the degree of abstraction It is only at a
this series as
:
from easy. must have been difficult.
involved
is
far
the Greeks and
And As
the discovery that I is a number for o, it is a very recent addition ;
Romans had no such
digit.
we had been earlier days, we If
embarking upon mathematical philosophy in should have had to start with something less abstract than the series of natural numbers, which we should reach as a stage on
our backward journey. When the logical foundations of mathe matics have grown more familiar, we shall be able to start further back, at what is now a late stage in our analysis. But for the
moment
the natural numbers seem to represent what
and most familiar
But though
in
is
easiest
mathematics.
familiar,
they are not understood.
Very few
people are prepared with a definition of what is meant by " " " I." It is not very difficult to see that, number," or o," or starting from o,
by
any other
repeated additions of
of the natural
numbers can be reached
I, but we shall have to define what " and what we mean
we mean by " adding I," These questions are by no means
by
repeated."
It was believed until easy. recently that some, at least, of these first notions of arithmetic
must be accepted as too simple and primitive to be defined. Since all terms that are defined are defined by means of other terms,
it is
to accept
clear that
some terms
human knowledge must always
be content
as intelligible without definition, in order
Introduction
4 to
to
Mathematical Philosophy
have a starting-point for its definitions. must be terms which are incapable
there
however
possible that, might go further
when
that,
still.
far
back we go in
On
It is not clear that
of definition
defining,
the other hand,
it
is
analysis has been pushed far enough,
:
it
is
we always
also possible
we can reach
terms that really are simple, and therefore logically incapable This is a of the sort of definition that consists in analysing. question which
purposes
it
is
not necessary for us to decide ; for our sufficient to observe that, since human powers it
is
known to us must always begin some for the moment, though perhaps undefined with terms where, not permanently. are finite, the definitions
All traditional pure mathematics, including analytical geom of propositions etry, may be regarded as consisting wholly
That is to say, the terms which means of the natural numbers, and occur can be defined by about the natural numbers.
the propositions can be deduced from the properties of the natural numbers with the addition, in each case, of the ideas
and propositions of pure logic. That all traditional pure mathematics can be derived from the natural numbers
long been suspected.
is
a fairly recent discovery, though it had who believed that not only
Pythagoras,
but everything else could be deduced from the discoverer of the most serious obstacle in was numbers, " " of mathematics. the way of what is called the arithmetising It was Pythagoras who discovered the existence of incommathematics,
mensurables, and, in particular, the incommensurability of the and the diagonal. If the length of the side is of inches in the diagonal is the square root number I inch, the
side of a square
of 2,
which appeared not to be a number at
all.
The problem
thus raised was solved only in our own day, and was only solved to logic, completely by the help of the reduction of arithmetic
be explained in following chapters. For the present, we shall take for granted the arithmetisation of mathematics,
which
will
though
this
was a
feat of the very greatest importance.
The Having reduced
5
traditional pure mathematics to the numbers, the next step in logical analysis
all
theory of the natural
was to reduce
of Natural Numbers
Series
this theory itself to the smallest set of premisses
and undefined terms from which
it
This work
could be derived.
was accomplished by Peano. He showed that the entire theory of the natural numbers could be derived from three primitive and
ideas
five primitive propositions in
These three ideas and
addition to those of
thus became, whole of traditional pure mathe If they could be defined and proved in terms of others, matics. " so could all pure mathematics. Their logical weight," if one
pure
as
logic.
five propositions
were, hostages for the
it
use such an expression,
may
is equal to that of the whole series have been deduced from the theory of the natural the truth of this whole series is assured if the truth
of sciences that
numbers
;
of the five primitive propositions is guaranteed, provided,
course, that there
of
nothing erroneous in the purely logical also involved. The work of analysing mathe is
apparatus which is matics is extraordinarily facilitated by this work of Peano's. The three primitive ideas in Peano's arithmetic are :
o,
By
number, successor.
" successor " he means the next number in the natural
That
order. I is 2,
to say, the successor of o
is
the class
is
I,
the successor of
" number " he means, in this connection, of the natural numbers. 1 He is not assuming that
and so on.
By
we know all the members of this class, but only that we know what we mean when we say that this or that is a number, just " as we know what we mean when we say Jones is a man," though we do not know all men individually. The five primitive propositions which Peano assumes are :
(1)
o
(2)
The
(3) 1
is
a number.
number is a number. No two numbers have the same successor. successor of any
We shall
use
wards the word
"
number "
will
in this sense in the present chapter.
be used in a more general sense.
After
6
Introduction
(4)
o
(5)
Any
is
to
Mathematical Philosophy
not the successor of any number.
property which belongs to o, and also to the successor number which has the property, belongs to all
of every
numbers.
The
We
the principle of mathematical induction.
last of these is
shall
have much to say concerning mathematical induction
in the sequel
for the present,
;
we
are concerned with
it
only
occurs in Peano's analysis of arithmetic. Let us consider briefly the kind of way in which the theory of the natural numbers results from these three ideas and five as
it
" the successor of o," begin with, we define I as " the successor of We can obviously go 2 as I," and so on.
To
propositions.
on as long as we (2),
like
with these definitions, since, in virtue of will have a successor, and, in
every number that we reach
virtue of
because, successor
(3), this if it ;
cannot be any of the numbers already defined,
were, two different numbers would have the
and
in virtue of (4)
in the series of successors can be o.
numbers we reach
Thus the
series of successors
gives us an endless series of continually of (5) all
numbers come
same
of the
none
in this series,
new numbers.
In virtue
which begins with o and
on through successive successors for (a) o belongs to this series, and (b) if a number n belongs to it, so does its successor, whence, by mathematical induction, every number belongs to travels
:
the series.
Suppose we wish to define the sum of two numbers. Taking any number m, we define m-\-o as m, and m-\-(n-{-i) as the successor of m-\-n.
the
sum
we can
of
m
and
In virtue of n,
(5)
this gives a definition of
whatever number n
define the product of
may
any two numbers.
be.
Similarly
The reader can
easily convince himself that
any ordinary elementary proposition can be proved by means of our five premisses, he has any difficulty he can find the proof in Peano.
of arithmetic
and
if
It is
time
now
make it Peano, who
to turn to the considerations which
necessary to advance beyond the standpoint of
The represents
the
of Natural Numbers
Series
last
perfection
mathematics, to that of Frege,
the
of
7
" arithmetisation "
who first succeeded in "
of
" logicising
mathematics, i.e. in reducing to logic the arithmetical notions which his predecessors had shown to be sufficient for mathematics.
We
shall not, in this chapter, actually give Frege's definition of
of particular numbers, but we shall give some of the Peano's treatment is less final than it appears to be. why In the first place, Peano's three primitive ideas namely, " o," " " " are capable of an infinite number number," and successor of different interpretations, all of which will satisfy the five
number and
reasons
primitive propositions.
We
will give
some examples.
" o " be taken " number " be to mean loo, and let (1) Let taken to mean the numbers from 100 onward in the series of
numbers.
natural satisfied,
99
99,
to the
is
Then
word " number."
our
all
even the fourth, " not a number "
are
propositions
primitive
though 100 is the successor of in the sense which we are now giving
It
for,
is
obvious that any number
may
be
substituted for 100 in this example. " " o have its usual meaning, (2) Let
but let " number " " mean what we usually call even numbers," and let the " successor " of a number be what results from adding two to it. Then " I " will stand for the number two, " 2 " will stand " numbers " for the number four, and so on ; the series of now will
be o,
two, four, six, eight
All Peano's five premisses are satisfied (3)
.
.
.
still.
Let " o " mean the number one,
let
" number " mean
the set !>
and
let
axioms
"successor"
will
fact,
1>
i TV
mean "half."
be true of this
It is clear that
In
i>
Then
all
Peano's
five
set.
such examples might be multiplied indefinitely.
given any series
Introduction
8
which
to
endless, contains
is
Mathematical Philosophy no
repetitions, has a beginning,
and
has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's This
axioms.
what
(2)
though the formal proof is some mean # let " number " mean the whole " successor " of # mean x the Then
easily seen,
" o" Let
long.
set of terms, (1)
is
and
let
,
n
n+l .
" o is a number," i.e. x is a member of the set. " The successor of any number is a number,"
any term xn " (3)
xn+l
in the set,
different
;
different
the same successor," i.e. if xm of the set, x m+l and xn+l are
members
from the fact that (by hypothesis) there
this results
are no repetitions in the set. " o is not the successor of any number," (4)
the set comes before x (5)
This becomes
no term
i.e.
in
.
Any
:
taking
also in the set.
is
No two numbers have
and xn are two
i.e.
property which belongs to x09 and belongs to xn belongs to all the x's.
belongs to xn+l provided it This follows from the corresponding property for numbers. series of the form ,
A
in
which there
there
is
no
is
a
term, a successor to each term (so that
first
last term),
no
repetitions,
reached from the start in a progression.
and every term can be
number
of steps,
is
called a
Progressions are of great importance in the princi
As we have
ples of mathematics. verifies
finite
Peano's five axioms.
just seen, every progression can be proved, conversely, Peano's five axioms is a pro
It
that every series which verifies Hence these five axioms gression. class of progressions
"
may "
be used to define the " those series which
are
progressions verify these five axioms." Any progression may be taken as we may give the name " o " the basis of pure mathematics :
:
name " number
"
to the whole set of its " successor " to the the name and next in the progression. terms, The progression need not be composed of numbers it may be to its first term, the
:
The composed
Series of
Natural Numbers
of points in space, or
terms of which there
is
an
progression will give rise to
moments
9
of time, or
any other
Each
different supply. a different interpretation of all the infinite
propositions of traditional pure mathematics
all
;
these possible
interpretations will be equally true. In Peano's system there is nothing to enable us to distinguish
between these It is
different interpretations of his primitive ideas.
meant by " o," and that symbol means 100 or Cleopatra's
assumed that we know what
we
is
shall not suppose that this Needle or any of the other things that it might mean. " " o" and " number " and "successor This point, that
cannot be defined by means of Peano's five axioms, but must be independently understood, is important. We want our numbers not merely to verify mathematical formulae, but to apply in the right
way
to
common
objects.
We
want
to
have "
A
"
I ten fingers and two eyes and one nose. system in which meant 100, and " 2 " meant 101, and so on, might be all right
pure mathematics, but would not suit daily life. We want " o " and " number " and " successor " to have meanings which
for
will give
We
us the right allowance of fingers and eyes and noses. (though not sufficiently " " " "
have already some knowledge
and 2 and what we mean by I numbers in arithmetic must conform to
articulate or analytic) of
so on,
and our use
of
We
cannot secure that this shall be the case knowledge. all method that we can do, if we adopt his method, Peano's ; by ' ' " we know what we mean * ' is to say by o and number and ' successor,' though we cannot explain what we mean in terms this
of other simpler concepts."
when we must, and object of
It is quite legitimate to
at some point
we
all
mathematical philosophy to put
as possible.
By
must
;
but
off
say this it is
it
the
as long
saying the logical theory of arithmetic we are able to
for a very long time. " " and o might be suggested that, instead of setting up " number " " " and as terms of which we know the successor
put
it off
It
meaning although we cannot define them, we might
let
them
io
Introduction
Mathematical Philosophy
to
stand for any three terms that verify Peano's five axioms. They will then no longer be terms which have a meaning that is definite
though undefined: they will be "variables," terms concerning which we make certain hypotheses, namely, those stated in the axioms, but which are otherwise undetermined. If we adopt our theorems will not be proved concerning an ascer " the natural tained set of terms called numbers," but concerning five
this plan,
all sets is
of terms
not fallacious
;
having certain properties. Such a procedure indeed for certain purposes it represents a
valuable generalisation.
But from two points
of
view
it
fails
In the first place, it to give an adequate basis for arithmetic. does not enable us to know whether there are any sets of terms verifying Peano's axioms
suggestion of
any way
;
it
does not even give the faintest whether there are such sets.
of discovering
In the second place, as already observed, we want our numbers to be such as can be used for counting common objects, and this requires that our
numbers should have a
definite
meaning, not
merely that they should have certain formal properties. This definite meaning is defined by the logical theory of arithmetic.
CHAPTER
NUMBER
DEFINITION OF
THE
" question
What
is
II
"
number
a
?
is
one which has been
often asked, but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen
Although this book is quite short, not difficult, the very highest importance, it attracted almost no attention, and the definition of number which it contains re
der Arithmetik*
and
of
mained practically unknown
until
it
was rediscovered by the
present author in 1901. In seeking a definition of number, the
about
is
what we may
first
thing to be clear
grammar of our inquiry. attempting to define number, are call
the
Many
really philosophers, when setting to work to define plurality, which is quite a different Number is what is characteristic of numbers, as man thing. is
what
is
characteristic of
A
men.
plurality
is
not an instance
number, but of some particular number. A trio of men, for example, is an instance of the number 3, and the number of
an instance of number
3 is
;
but the
trio is
not an instance of
This point may seem elementary and scarcely worth mentioning ; yet it has proved too subtle for the philosophers,
number.
with few exceptions.
A
particular number is not identical with any collection of terms having that number the number 3 is not identical with :
1
The same answer
is
given more fully and with more development in
his Grundgesetze der Arithmetik, vol.
i.,
1893.
12
Introduction
to
Mathematical Philosophy
The number
the trio consisting of Brown, Jones, and Robinson.
something which all trios have in common, and which dis tinguishes them from other collections. A number is something that characterises certain collections, namely, those that have 3 is
that number.
"
Instead of speaking of a collection," " set." " of a class," or sometimes a
mathematics fold."
We
we
shall as a rule speak Other words used in " and " mani
same thing are " aggregate have much to say later on about
for the
shall
the present,
we
will
say as little as possible.
classes.
For
But there are
some remarks that must be made immediately. A class or collection may be defined in two ways that at first We may enumerate its members, as sight seem quite distinct. when we say, " The collection I mean is Brown, Jones, and Robinson." Or we may mention a defining property, as when we speak of " mankind " or " the inhabitants of London." The definition
sion,"
which enumerates
is
called a definition
by
" exten
and the one which mentions a defining property is called " by intension." Of these two kinds of definition,
a definition
the one by intension is logically more fundamental. This is shown by two considerations (i) that the extensional defini :
tion can always be reduced to an intensional one; (2) that the intensional one often cannot even theoretically be reduced to
Each
the extensional one.
of
these points needs a
word
of
explanation.
Brown, Jones, and Robinson all of them possess a certain property which is possessed by nothing else in the whole universe, (i)
namely, the property of being either Brown or Jones or Robinson. This property can be used to give a definition by intension of
Brown and Jones and Robinson. Con " x is Brown or x is Jones or x is Robinson."
the class consisting of sider such a formula as
This formula will be true for just three x's, namely, Brown and Jones and Robinson. In this respect it resembles a cubic equa tion with its three roots. It may be taken as assigning a property
common
to the
members
of the class consisting of these three
Definition of
men, and peculiar to them.
A
Number
13
similar treatment can obviously
be applied to any other class given in extension. (2) It is obvious that in practice we can often
know
deal about a class without being able to enumerate
No
its
a great
members.
man
could actually enumerate all men, or even all the inhabitants of London, yet a great deal is known about each of
one
This
these classes.
enough to show that definition by extension
is
not necessary to knowledge about a class. But when we come to consider infinite classes, we find that enumeration is not even
is
who only live for a finite time. the natural numbers : they are o, I, 2,
theoretically possible for beings
We
cannot enumerate
3, and so on. " and so on."
numbers, or
all
At some point we must content ourselves with
We cannot enumerate all fractions or all irrational of
ledge in regard to definition
by
all
any other all
infinite collection.
Thus our know
such collections can only be derived from a
intension.
These remarks are relevant, when we are seeking the definition of number, in three different ways. In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. In the second place, the collections
having a given number of terms themselves presumably form an it is to be presumed, for example, that there
infinite collection
:
are an infinite collection of trios in the world, for
if
this
were
not the case the total number of things in the world would be In the third finite, which, though possible, seems unlikely. " number " we wish to define such a in place, way that infinite
numbers may be possible thus we must be able to speak of the number of terms in an infinite collection, and such a collection ;
must be defined by intension, i.e. by a property common to members and peculiar to them. For many purposes, a class and a defining characteristic
all
its
it
are practically interchangeable.
The
the two consists in the fact that there
is
vital difference
of
between
only one class having a
given set of members, whereas there are always many different characteristics by which a given class may be defined. Men
Introduction
14
to
Mathematical Philosophy
be defined as featherless bipeds, or as rational animals, or (more correctly) by the traits by which Swift delineates the
may
Yahoos.
It is this fact that a defining characteristic is
unique which makes
classes
useful
;
otherwise
never
we could be
the properties common and peculiar to their Any one of these properties can be used in place
content with
members. 1
whenever uniqueness is not important. Returning now to the definition of number, it
of the class
number
a
is
way
those that have a given all
couples in
way we
this
is
clear that
of bringing together certain collections, namely,
one bundle,
We
number
of terms.
all trios
in another,
can suppose In
and so on.
obtain various bundles of collections, each bundle have a certain number of
consisting of all the collections that
Each bundle
terms. i.e.
classes
;
a class whose
is
thus each
is
members are collections, The bundle con
a class of classes.
sisting of all couples, for example, is a class of classes is
couple
is
couples
which
is
How
each
a class with an infinite
number
of
members, each
of
a class of two members. shall
we
decide whether two collections are to belong ? The answer that suggests itself is " Find
same bundle
to the
:
a class with two members, and the whole bundle of
:
how many members
each has, and put them in the same they have the same number of members." But this presupposes that we have defined numbers, and that we know out
bundle
how
if
to discover
how many terms
a collection has.
We
are so
used to the operation of counting that such a presupposition might easily pass unnoticed. In fact, however, counting,
more complex operation means of discovering how many terms a collection has, when the collection is finite. Our defini tion of number must not assume in advance that all numbers are finite ; and we cannot in any case, without a vicious circle,
though over
familiar,
it is
is
logically a very
;
only available, as a
1 As will be explained later, classes may be regarded as logical fictions, manufactured out of denning characteristics. But for the present it will simplify our exposition to treat classes as if they were real.
Number
Definition of
1
5
use counting to define numbers, because numbers are used in counting. We need, therefore, some other method of deciding when two collections have the same number of terms.
In actual fact,
it is
simpler logically to find out whether two
have the same number
collections
what that number
is.
An
of terms
illustration
than
will
it is
make
to define
this
clear.
there were no
polygamy or polyandry anywhere in the world, it is clear that the number of husbands living at any moment would be exactly the same as the number of wives. We do If
not need a census to assure us of
this,
nor do
we need
to
know know
number of husbands and of wives. We number must be the same in both collections, because each husband has one wife and each wife has one husband. The " relation of husband and wife is what is called one-one." what
is
the actual
the
A
relation
is
said to be
"
one-one
"
when,
if
x has the relation
no other term x' has the same relation to y, and x does not have the same relation to any term y' other than y. When only the first of these two conditions is fulfilled, the relation is called " one-many " ; when only the second is
in question to y,
the
number
I is
"
many-one." It should be observed that not used in these definitions.
fulfilled, it is called
In Christian countries, the relation of husband to wife is one-one ; in Mahometan countries it is one-many ; in Tibet it is many-one. The relation of father to son is one-many ; that of son to father is
one-one.
one-one
;
so
is
many-one, but that of eldest son to father
If
n
is
the relation of n to 2n or to
is
any number, the
relation of
3.
n
to
-|-i
When we
considering only positive numbers, the relation of one-one ; but when negative numbers are admitted,
is
are 2
n
to
it
becomes
is
n have the same square. These instances two-one, since n and should suffice to make clear the notions of one-one, and many-one
relations,
one-many, which play a great part in the princi
mathematics, not only in relation to the definition of numbers, but in many other connections. Two classes are said to be " similar " when there is a one-one
ples of
1
6
Introduction
Mathematical Philosophy
to
which correlates the terms of the one
relation
one term of the other
class, in the
class
same manner
in
each with
which the
husbands with wives.
relation of marriage correlates
A
few
preliminary definitions will help us to state this definition more The class of those terms that have a given relation precisely. to something or other
domain
called the
is
of that relation
:
thus fathers are the domain of the relation of father to child, husbands are the domain of the relation of husband to wife,
wives are the domain of the relation of wife to husband, and husbands and wives together are the domain of the relation of marriage. The relation of wife to husband is called the converse of the relation of husband to wife. Similarly less is the converse of greater, later is the converse of earlier, and so on. Generally, the converse of a given relation
is
that relation which holds
between y and x whenever the given relation holds between x and y. The converse domain of a relation is the domain of its
converse
of the
:
thus the class of wives
husband to
relation of
definition of similarity as follows
One
class is said to be
"
if
similar
"
to
that
if
relation
a
is
is
is
when
a
(3)
the
(i)
similar to
j3
and
j8
said to be reflexive
it
is
that every class is similar to itself, similar to a class j3, then j3 is similar to a,
is
domain.
when
to y, then
when
it
a
(2)
A
similar to y. possesses the first of these is
possesses the second, and transi It is obvious that a relation possesses the third.
properties, symmetrical tive
another when there
the
prove
a class a
state our
domain, while
other is the converse
that
the converse domain
:
one-one relation of which the one class
It is easy to
is
We may now
wife.
it
is symmetrical and transitive must be reflexive throughout domain. Relations which possess these properties are an
which its
important kind, and it is worth while to note that similarity one of this kind of relations. It
is
obvious to
the same
The
number
common
of terms
if
is
sense that two finite classes have
they are similar, but not otherwise.
act of counting consists in establishing a one-one correlation
Definition of
Number
17
between the set of objects counted and the natural numbers (excluding o) that are used up in the process. Accordingly
common
sense concludes that there are as
many
objects in the
number we confine ourselves to finite numbers, there are just n numbers from I up to n. Hence it follows that the last number used in counting a collection is the number of terms in the collection, provided the collection is finite. But this result, besides being
set to
be counted as there are numbers up to the
used in the counting.
And we
also
know
last
that, so long as
only applicable to finite collections, depends upon and assumes the fact that two classes which are similar have the same number of terms
for
;
what we do when we count
show that the set of these I to 10. The notion of
is
objects
(say) 10 objects
similar to the set of is
similarity
logically
is
to
numbers
presupposed in
the operation of counting, and is logically simpler though less In counting, it is necessary to take the objects counted familiar. in a certain order, as first, second, third, etc., but order is not of the essence of
number
:
an irrelevant addition, an un
it is
The logical point of view. notion of similarity does not demand an order for example, we saw that the number of husbands is the same as the number necessary complication from the
:
without having to establish an order of precedence The notion of similarity also does not require
of wives,
among them.
that the classes which are similar should be
finite.
Take, for
example, the natural numbers (excluding o) on the one hand, and the fractions which have I for their numerator on the other
hand
:
it is
obvious that
we can
so on, thus proving that the
two
correlate 2 with J, 3 with J,
and
classes are similar.
thus use the notion of " similarity " to decide when two collections are to belong to the same bundle, in the sense
We may
which we were asking this question earlier in this chapter. We want to make one bundle containing the class that has no
in
members
:
this will
be for the number
of all the classes that
number
I.
o.
have one member
Then, for the number
2,
Then we want :
this will
we want
a bundle
be for the
a bundle consisting 2
1
Introduction
8
of all couples tion,
;
we can
to
then one of
Mathematical Philosophy all trios
define the bundle
of all those collections that are
to see that
;
it is
"
and so on.
Given any
collec
to belong to as being the class
similar
"
to it. It is very easy has three a collection members, the (for example) those collections that are similar to it will be the
if
class of all
class of trios.
And whatever number
terms a collection
of
may
"
"
similar to it will have the same have, those collections that are number of terms. We may take this as a definition of " having the same number of terms." It is obvious that it gives results
conformable to usage so long as we confine ourselves to
finite
collections.
we have not suggested anything in the slightest degree paradoxical. But when we come to the actual definition of numbers we cannot avoid what must at first sight seem a paradox, So
far
though this impression will soon wear off. We naturally think that the class of couples (for example) is something different from the number 2. But there is no doubt about the class of it couples the number :
is
indubitable and not difficult to define, whereas
any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it
down.
2, in
It is therefore
more prudent
to content ourselves with
the class of couples, which we are sure of, than to hunt for a problematical number 2 which must always remain elusive.
Accordingly
we
set
The number of a similar
up the following
definition
:
class is the class of all those classes that are
to it.
Thus the number
of a couple will be the class of all couples. In fact, the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition
secures definiteness
and indubitableness
;
and
it is
not
difficult
numbers so defined have all the properties that we numbers to have. expect We may now go on to define numbers in general as any one of the bundles into which similarity collects classes. A number will be a set of classes such as that any two are similar to each to prove that
Definition of
Number
1
9
any inside the set. which is In other words, a number (in general) any the number of one of its members or, more simply still and none outside the
other,
set are similar to
collection
is
:
;
A
anything which is the number of some class. Such a definition has a verbal appearance of being circular, " " the number of a given class but in fact it is not. We define without using the notion of number in general ; therefore we may " the number of a define number in general in terms of given " class without committing any logical error.
number
is
The class Definitions of this sort are in fact very common. be defined first of fathers, for example, would have to defining by what will
it is
be
all
to be the father of
those
who
to define square
mean by saying
somebody
;
then the class of fathers
are somebody's father.
that
Similarly if we want define what we
(say), we must first one number is the square
numbers
of another,
and
then define square numbers as those that are the squares of other numbers. This kind of procedure is very common, and it
is
important to
realise that it
is
legitimate and even often
necessary.
We
have now given a definition of numbers which
for finite collections.
for infinite collections.
by
"
finite
"
and "
It
remains to be seen
But
infinite,"
first
we must
how
will serve will serve
it
decide what
we mean
which cannot be done within the
limits of the present chapter.
CHAPTER
III
FINITUDE AND MATHEMATICAL INDUCTION
THE
numbers, as we saw in Chapter I., can all be defined if we know what we mean by the three terms " o," " " But we may go a step farther successor." number," and we can define all the natural numbers if we know what we mean " o " and " us to understand the successor." It will series of natural
:
by
help
difference
between
finite
and
infinite to see
and why the method by which beyond the cessor
"
We will not yet
finite.
these
this
can be done,
moment assume that terms mean, and show how thence all other
are to be defined
we know what
how
done cannot be extended " o " and " consider how suc
it is
we
:
will for the
natural numbers can be obtained. It is easy to see that
we can reach any
assigned number, say " as " the successor of o," then we " " " define 2 as the successor of I," and so on. In the case of 30,000.
We
first
define
"
I
an assigned number, such as 30,000, the proof that we can reach by proceeding step by step in this fashion may be made, if we
it
have the patience, by actual experiment
:
we can go on
until
But although the method of at 30,000. each for available particular natural number, it experiment the is not available for general proposition that all such proving
we
arrive
actually
is
numbers can be reached in this way, i.e. by proceeding from o Is there step by step from each number to its successor. any other way by which this can be proved ? Let us consider the question the other way round. What are "o" and the numbers that can be reached, given the terms
and Mathematical
Finitude
" successor " whole
class of
such numbers
the successor of
2, as
any way by which we can define the We reach I, as the successor of o ; ?
Is there
?
21
Induction
I
3,
;
as the successor of 2
;
and so on.
It
"
"
that we wish to replace by something less and so on this " and vague and indefinite. We might be tempted to say that " means that the process of proceeding to the successor so on is
be repeated any finite number of times
may
upon which we
but the problem ; " the problem of defining finite must not use this notion in our defini
are engaged
is
number," and therefore we Our definition must not assume that we know what a tion.
number is. The key to our problem lies in mathematical induction. It will be remembered that, in Chapter I., this was the fifth of the five primitive propositions which we laid down about the natural finite
numbers.
It stated that
to the successor of
any property which belongs to
o,
and
any number which has the property, belongs
This was then presented as a principle, but we shall now adopt it as a definition. It is not difficult to see that the terms obeying it are the same as the numbers to
the natural numbers.
all
that can be reached from o next, but as the point in
some
We
is
successive steps from next to important we will set forth the matter
by
detail.
shall
do well to begin with some
definitions,
which
will
be
useful in other connections also.
A
property
series
if,
is
said to be
whenever
in the natural-number hereditary belongs to a number , it also belongs to
it
n-j-i, the successor of n.
" tary
if,
whenever n
easy to see,
"
"
is
though we
"
heredi Similarly a class is said to be a member of the class, so is n+i. It is
are not yet supposed to know, that to say
a property is hereditary is equivalent to saying that it belongs to all the natural numbers not less than some one of them, e.g. it
must belong to
less
less
A
than 1000, or than o, i.e. to property
is
all it
all
that are not less than 100, or
may
be that
it
belongs to
without exception. " "
said to be
inductive
when
all
it is
all
that are
that are not
a
hereditary
22
Introduction
to
Mathematical Philosophy
" " inductive property which belongs to o. Similarly a class is when it is a hereditary class of which o is a member.
Given a hereditary that
I is
member
a
of
class of it,
which o
is
a member,
it
follows
because a hereditary class contains the
members, and I is the successor of o. Similarly, given a hereditary class of which I is a member, it follows that 2 is a member of it ; and so on. Thus we can prove by a stepsuccessors of
its
by-step procedure that any assigned natural number, say 30,000, a member of every inductive class. We will define the " posterity " of a given natural number with respect to the relation " immediate predecessor " (which
is
" successor ") as all those terms that belong to every hereditary class to which the given number belongs. It is again easy to see that the posterity of a natural number con is
the converse of
sists of itself
and
all
greater natural
numbers
do not yet officially know. By the above definitions, the posterity of o terms which belong to every inductive class. It is
o
now not same
difficult to
make
it
;
but
this also
we
will consist of those
obvious that the posterity of
terms that can be reached from o by successive steps from next to next. For, in the first place, o belongs to both these sets (in the sense in which we have defined is
the
our terms)
set as those
in the second place,
;
if
n belongs
to both sets, so does
be observed that we are dealing here with the n+i. kind of matter that does not admit of precise proof, namely, the It is to
comparison of a relatively vague idea with a relatively precise one. The notion of " those terms that can be reached from o " by successive steps from next to next is vague, though it seems " the as if it conveyed a definite meaning ; on the other hand, " is posterity of o precise and explicit just where the other idea is
hazy.
It
may
when we spoke
be taken as giving what we meant to mean terms that can be reached from o by
of the
successive steps.
We now
lay
down
the following definition
The " natural numbers " are
the -posterity of
:
o with
respect to the
Finitude
and Mathematical
" immediate relation predecessor " successor " ).
We
have thus arrived at a
Induction
" (which
is
two
As a
a
is
of
number and the one
result of this
namely, the one
of his primitive propositions
asserting that o
converse
definition of one of Peano's three
primitive ideas in terms of the other two. definition,
the
23
asserting mathematical
become unnecessary, since they result from the defini The one asserting that the successor of a natural number " is a natural number is every only needed in the weakened form natural number has a successor." We can, of course, easily define " o " and " successor " by means of the definition of number in general which we arrived at in Chapter II. The number o is the number of terms in a class induction tion.
which has no members,
By the general
class."
in the null-class i.e.
in the class
definition of
the set of
all
which
is
called the
number, the number
" null-
of
terms
classes similar to the null-class,
proved) the set consisting of the null-class all the class whose only member is the null-class. (This
(as is easily i.e.
alone, is
is
i.e.
not identical with the null-class
it
:
has one member, namely ?
the null-class, whereas the null-class itself has no members. class
which has one member
we
member,
as
classes.)
Thus we
o It
when we come
to the theory of have the following purely logical definition :
shall explain
whose only member is remains to define " successor." class
not a
A
never identical with that one
is the class
a be a is
is
the null-class.
Given any number n, let let x be a term which
which has n members, and
member
added on
will
definition
:
of a.
have
n-\-i
Then the members.
a with x Thus we have the following
class consisting of
The successor of the number of terms in of terms in the class consisting of term not belonging to the class.
a
Certain niceties are required to
but they need not concern us. 1 1
the class
a
is the
number
together with x, where x is any
make
It will
this definition perfect,
be remembered that
See Principia Mathematical, vol.
ii.
*
no,
we
Introduction
24
have already given
number
(in
to
Mathematical Philosophy
Chapter
of terms in a class,
II.)
a logical definition of the defined it as the set of all
namely, we
classes that are similar to the given class.
We
have thus reduced Peano's three primitive ideas to ideas we have given definitions of them which make them
of logic
definite,
:
no longer capable of an infinity of different meanings, when they were only determinate to the extent of
as they were
obeying Peano's five axioms. We have removed them from the fundamental apparatus of terms that must be merely appre hended, and have thus increased the deductive articulation of
mathematics.
As regards the five primitive propositions, we have already succeeded in making two of them demonstrable by our definition " natural number." How stands it with the of remaining three ? very easy to prove that o
not the successor of any number, and that the successor of any number is a number. But there is a difficulty about the remaining primitive proposition, namely, " no two numbers have the same successor." The It is
is
difficulty
number of individuals in the for given two numbers m and n, neither of universe is finite which is the total number of individuals in the universe, it is easy to prove that we cannot have m-\-i=n-{-i unless we have mn. But let us suppose that the total number of individuals does not arise unless the total ;
in the universe were (say) 10
;
then there would be no class of
and the number
1 1 would be the null-class. So Thus we should have 11 = 12 therefore the successor of 10 would be the same as the successor of n, although 10 would not be the same as n. Thus we should have two different numbers with the same successor. This failure of the third axiom cannot arise, however, if the number of indi
II individuals,
would the number
12.
viduals in the world
is
;
not
finite.
We
shall return to this topic
at a later stage. 1
Assuming that the number of individuals in the universe is not finite, we have now succeeded not only in defining Peano's *
See Chapter
XIH,
and Mathematical
Finitude
three primitive ideas, but in seeing propositions, ing to logic. as
by means
how
Induction
25
to prove his five primitive
and propositions belong
of primitive ideas
It follows that all
pure mathematics, in so far deducible from the theory of the natural numbers, is only
it is
The extension
a prolongation of logic.
modern branches
of
of this result to those
mathematics which are not deducible from
the theory of the natural numbers offers no difficulty of principle, as we have shown elsewhere. 1
The process of mathematical induction, by means of which we defined the natural numbers, is capable of generalisation.
We
defined the natural
numbers
as the
" posterity
" of o with
respect to the relation of a number to its immediate successor. If we call this relation N, any number will have this relation
m
A
w+i.
to
property
"
simply
m has
whenever the property belongs to a m-fi, i.e. to the number to which And a number n will be said to belong to
N-hereditary,"
number m,
it
"hereditary with respect to N," or
is if,
also belongs to
the relation N.
"
"
m
with respect to the relation N if n has every N-hereditary property belonging to m. These definitions can all be applied to any other relation just as well as to N. Thus
the
if
R
posterity
is
any relation whatever, we can lay down the following
definitions
A
2 :
property
a term x,
A
of
is
called
and x has the
class
is
"
"
R-hereditary relation
R
R-hereditary when
when,
to y, then its
it
defining
if it
belongs to
belongs to y.
property
is
R-
hereditary.
A
term x
said to be an
"
R-ancestor
"
of the term y if y has every R-hereditary property that x has, provided x is a term which has the relation R to something or to which something has the relation R. (This is only to exclude trivial cases.) is
1 For geometry, in so far as it is not purely analytical, see Principles of Mathematics, part vi. ; for rational dynamics, ibid., part vii. 2 These definitions, and the generalised theory of induction, are due to Frege, and were published so long ago as 1879 in his Begriffsschrift. In spite of the great value of this work, I was, I believe, the first person who ever read it more than years after its
twenty
publication.
26
Introduction
The " R-posterity "
Mathematical Philosophy
to
of
x
is all
the terms of which x
an R-
is
ancestor.
We
have framed the above
definitions so that
a term
if
is
the
ancestor of anything it is its own ancestor and belongs to its own This is merely for convenience. posterity. the relation " parent," It will be observed that if we take for " " ancestor " and " will have the usual meanings, posterity
R
except that a person will be included among his own ancestors and posterity. It is, of course, obvious at once that " ancestor " must be capable of definition in terms of " parent," but until Frege developed his generalised theory of induction, no one could have defined " ancestor " precisely in terms of " parent." A brief consideration of this point will serve to show the importance of the theory.
problem
A
person confronted for the first time with the " would " ancestor " in terms of " parent
of defining
A
naturally say that there are a certain
B
is
is
an ancestor of
number
a child of A, each
is
Z
if,
between
of people, B, C,
.
.
.,
a parent of the next, until the
A of
and
Z,
whom
last,
who
a parent of Z. But this definition is not adequate unless we add that the number of intermediate terms is to be finite. Take,
is
for example, such a series as the following I,
Here we have and then a
we say
first
f,
89
J,
.
:
g>
>
2?
M
a series of negative fractions with no end, with no beginning. Shall
series of positive fractions
that, in this series,
J
is
an ancestor of J
?
It will
be
so according to the beginner's definition suggested above, but it will not be so according to any definition which will give the
kind of idea that we wish to define.
For
this purpose, it is
essential that the number of intermediaries should be finite. " finite " is to be defined But, as we saw, by means of mathe
matical induction, and
it is
simpler to define the ancestral relation
case of the generally at once than to define it first only for the cases. it to other then extend n to and relation of Here, n-f-i, as constantly elsewhere, generality
from the
first,
though
it
may
Finitude
and Mathematical
Induction
27
require more thought at the start, will be found in the long run to economise thought and increase logical power.
The use
mathematical induction in demonstrations was, something of a mystery. There seemed no reason
of
in the past,
able doubt that
knew why
it
was a valid method
it
was
Some
valid.
of induction, in the sense in *
Poincare
considered
by means
ance,
of
it
of proof, but
believed
it
to be really a case
which that word
is
to be a principle of the
which an
logic.
number of syllogisms could be We now know that all such views is
a definition,
There are some numbers to which
applied, and there are others to which it cannot be applied. as those to
used in
utmost import
infinite
condensed into one argument. are mistaken, and that mathematical induction not a principle.
no one quite
(as
we
shall see in
it
can be
Chapter VIII.)
We define the " natural numbers "
which proofs by mathematical induction can be as those that possess all inductive properties.
It applied, follows that such proofs can be applied to the natural numbers, not in virtue of any mysterious intuition or axiom or principle, " but as a purely verbal proposition. If " quadrupeds are i.e.
defined as animals having four legs, that have four legs are quadrupeds
it will ;
follow that animals
and the case
numbers
of
that obey mathematical induction is exactly similar. " " shall use the phrase inductive numbers to
We
mean
the
same set as we have hitherto spoken of as the " natural numbers." The phrase " inductive numbers " is preferable as affording a reminder that the definition of this set of numbers
is
obtained
from mathematical induction. Mathematical induction the essential characteristic
from the
The
infinite.
affords, more than anything else, by which the finite is distinguished
principle
of
mathematical induction
" what can be might be stated popularly in some such form as inferred from next to next can be inferred from first to last." This
is
first
and
true
when the number
last is
finite, 1
of intermediate steps
not otherwise.
between
Anyone who has ever
Science and Method, chap.
iv.
28
Introduction
to
Mathematical Philosophy
watched a goods train beginning to move
will
have noticed how
the impulse is communicated with a jerk from each truck to the next, until at last even the hindmost truck is in motion.
When
the train
truck moves.
an
a very long time before the last If the train were infinitely long, there would be is
very long,
it is
infinite succession of jerks,
when the whole
train
and the time would never come
would be
in motion.
Nevertheless,
if
there were a series of trucks no longer than the series of inductive
numbers (which,
as
we
shall see, is
an instance of the smallest
would begin to move sooner or later if the engine persevered, though there would always be other trucks further back which had not yet begun to move. This image will help to elucidate the argument from next to next, and its connection with finitude. When we come to infinite of infinites), every truck
numbers, where arguments from mathematical induction will be no longer valid, the properties of such numbers will help to
make of
clear,
by
contrast, the almost unconscious use that
mathematical induction where
finite
is
made
numbers are concerned.
CHAPTER IV THE DEFINITION OF ORDER
WE have now carried our analysis of the series of natural numbers we have obtained
to the point where
members
of this series, of the
of the relation of a
number
must now consider the in the order o,
I,
whole to its
logical definitions of the
members, and immediate successor. We class of its
numbers
serial character of the natural .
2, 3,
.
.
We
ordinarily think of the
num
an essential part of the work " order " " series " of analysing our data to seek a definition of or bers as in this order,
and
it is
in logical terms.
The notion
one which has enormous importance
of order is
Not only the integers, but also rational frac tions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The
in mathematics.
order of points on a line slightly
is
essential to
more complicated order
of lines
geometry
;
so
is
the
through a point in a
Dimensions, in geometry, plane, or of planes through a line. are a development of order. The conception of a limit, which a serial conception. There are parts of mathematics which do not depend upon the notion of order, but they are very few in comparison with the parts
underlies
in
which
all
higher mathematics,
this
notion
is
is
involved.
In seeking a definition of order, the first thing to realise is that no set of terms has just one order to the exclusion of others.
A
set of
terms has
times one order
is
all
so
the orders of which
much more
it is
familiar
capable.
Some
and natural to our
Introduction
30
to
Mathematical Philosophy
thoughts that we are inclined to regard it as the order of that set of terms ; but this is a mistake. The natural numbers " inductive " or the numbers, as we shall also call them occur to us of
an
most readily infinite
in order of
number
magnitude
;
but they are capable
We might, for the odd numbers and then all the
of other arrangements.
example, consider first even numbers ; or first
all I,
then
all
the even numbers, then
all
the odd multiples of 3, then all the multiples of 5 but not of 2 or 3, then all the multiples of 7 but not of 2 or 3 or 5, and so
on through the whole series of primes. When we say that we " " the numbers in these various orders, that is an arrange inaccurate expression what we really do is to turn our attention to certain relations between the natural numbers, which them selves generate such-and-such an arrangement. We can no more " arrange " the natural numbers than we can the starry heavens ; but just as we may notice among the fixed stars :
either their order of brightness or their distribution in the sky, so there are various relations among numbers which may be
observed, and which give rise to various different orders among numbers, all equally legitimate. And what is true of numbers equally true of points on a line or of the moments of time one order is more familiar, but others are equally valid. We
is
:
might, for example, take first, on a line, all the points that have integral co-ordinates, then all those that have non-integral rational co-ordinates, then
rational co-ordinates, tions
we
please.
all
those that have algebraic non-
and so on, through any
The
resulting
set of
complica
order will be one which the
points of the line certainly have, whether we choose to notice or not ; the only thing that is arbitrary about the various
it
orders of a set of terms
have always
all
is
our attention, for the terms themselves
the orders of which they are capable. result of this consideration is that
One important
we must
not look for the definition of order in the nature of the set of
terms to be ordered, since one set of terms has many orders. The order lies, not in the class of terms, but in a relation among
The Definition of Order
3
.
1
members of the class, in respect of which some appear as The fact that a class may have many earlier and some as later. the
orders
among
due to the fact that there can be
is
the
members
one single
of
class.
many relations holding What properties must
a relation have in order to give rise to an order ? The essential characteristics of a relation which
is
to give rise
may be discovered by considering that in respect of such a relation we must be able to say, of any two terms in " " and the the class which is to be ordered, that one precedes " be able to use other order that we to order
follows."
Now,
in
may
which we should naturally understand them, we require that the ordering relation should have three these words in the
properties
way
in
:
This is an y, y must not also precede x. obvious characteristic of the kind of relations that lead to series. (1) If
x
If
x precedes
is less
time than
than y,
y
y is not also less than not also earlier than x.
y,
is
If
x.
If
x
x
is
is earlier
in
to the left of
y is not to the left of x. On the other hand, relations which do not give rise to series often do not have this property. If
y,
x
a brother or sister of y, y is a brother or sister of x. same height as y, y is of the same height as x. If
is
of the
different height
these cases,
all
from
y,
when the
holds between y and x.
cannot happen.
A
y
is
But with
relation
from
of a different height
relation holds
having
x
between x and
If
x
is
of a
is
x.
In
y, it
also
such a thing
serial relations
this first property is called
asymmetrical. (2)
may left
If x precedes y and y precedes z, x must precede z. This be illustrated by the same instances as before less, earlier, :
of.
But
as instances of relations
property only two
x
which do not have
this
of our previous three instances will serve.
brother or sister of y, and y of z, x may not be brother The same z, since x and z may be the same person. applies to difference of height, but not to sameness of height, which has our second property but not our first. The relation If
is
or sister of
"
father," on the
other hand, has our
first
property but not
Introduction
32
our second.
A
Mathematical Philosophy
to
relation having our second property
is
called
transitive.
Given any two terms of the class which is to be ordered, must be one which precedes and the other which follows. For example, of any two integers, or fractions, or real numbers, one is smaller and the other greater but of any two complex (3)
there
;
numbers must be
this is earlier
Of any two moments in time, one than the other ; but of events, which may be not true.
simultaneous, this cannot be said. Of two points on a line, one must be to the left of the other. A relation having this third property
called connected.
is
When
a relation possesses these three properties, it is of the sort to give rise to an order among the terms between which it holds
;
and wherever an order
exists,
some
three properties can be found generating
Before
illustrating
this
thesis,
we
relation having these
it.
introduce
will
a
few
definitions. (1)
A
relation
said to be an aliorelative, 1 or to be contained
is
in or imply diversity,
if
no term has
"
for example,
"
this
relation
different in size,"
to
itself.
"
brother," Thus, greater," " father " are aliorelatives " but " equal," " born ; husband," " dear friend " are not. of the same parents," (2)
The square
of a relation
when
is
that relation which holds between
an intermediate term y such that the given relation holds between x and y and between " " " y and z. Thus paternal grandfather is the square of father," " " " is the square of greater by I," and so on. greater by 2 of all those terms that consists a relation The of domain (3)
two terms x and
z
there
is
have the relation to something or other, and the converse domain consists of all those terms to which something or other has the relation.
These words have been already defined,
recalled here for the sake of the following definition
The field of a domain together. (4)
1
relation consists of its
This term
is
due to
but
are
:
domain and converse
C. S. Peirce.
The Definition of Order
One
(5) it
relation
is
33
said to contain or be implied by another
if
holds whenever the other holds.
be seen that an asymmetrical relation
It will
is
the same thing
whose square is an aliorelative. It often happens that a relation is an aliorelative without being asymmetrical, though an asymmetrical relation is always an aliorelative. For as a relation
" but is symmetrical, is an aliorelative, spouse the spouse of y, y is the spouse of x. But among
" example, since
x
if
transitive
is
relations,
aliorelatives
all
are asymmetrical
as
well
as vice versa.
From is
the definitions
one which "
tains
its
is
it will
implied by
square.
its
be seen that a
Thus " ancestor "
an ancestor's ancestor
is
an ancestor
;
transitive relation
we
square, or, as
one which contains
in diversity
or,
;
A transitive
is
transitive,
A
asymmetry
is
is
when
a relation
equivalent to being an aliorelative.
is
or between the second
that both
is
because,
connected when, given any the relation holds between the
relation
of its field,
square and is contained the same thing, one whose
its
what comes to it and diversity
square implies both
con
transitive, because " but " father is not
is
transitive, because a father's father is not a father.
aliorelative is
"
also say,
and the
may happen, though
first
two
different terms
first
and the second
(not excluding the possibility
both cannot happen
if
the relation
asymmetrical). " It will be seen that the relation ancestor," for example, an aliorelative and transitive, but not connected ; it is because
it is
not connected that
it
does not suffice to arrange the
human
race in a series.
The
"
less than or equal to," among numbers, is and connected, but not asymmetrical or an aliorelative. The relation " greater or less " among numbers is an alio relative and is connected, but is not transitive, for if x is greater
relation
transitive
or less than y, and y is greater or less than that x and z are the same number.
Thus the three properties
of
being
(i)
z, it
an
may happen
aliorelative,
3
(2)
34
Introduction
transitive,
and
a relation
may
Mathematical Philosophy
to
connected, are mutually independent, since have any two without having the third. (3)
We now lay down the following definition A relation is serial when it is an aliorelative, :
connected
or,
;
transitive,
is
when
equivalent,
it
the
same thing
and
asymmetrical,
as a serial relation.
might have been thought that a
should be the field
series
But
of a serial relation, not the serial relation itself.
be an
transitive,
is
and connected.
A series is It
what
error.
would
this
For example,
I, 2, 3
;
2
i, 3,
;
2, 3, I
are six different series which
;
2, i, 3
all
;
3, I,
2
;
have the same
3, 2, I
If
field.
the
were the series, there could only be one series with a given field. What distinguishes the above six series is simply the field
Given the ordering the field and order Thus the are both determinate. relation,
different ordering relations in the six cases.
the ordering relation cannot be so taken.
Given any
shall write
We
say P,
x " precedes " y
which P must have (1)
be taken to be the
serial relation,
of this relation,
which we
may
"
xPy
"
we if
shall
but the
The
in order to be serial are
i.e.
field
say that, in respect
x has the relation
for short.
must never have xPx,
series,
P
to y,
three characteristics
:
no term must precede
itself.
(2)
P 2 must imply precede
(3)
If
P,
i.e. if
x precedes y and y precedes
z,
x must
z.
x and y are two different terms in the field of P, we shall have xPy or yPx, i.e. one of the two must precede the other.
The reader can
easily convince himself that,
where these three
properties are found in an ordering relation, the characteristics are we expect of series will also be found, and vice versa.
We
therefore justified in taking the above as a definition of order
The Definition of Order
And
or series.
35
be observed that the definition
it will
is
effected
in purely logical terms.
Although a transitive asymmetrical connected relation always exists wherever there is a series, it is not always the relation which would most naturally be regarded as generating the series.
The natural-number relation we assumed
series
may
The
serve as an illustration.
numbers was
in considering the natural
the relation of immediate succession,
i.e.
the relation between
consecutive integers. This relation is asymmetrical, but not We can, however, derive from it, transitive or connected. " " mathematical ancestral the method of induction, the by relation
which we considered
relation will
This than or equal to " among
in the preceding chapter.
be the same as "
less
For purposes of generating the series of inductive integers. " less than," excluding natural numbers, we want the relation " This is the relation oimton when is an ancestor to." equal
m
of
n but not
comes to the same thing) an ancestor of n in the sense in which
identical with n, or (what
when the successor of m is a number is its own ancestor. the following definition : An inductive number
m is
That
is
to say,
we
shall lay
said to be less than another
down
number
n when n possesses every hereditary property possessed by the successor of m. It is easy to see,
"
and not
difficult to
prove, that the relation
asymmetrical, transitive, and con and has the inductive numbers for its field. Thus by nected, means of this relation the inductive numbers acquire an order less
than," so defined,
in the sense in is
is
which we defined the term " order," and
this order
the so-called " natural " order, or order of magnitude.
The generation
by means
of relations more or less The series of the common. very example, is generated by relations of each
of series
resembling that of n to n-j-i
Kings of England, for This
to his successor. applicable,
is
is
probably the easiest way, where In of a series.
of conceiving the generation
method we pass on from each term
it is
this
to the next, as long as there
Introduction
36
to
Mathematical Philosophy
is a next, or back to the one before, as long as there is one before. This method always requires the generalised form of mathe " " matical induction in order to enable us to define and earlier " later " in a series so " On the of generated. proper analogy " fractions," let us give the name proper posterity of x with respect " to R to the class of those terms that belong to the R-posterity of some term to which x has the relation R, in the sense which we gave before to " posterity," which includes a term in its own
Reverting to the fundamental definitions, we find that " proper posterity may be defined as follows " The proper posterity " of x with respect to R consists of
posterity.
the
"
:
terms that possess every R-hereditary property possessed by every term to which x has the relation R.
all
It is to
be observed that this definition has to be so framed
be applicable not only when there is only one term to which x has the relation R, but also in cases (as e.g. that of father and as to
child)
where there
We
R.
A
may be many
define further
term x
"
terms to which x has the relation
:
"
of y with respect to to the of x with belongs proper posterity respect to R. is
a
proper ancestor
"
We shall speak for short of " when
R-posterity these terms seem more convenient.
Reverting
now
to the generation of series
and connected.
if
y
and " R-ancestors "
by the
relation
between consecutive terms, we see that, if this method " " must be an possible, the relation proper R-ancestor tive, transitive,
R
is
R
to be
aliorela-
Under what circumstances
will
no matter what sort always be transitive " " and " proper R-ancestor " R-ancestor of relation R may be, But it is only under certain circum are always both transitive. stances that it will be an aliorelative or connected. Consider, this occur
?
It will
:
for example, the relation to one's left-hand neighbour at a
round
dinner-table at which there are twelve people. If we call this relation R, the proper R-posterity of a person consists of all who
can be reached by going round the table from right to
left.
This
includes everybody at the table, including the person himself, since
The Definition of Order
37
twelve steps bring us back to our starting-point. Thus in such " " is connected, a case, though the relation proper R-ancestor and though R itself is an aliorelative, we do not get a series " " is not an aliorelative. It is for because proper R-ancestor this reason that we cannot say that one person comes before " another with respect to the relation " right of or to its ancestral derivative.
The above was an instance
in
which the ancestral relation was
connected but not contained in diversity. An instance where it is contained in diversity but not connected is derived from the sense of the word " ancestor." If x is a proper ancestor ordinary of y, x and y cannot be the same person ; but it is not true that
any two persons one must be an ancestor of the other. The question of the circumstances under which series can be generated by ancestral relations derived from relations of consecutiveness is often important. Some of the most important cases are the following Let R be a many-one relation, and let us confine our attention to the posterity of some term x. When " " so confined, the relation proper R-ancestor must be connected of
:
;
therefore
all
that remains to ensure
be contained in diversity.
This
is
its
being serial
is
that
it
shall
a generalisation of the instance
Another generalisation consists in taking to be a one-one relation, and including the ancestry of x as
of the dinner-table.
R
well as the posterity. Here again, the one condition required " to secure the generation of a series is that the relation proper
R-ancestor
"
shall
The generation
be contained in diversity. of order
by means
own
ness, though important in its
method which uses a
of relations of consecutive-
sphere,
is less
general than the
transitive relation to define the order.
often happens in a series that there are an infinite
number of
It
inter
mediate terms between any two that near together these of
magnitude.
may be.
may be selected, however for Take, instance, fractions in order
Between any two
example, the arithmetic
no such thing as a pair
mean
fractions there are others
of the two.
for
Consequently there
of consecutive fractions.
If
is
we depended
Introduction to Mathematical Philosophy
38
upon consecutiveness
for defining order,
we should not be
able
magnitude among fractions. But in fact greater and less among fractions do not demand
to define the order of
the relations of
generation from relations of consecutiveness, and the relations of greater and less among fractions have the three characteristics
which we need
In
for defining serial relations.
the order must be defined
by means
all
such cases
of a transitive relation, since
only such a relation is able to leap over an infinite number of intermediate terms. The method of consecutiveness, like that of counting for discovering the
priate to the finite
;
it
may
number
of a collection, is
even be extended to certain
appro infinite
namely, those in which, though the total number of terms is infinite, the number of terms between any two is always finite ; series,
it must not be regarded as general. Not only so, but care must be taken to eradicate from the imagination all habits of
but
thought resulting from supposing it general. series in which there are no consecutive terms
and puzzling.
And
such
series are of vital
understanding of continuity, space, time,
There are
many ways
in
which
series
If this is
will
not done,
remain
difficult
importance for the
and motion.
may
be generated, but
depend upon the finding or construction of an asymmetrical Some of these ways have con
all
transitive connected relation.
siderable importance. tion of series
"
call
may
by means
We may
take as illustrative the genera
of a three-term relation
This method
which we
may
very useful in geometry, and serve as an introduction to relations having more than two
terms
between."
;
it
is
is
best introduced in connection with elementary
geometry.
Given any three points on a straight line in ordinary space, must be one of them which is between the other two. This
there will
not be the case with the points on a
circle or
any other closed
curve, because, given any three points on a circle, we can travel from any one to any other without passing through the third. " is characteristic of In fact, the notion " between open series or series in the strict sense
as opposed to
what may be
called
The Definition of Order
39
"
"
where, as with people at the dinner-table, a This sufficient journey brings us back to our starting-point. " between " as the fundamental notion chosen be notion of may of ordinary geometry ; but for the present we will only consider its application to a single straight line and to the ordering of the series,
cyclic
1 Taking any two points #, b, the line points on a straight line. : of three consists (ab) parts (besides a and b themselves)
(1)
Points between a and
(2)
Points x such that a
is
between x and
b.
(3)
Points y such that b
is
between y and
a.
Thus the
line
b.
can be defined in terms of the relation
(ab)
" between." " between " In order that this relation of the line in
an order from
left to right,
namely, the following If anything is between a and
tions,
may
arrange the points
we need
certain
assump
:
(1)
a and b are not identical.
b,
Anything between a and b is also between b and a. Anything between a and b is not identical with a
(2)
(3)
consequently, with If
(4)
x
If
x
is
(nor,
in virtue of (2)).
between a and
is
between a and (5)
b,
b,
anything between a and x
b,
and
is
also
b.
between a and
b is
between x and
y,
then b
between a and y. (6) If x and y are between a and b, then either x and y are identical, or x is between a and y, or x is between y and b.
is
If b is
(7) a:
b
and y are and #.
between a and x and identical, or x
is
also
between a and
between
and
y,
y, or y
then either is
between
These seven properties are obviously verified in the case of points on a straight line in ordinary space. Any three-term relation
which
verifies
them
gives rise to series, as
following definitions. 1
394
Cf (
.
For the sake
Rivista di Matematica, iv. pp. 55
375).
may
be seen from the
of definiteness, let us ft.
;
assume
Principles of Mathematics, p.
Introduction
4-O
that a
to the left of
is
Then the
b.
those between which and of a
;
(2)
a
itself
;
(3)
Mathematical Philosophy
to
lies
(ab),
we
When x and x and a
(2)
When
(3)
When
x x
y are both to the
to the left of a,
is a,
;
(5)
and y
left of a,
is
between
and y
is
and y
is
a or b or between a and
;
between a and b or
is
b or
is
to the
;
When x and # and b
(7)
(4) b itself
;
will call to the right
;
is
right of b
(6)
we
:
b or to the right of b
(5)
these
define generally that of two points x, y, on " to the left of " shall say that x is y in any
of the following cases
(4)
b
We may now
the line
(1)
lies
those between a and b
those between which and a of b.
a
b,
points of the line (ab) are (i) these we will call to the left
y are both between a and
,
and y
is
between
;
When x is between and b, and y is 3 or to the right of b When x is and y is to the right of b When x and y are both to the right of b and x is between
;
;
b
and
y.
be found that, from the seven properties which we have " between," it can be deduced that the assigned to the relation " to the relation left of," as above defined, is a serial relation as It will
we
defined that term.
important to notice that nothing in the definitions or the argument depends upon our meaning " " by between the actual relation of that name which occurs in empirical space
:
It is
any three-term
relation having the
above seven
purely formal properties will serve the purpose of the argument equally well. Cyclic order, such as that of the points on a circle, cannot be " between." generated by means of three-term relations of
We
need a relation of four terms, which may be called " separation of couples." The point may be illustrated by considering a journey round the world. One may go from England to New Zealand by way of Suez or by way of San Francisco ; we cannot
The Definition of Order
41
" between " say definitely that either of these two places is England and New Zealand. But if a man chooses that route
round the world, whichever way round he goes, his times in England and New Zealand are separated from each other by his to go
times in Suez and San Francisco, and conversely. Generalising, if we take any four points on a circle, we can separate them into
two couples, say a and b and x and y, such that, in order to get b one must pass through either x or y, and in order to from to y one must pass through either a or b. Under these x get from a to
we say that the couple (a, b) are " separated " by the couple (x, y). Out of this relation a cyclic order can be gen erated, in a way resembling that in which we generated an open
circumstances
order from " between," but somewhat more complicated. 1 The purpose of the latter half of this chapter has been to suggest the subject which one
When
"
generation of serial relations." such relations have been defined, the generation of them
may
call
from other relations possessing only some of the properties required for series becomes very important, especially in the philosophy of geometry and physics. But we cannot, within the limits of the present volume, do more than make the reader aware that such a subject exists. 1
Cf. Principles of
Mathematics, p. 205
(
194),
and references there given.
CHAPTER V KINDS OF RELATIONS
A GREAT part relations,
of the philosophy of mathematics is concerned with different kinds of relations have different
and many
kinds of uses.
It often
to all relations
is
sorts
happens that a property which belongs
only important as regards relations of certain
in these cases the reader will not see the bearing of the
;
proposition asserting such a property unless he has in mind the For reasons of this which it is useful.
sorts of relations for
description, as well as it
well
is
from the
intrinsic interest of the subject,
have in our minds a rough
to
list
of
the more
mathematically serviceable varieties of relations. dealt in the preceding chapter with a supremely important Each of the three properties which class, namely, serial relations.
We
we combined in
defining series
and connexity has something on each is
namely, asymmetry, transitiveness,
We will begin by saying
own importance.
of these three.
the property of being incompatible with the a characteristic of the very greatest interest and
Asymmetry, converse,
its
i.e.
importance. In order to develop its functions, we will consider The relation husband is asymmetrical, and various examples. so is the relation wife ; i.e. if a is husband of b, b cannot be husband
and
On
the other hand, the spouse of b, then b is spouse of a. Suppose now we are given the relation spouse, and we wish to derive the relation husband. Husband is the same as of a,
relation
similarly in the case of wife.
"
"
spouse
is
symmetrical
male spouse or spouse of a female
;
4*
:
if
a
is
thus the relation husband can
Kinds of Relations
43
be derived from spouse either by limiting the domain to males We see from this instance or by limiting the converse to females.
when a symmetrical relation is given, it is sometimes possible, without the help of any further relation, to separate it into two that,
asymmetrical relations. rare
and exceptional
:
exclusive classes, say
But the
cases where this
is
possible are
they are cases where there are two mutually a and j3, such that whenever the relation
holds between two terms, one of the terms is a member of a and the other is a member of )3 as, in the case of spouse, one term of the relation belongs to the class of males and one to the class of females.
In such a case, the relation with
its
domain confined
be asymmetrical, and so will the relation with its domain But such cases are not of the sort that occur confined to j3.
to
a
will
more than two terms ; for in a series, all terms, except the first and last (if these exist), belong both to the domain and to the converse domain of the generating relation, so that a relation like husband, where the domain and
when we
are dealing with series of
converse domain do not overlap,
is
excluded.
The question how to construct relations having some useful property by means of operations upon relations which only have rudiments of the property
is
one of considerable importance.
Transitiveness and connexity are easily constructed in where the originally given relation does not possess
many cases them
:
for
example, if R is any relation whatever, the ancestral relation derived from R by generalised induction is transitive ; and if R is
a
many-one
relation, the ancestral relation will
be connected
confined to the posterity of a given term. But asymmetry is a much more difficult property to secure by construction. The
if
method by which we derived husband from spouse is, as we have most important cases, such as greater, to the where domain and converse domain overlap. before, right of,
seen, not available in the
In
we can
symmetrical relation by adding together the given relation and its converse, but we cannot pass back from this symmetrical relation to the original all
these cases,
of course obtain a
asymmetrical relation except by the help of some asymmetrical
Introauction
44 relation.
to
Mathematical Philosophy
Take, for example, the relation greater : the relation is i.e. unequal symmetrical, but there is nothing
greater or less
show that it is the sum of two asymmetrical Take such a relation as " differing in shape." This is not the sum of an asymmetrical relation and its converse, since but there is nothing to show shapes do not form a single series
in this relation to relations.
;
from "
"
if we did not already that magnitudes have relations of greater and less. This illustrates the fundamental character of asymmetry as a property
that
it differs
differing in
magnitude
know
of relations.
From
the point of view of the classification of relations, being asymmetrical is a much more important characteristic than
implying diversity. Asymmetrical relations imply diversity, but the converse is not the case. " Unequal," for example, implies diversity, but is symmetrical. Broadly speaking, we
may
say that,
if
we wished
as far as possible to dispense with
and replace them by such as ascribed to we could succeed in this so long as we predicates subjects, confined ourselves to symmetrical relations : those that do not relational propositions
imply diversity, ing a
common
they are transitive, may be regarded as assert predicate, while those that do imply diversity if
be regarded as asserting incompatible predicates.
may
example,
consider
the
relation
of
similarity
between
For
classes,
by means of which we defined numbers. This relation is sym It would metrical and transitive and does not imply diversity. be possible, though less simple than the procedure we adopted, to regard the number of a collection as a predicate of the collec then two similar classes will be two that have the same tion numerical predicate, while two that are not similar will be two that have different numerical predicates. Such a method of :
replacing relations by predicates is formally possible (though often very inconvenient) so long as the relations concerned are symmetrical ; but it is formally impossible when the relations are asymmetrical, because both sameness and difference of predi are symmetrical. Asymmetrical relations are, we may
cates
Kinds of Relations
45
most characteristically relational of relations, and the most important to the philosopher who wishes to study the
say, the
ultimate logical nature of relations. Another class of relations that is of the greatest use class of
one-many
relations,
i.e.
relations
is
the
which at most one
term can have to a given term. Such are father, mother, husband (except in Tibet), square of, sine of, and so on. But It is possible, parent, square root, and so on, are not one-many. relations to all relations by means formally, replace by one-many
Take
of a device.
(say) the relation less
among
the inductive
Given any number n greater than I, there will not be only one number having the relation less to n, but we can form the whole class of numbers that are less than n. This numbers.
is
one
class,
and
its
relation to
n
is
not shared by any other
class.
We may call the class of numbers that are less than n the " proper "
which we spoke of ancestry and mathematical induction. Then connection with posterity " " is a one-many relation (one-many will always proper ancestry be used so as to include one-one), since each number determines ancestry
of n, in the sense in
in
a single class of numbers as constituting its proper ancestry. relation less than can be replaced by being a member of the proper ancestry of. In this way a one-many relation in which
Thus the
a class, together with membership of this class, can always formally replace a relation which is not one-many. Peano, who for some reason always instinctively conceives of a relation the one
is
one-many, deals in this way with those that are naturally so. Reduction to one-many relations by this method, however, though possible as a matter of form, does not represent as
not
a technical simplification, and there is every reason to think that it does not represent a philosophical analysis, if only because classes must be regarded as " logical fictions." We shall there fore continue to regard
one-many
relations as a special kind of
relations.
One-many
relations are involved in all phrases of the
" the so-and-so of such-and-such."
form
" The King of England,"
Introduction
46
to
Mathematical Philosophy
" the father of " the wife of Socrates," John Stuart Mill," and so on,
some person by means of a one-many relation A person cannot have more than one father, " " the father of John Stuart Mill described some one describe
all
to a given term.
therefore
person, even if we did not know whom. There is much to say on the subject of descriptions, but for the present it is relations that we are concerned with, and descriptions are only
relevant as exemplifying the uses of one-many relations. It should be observed that all mathematical functions result from relations
one-many
:
the logarithm of
are, like the father of x, terms described
x,
the cosine of
by means
x, etc.,
of a
one-many The to a given term (x).
relation
(logarithm, cosine, etc.) notion of function need not be confined to numbers, or to the uses to which mathematicians have accustomed us ; it can be " " extended to all cases of one-many relations, and the father of x is
is
just as legitimately a function of which x is the argument as " the logarithm of x." Functions in this sense are descriptive
As we shall see later, there are functions of a still more general and more fundamental sort, namely, prepositional
functions.
functions
;
but for the present we shall confine our attention " the term having the relation R
to descriptive functions, i.e. " the to x," or, for short,
R
of
x"
where
R
is
any one-many
relation. It will
be observed that
if
"
the
R of x " is to describe a definite
term, x must be a term to which something has the relation R, and there must not be more than one term having the relation " to x, since the," correctly used, must imply uniqueness. " " Thus we may speak of the father of x if x is any human being " the father except Adam and Eve ; but we cannot speak of " if x is a table or a chair or anything else that does not of x " exists " when have a father. We shall say that the R of x to x. there is just one term, and no more, having the relation
R
R
Thus
if
R
is
a one-many relation, the
R
of
x
exists
whenever
x belongs to the converse domain of R, and not otherwise. " the R of x " as a function in the mathematical Regarding
Kinds of Relations
we say that x
sense,
is
the
"
argument
"
47
of the function,
and
if
R
R
of x, to x, i.e. if y is the y is the term which has the relation " value " of the function for the If x. then y is the argument to is a the of range possible arguments one-many relation,
R
the converse domain of R, and the range of values is the domain. Thus the range of possible arguments to the " function " the father of x is all who have fathers, i.e. the con the function
is
domain
verse
of the relation father, while the range of possible
values for the function
Many
of the
is all
fathers,
i.e.
the domain of the relation.
most important notions in the
logic of relations
converse, domain, con example Other examples will occur as we proceed.
are descriptive functions, for verse domain, field.
Among one-many
:
relations, one-one relations are a specially
We
have already had occasion to speak of important one-one relations in connection with the definition of number, but it is necessary to be familiar with them, and not merely class.
to
know
their formal definition.
Their formal definition
be derived from that of one-many relations defined as
relations
one-many
one-many relations, i.e. as and many-one. One-many
which are
:
they
may may be
also the converses of
which are both one-many relations may be defined as relations relations
such that, if x has the relation in question to y, there is no other term x' which also has the relation to y. Or, again, they may be defined as follows
:
Given two terms x and
x',
the terms to
which x has the given relation and those to which x' has it have no member in common. Or, again, they may be defined as such that the relative product of one of them and " " converse implies identity, where the relative product and S is that relation which holds between of two relations relations
its
R
x and
2
when
there
is
an intermediate term
y,
such that x has
R to y and y has the relation S to 2. Thus, for if R is the relation of father to son, the relative product
the relation
example, of
R
and
its
x and a man of y
and y
is
converse will be the relation which holds between 2
when
there
the son of
2.
is
a person It is
y,
such that x
is
the father
obvious that x and z must be
Introduction
48
the same person. of parent
and
to
Mathematical Philosophy
on the other hand, we take the relation is not one-many, we can no longer
If,
which
child,
argue that, if x is a parent of y and y is a child of z, x and z must be the same person, because one may be the father of y and the other the mother.
This illustrates that
it
characteristic of
is
when
the relative product of a relation and one-many In the case of one-one relations its converse implies identity. this happens, and also the relative product of the converse and relations
the relation implies identity. Given a relation R, it is convenient, to y, to think of y as being reached from x has the relation
R
if
" " R-vector." In the same case x will x by an " R-step or an " backward be reached from y by a R-step." Thus we may state the characteristic of one-many relations with which we
have been dealing by saying that an R-step followed by a back ward R-step must bring us back to our starting-point. With other relations, this
is
by no means the case
;
for example,
if
R is its
the relation of child to parent, the relative product of R and " self or brother or sister,'* and if converse is the relation
R
is the relation of grandchild to grandparent, the relative product of and its converse is " self or brother or sister or first cousin."
R
be observed that the relative product of two relations not in general commutative, i.e. the relative product of R and S is not in general the same relation as the relative product
It will is
E.g. the relative product of parent and brother is the but relative product of brother and parent is parent. uncle, One-one relations give a correlation of two classes, term for of S
and R.
term, so that each term in either class has other. classes
its
correlate in the
Such correlations are simplest to grasp when the two in common, like the class of husbands
have no members
and the
class of
wives
;
for in that case
we know
at once
whether
a term is to be considered as one from which the correlating It is convenient relation R goes, or as one to which it goes.
word referent for the term from which the relation term relatum for the term to which it goes. Thus the and goes, if x and y are husband and wife, then, with respect to the relation to use the
Kinas of Relations "
husband," x "
relation
is
referent is
wife," y
49
and y relatum, but with respect to the and x relatum. We say that a
referent
" senses " thus the converse have opposite ; " sense of a relation that goes from x to y is the opposite of that of the corresponding relation from y to x. The fact that a " " is fundamental, and is part of the reason sense relation has a
and
relation
its
"
order can be generated by suitable relations. It will be observed that the class of all possible referents to a given relation
why
domain, and the domain.
is its
But
it
domain the
first
class of all possible relata is its
converse
very often happens that the domain and converse a one-one relation overlap. Take, for example, ten integers (excluding o), and add I to each ; thus
of
instead of the
ten integers
first
2, 3, 4> 5
6>
we now have
the integers
9 I0 >
7> 8 >
These are the same as those we had before, except that I has been cut off at the beginning and II has been joined on at the ten integers they are correlated with the previous ten by the relation of n to n-{-i, which is a one-one relation. Or, again, instead of adding I to each of our original
There are
end.
ten integers,
still
:
we could have doubled each
of
them, thus obtaining
the integers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Here we
still
have
2, 4, 6, 8, 10.
The
five of
our previous set of integers, namely,
correlating relation in this case
is
the relation
number to its double, which is again a one-one relation. Or we might have replaced each number by its square, thus of a
obtaining the set i,
On I,
4, 9, 16, 25, 36, 49, 64, 81, 100.
only three of our original set are left, namely, Such 4, 9. processes of correlation may be varied endlessly. The most interesting case of the above kind is the case where this occasion
our one-one relation has a converse domain which
is
part, but
4
Introduction
50
Mathematical Philosophy
to
not the whole, of the domain. If, instead of confining the domain to the first ten integers, we had considered the whole of the inductive numbers, the above instances would have illustrated this case. We may place the numbers concerned in two rows, the correlate directly under the number whose correlate putting it is.
Thus when the
have the two rows
correlator
2,
3>4>
the correlator
we have
is
the two rows
6
5>
"+
>
the correlator
all
-
-
the relation of a
number
to its double,
5,
is
... n ... 2w ... .
the relation of a
.
.
number
to its square,
:
4,
5,
...
i, 4, 9, 1 6,
25,
..... n2 ...
i, 2, 3,
In
1
:
1, 2, 3, 4,
the rows are
we
... n ...
2, 4, 6, 8, 10,
When
the relation of n to n-{-i y
:
1, 2, 3, 4, 5,
When
is
these cases,
and only some
all
in the
inductive numbers occur in the top row,
bottom row.
" Cases of this sort, where the converse domain is a proper " of the domain (i.e. a part not the whole), will occupy us part again when we come to deal with infinity. For the present, we wish only to note that they exist and demand consideration.
Another
class
the class called
domain are
"
which are often important is permutations," where the domain and converse of
correlations
identical.
arrangements
Consider, for example, the six possible
of three letters
:
a,
b,
c
a,
c,
b
b,
c,
a
b,
a,
c
c,
a,
b
c,
b,
a
Kinds of Relations
51
Each of these can be obtained from any one of the others by means of a correlation. Take, for example, the first and last, Here a is correlated with c, b with itself, (a, b, c) and (c, b, a). and c with a. It is obvious that the combination of two permu tations class
is
again a permutation,
form what
is
called a
i.e.
the permutations of a given
" group."
These various kinds of correlations have importance in various connections, some for one purpose, some for another. The general notion of one-one correlations has boundless importance we have partly seen already,
in the philosophy of mathematics, as
but shall see will
much more
fully as
occupy us in our next chapter.
we
proceed.
One
of its uses
CHAPTER
VI
SIMILARITY OF RELATIONS
WE
saw
in Chapter II. that
when they
of terms
whose domain
relation
domain
are
similar," is
have the same number
classes
the one
when
i.e.
class
" one-one " between the two correlation
there
a one-one
is
and whose converse
we say
In such a case
the other.
is
two
"
that there
is
a
classes.
In the present chapter we have to define a relation between relations,
which
will
play the same part for them that similarity " We will call this
of classes plays for classes.
of relations," or different
relation
" likeness " when
it
word from that which we use
likeness to be defined
similarity
seems desirable to use a for classes.
How
is
?
We shall employ still the notion of correlation we shall assume that the domain of the one relation can be correlated :
with the domain of the other, and the converse domain with the converse domain ; but that is not enough for the sort of resem blance which we desire to have between our two relations.
What we
desire is that, whenever either relation holds between two terms, the other relation shall hold between the correlates The easiest example of the sort of thing of these two terms. is a map. When one place is north of another, the the to the one is above the place on map corresponding place on the map corresponding to the other when one place is west
we
desire
;
on the map corresponding to the one is the place on the map corresponding to the other ;
of another, the place
to the left of
and so on.
The
structure of the 52
map
corresponds with that of
Similarity of Relations
the country of which it " "
map have
We
space-relations in the
the space-relations in the country this kind of connection between relations that
likeness
It is
mapped.
we wish
The
a map.
is
53
to
to define.
may,
profitably introduce a certain will confine ourselves, in defining likeness, to
in the first place,
We
restriction.
such relations as have
"
fields," i.e. to such as permit of the formation of a single class out of the domain and the converse domain. This is not always the case. Take, for example, " the relation domain," i.e. the relation which the domain of a
This relation has
relation has to the relation.
domain, since every class it
has
all
is
the domain of
all
some
classes for its
relation
;
and
relations for its converse domain, since every relation
has a domain.
But
classes
gether to form a
new
"
We
types." doctrine of types, but
logical
and
relations cannot be
added to
single class, because they are of different
do not need to enter upon the it is
know when we
well to
difficult
are abstaining
it. We may say, without entering upon " " the the grounds for field assertion, that a relation only has a
from entering upon
what we call " homogeneous," i.e. when its domain and converse domain are of the same logical type and as a " what we mean a indication of rough-and-ready by type,"
when
it is
;
we may say between
that individuals, classes of individuals, relations
notion of likeness are not
is
between
relations
individuals,
classes,
relations
of
Now
and so on, are
classes to individuals,
different types. the not very useful as applied to relations that
homogeneous
we
;
shall, therefore, in defining likeness,
" " field of one of the simplify our problem by speaking of the relations concerned. This somewhat limits the generality of our definition, but the limitation tance.
And having been
We may having
define
two
" likeness,"
stated,
it
relations
when
there
is
not of any practical impor
need no longer be remembered.
P and Q is
as
"
similar," or as
a one-one relation S
domain is the field of P and whose converse domain of Q. and which is such that, if one term has the
is
whose
the field
relation
P
Introduction
54
Mathematical Philosophy
to
Q
to another, the correlate of the one has the relation
and
correlate of the other,
figure will
y >
.
w, such that x has the rela
tion S to
S to
Q
>
and
and
y,
as z
and w,
P
Q
w
Q if
to
zv,
z,
and
y has the relation z has the relation
If this
20.
happens with
.
every pair of
terms such as x
the converse happens with every pair of terms such clear that for every instance in which the relation
it is
holds there
and
holds,
this
terms having the relation P. Then there are to be two terms
.
z,
z
to the
make
Let x and v be two
clearer.
P
x,
A
vice versa.
a corresponding instance in which the relation and this is what we desire to secure by ;
is
vice versa
We
our definition.
can eliminate some redundancies in the
above sketch of a definition, by observing that, when the above conditions are realised, the relation P is the same as the relative product of S and Q and the converse of S, i.e. the P-step from x to y may be replaced by the succession of the S-step from
x to
w
z,
the Q-step from z to w, and the backward S-step from
to y. Thus we may set up the following definitions " correlator " or an " ordinal relation S is said to be a :
A
correlator
"
Q
field of
two
of
relations
for its converse
relative product of S
Two
relations
" likeness,"
there
These definitions
if
domain, and
S is
is
one-one, has the
such that
Q and the converse of S. Q are said to be " similar,"
P
is
the
and
P and
when
P and Q
will
is
at least one correlator of
or to have
P and
Q.
be found to yield what we above decided
to be necessary. It will
share
all
be found that, when two relations are similar, they properties which do not depend upon the actual terms
For instance,
in their fields.
the other nected, so
Again,
if
;
if
one
is
the other.
is
one
is
if
one implies diversity, so does is the other ; if one is con
transitive, so
Hence
one-many
one
so
is
the other.
or one-one, the other
is
one-many
if
is serial,
Similarity of Relations or one-one
;
and so on, through
all
55
the general properties of actual terms of the
Even statements involving the
relations.
field of a relation,
though they
may
not be true as they stand
when
applied to a similar relation, will always be capable of translation into statements that are analogous. are led
We
by such considerations to a problem which has, in mathematical philosophy, an importance by no means adequately recognised Our problem may be stated as follows
hitherto.
:
Given some statement in a language of which we know the grammar and the syntax, but not the vocabulary, what are the possible meanings of such a statement,
and what are the mean
unknown words that would make it true ? The reason that this question is important is that it represents, much more nearly than might be supposed, the state of our
ings of the
knowledge of nature.
We know
that
certain
scientific
pro which, in the most advanced sciences, are expressed in mathematical symbols are more or less true of the world, positions
we
are very much at sea as to the interpretation to be put the terms which occur in these propositions. We know upon much more (to use, for a moment, an old-fashioned pair of
but
about the form of
terms)
Accordingly, what we of nature
is
really
only that there
our terms which will
nature
make
than
about
know when we
the
matter.
enunciate a law
is probably some interpretation of the law approximately true. Thus to the question What are the
importance attaches meanings of a law expressed in terms of which we do not know the substantive meaning, but only the grammar and ? And this is the one syntax suggested above. question For the present we will ignore the general question, which
great
:
possible
will
occupy us again at a later stage; must first be further investigated.
the subject of likeness
itself
when two relations are similar, their same except when they depend upon the being composed of just the terms of which they are com
Owing
to the fact that,
properties are the fields
posed,
it
is
desirable to have a nomenclature which collects
Introduction
56
all
together
we
Just as
to
Mathematical Philosophy
the relations that are similar to a given relation. called the set of those classes that are similar to a
" number " of that class, so we may call the set given class the of all those relations that are similar to a given relation the " number " of that relation. But in order to avoid confusion with the numbers appropriate to classes, we will speak, in this case, of " relation-number." Thus we have the a following definitions " " The relation-number of a given relation is the class of all those relations that are similar to the given relation. " Relation-numbers " are the set of all those classes of relations :
that are relation-numbers of various relations
same
the
thing, a relation
number
of all those relations that are similar to
When
it
is
or,
what comes to
one member of the
class.
necessary to speak of the numbers of classes in makes it impossible to confuse them with relation-
way which numbers, we shall a
;
a class of relations consisting
is
them " cardinal numbers." Thus cardinal numbers are the numbers appropriate to classes. These include the ordinary integers of daily life, and also certain infinite call
numbers, of which we shall speak later. When we speak of " numbers " without qualification, we are to be understood as
meaning cardinal numbers. The definition be remembered, is as follows
it will
of a cardinal
number,
:
The "
cardinal
number "
of
a
given class
is
the set of
all
those classes that are similar to the given class.
The most obvious
application of relation-numbers
is
to series.
may be regarded as equally long when they have the same relation-number. Two finite series will have the same relation-number when their fields have the same cardinal
Two
series
number
of terms,
and only then
i.e.
a series of (say) 15 terms
have the same relation-number as any other series of fifteen terms, but will not have the same relation-number as a series
will
6 terms, nor, of course, the same relation-number as a relation which is not serial. Thus, in the quite special case
of
14 or
1
of finite series, there
numbers.
is
parallelism between cardinal
The relation-numbers
and
applicable to series
relation-
may
be
Similarity of Relations called
"
serial
numbers " are a sub-class in the field of a series
n
If
of these)
we know
determinate when
is
57
" ordinal " numbers (what are commonly called ;
number
thus a finite serial
number number in
of terms
the cardinal
having the
serial question. relation-number of a series the a finite cardinal number,
is
which has n terms
called the
is
"
ordinal
" number n.
(There
them we shall speak When the cardinal number of terms in
are also infinite ordinal numbers, but of in a later chapter.)
the field of a series
is infinite,
the relation-number of the series
not determined merely by the cardinal number, indeed an infinite number of relation-numbers exist for one infinite cardinal
is
as
number,
When its
we
a series
when we come to infinite, what we may
shall see is
may
relation-number,
number
We
but when a
;
can
numbers
define
series is finite, this
cannot happen.
and multiplication for relationcardinal numbers, and a whole arithmetic
addition
as well as for
of relation-numbers
be done
consider infinite series. " call its length," i.e. the without cardinal in change vary
The manner
can be developed.
in
which
easily seen by considering the case of series. Suppose, for example, that we wish to define the sum of two non-overlapping series in such a way that the relation-number this is to
is
sum shall be capable of being defined as the sum of the relation-numbers of the two series. In the first place, it is clear of the
that there
is
an order involved as between the two
series
:
one
them must be placed before the other. Thus if P and Q are the generating relations of the two series, in the series which
of
sum with P put before Q, every member of the field of precede every member of the field of Q. Thus the serial relation which is to be defined as the sum of P and Q is not " P or Q " simply, but " P or Q or the relation of any member of the field of P to any member of the field of Q." Assuming that P and Q do not overlap, this relation is serial, but " P or Q "
is
their
P
will
is
not
serial,
being not connected, since
member of the field the sum of P and Q, a
of
P and
as
above defined,
a
it
member is
does not hold between
of the field of Q.
what we need
Thus
in order
Introduction
58
sum
to define the
of
to
Mathematical Philosophy
two relation-numbers.
needed for products and powers. metic does not obey the commutative law
Similar modifica
The
tions are
:
the
resulting arith
sum
or product
two relation-numbers generally depends upon the order in which they are taken. But it obeys the associative law, one of
form of the distributive law, and two
of the formal laws for
powers, not only as applied to serial numbers, but as applied to relation-numbers generally. Relation-arithmetic, in fact, though recent, It
is
a thoroughly respectable branch of mathematics.
must not be supposed, merely because
series
afford the
most obvious application of the idea of likeness, that there are no other applications that are important. We have already mentioned maps, and we might extend our thoughts from this illustration to
geometry generally.
If
the system of relations
by which a geometry
is applied to a certain set of terms can be into relations of likeness with a system applying brought fully to another set of terms, then the geometry of the two sets is
indistinguishable from the mathematical point of view, i.e. all the propositions are the same, except for the fact that they are
applied in one case to one set of terms and in the other to another. illustrate this by the relations of the sort that may be
We may "
between," which we considered in Chapter IV. We there saw that, provided a three-term relation has certain formal
called
logical properties, it will give rise to series,
"
and may be
called
Given any two points, we can use the between-relation to define the straight line determined by those a
between-relation."
two points
;
it
consists of a
and
b together
with
all
points x,
such that the between-relation holds between the three points It has been shown by 0. Veblen a, b, x in some order or other.
we may
regard our whole space as the field of a three-term between-relation, and define our geometry by the properties we that
1 assign to our between-relation. 1
Now
likeness
is
just as easily
This does not apply to elliptic space, but only to spaces in which Modern Mathematics, edited by the straight line is an open series. " The Foundations of J. W. A. Young, pp. 3-51 (monograph by O. Veblen on
Geometry").
Similarity of Relations definable
59
between three-term relations as between two-term
B and B' are two between-relations, so that " means x is between y and z with respect to B," xB(y, z) we shall call S a correlator of B and B if it has the field of B' for its converse domain, and is such that the relation B holds relations.
If
"
"
7
between three terms when B' holds between their S-correlates, and only then. And we shall say that B is like B' when there is
at least one correlator of
B
with B'.
The reader can
easily
B' in this sense, there can be convince himself that, if B no difference between the geometry generated by B and that is like
generated by B'. It follows
from
this that the
mathematician need not concern
himself with the particular being or intrinsic nature of his points,
and planes, even when he
is speculating as an applied say that there is empirical evidence of the approximate truth of such parts of geometry as are not matters of definition. But there is no empirical evidence as to what a " point " is to be. It has to be something that as nearly " as possible satisfies our axioms, but it does not have to be very " " small or without parts." Whether or not it is those things
lines,
mathematician.
is
We may
a matter of indifference, so long as it satisfies the axioms. If can, out of empirical material, construct a logical structure,
we
no matter how complicated, which
will satisfy
our geometrical "
legitimately be called a point." must not say that there is nothing else that could legitimately be called a " point " ; we must only say : " This object we have
axioms, that structure
may
We
constructed
many
geometer ; it may be one of objects, any of which would be sufficient, but that is no is
sufficient for the
concern of ours, since this object is enough to vindicate the empirical truth of geometry, in so far as geometry is not a
matter of definition."
This is only an illustration of the general that what in mathematics, and to a very great matters principle extent in physical science, is not the intrinsic nature of our terms, but the logical nature of their interrelations.
We may
say, of
two similar
relations, that they
have the same
60
Introduction
Mathematical Philosophy
to
" structure."
For mathematical purposes (though not for those of pure philosophy) the only thing of importance about a relation is the cases in which it holds, not its intrinsic nature. Just as a
may be defined by various different but co-extensive concepts " man " and " featherless biped," so two relations which e.g.
class
may hold in the same set of instances. " instance " in which a relation holds is to be conceived as a
are conceptually different
An
couple of terms, with an order, so that one of the terms comes and the other second ; the couple is to be, of course, such that its first term has the relation in question to its second. Take (say) the relation " father " : we can define what we may first
the
call
" extension " of this relation as the
class of all
ordered
From y. couples (Xy y) the mathematical point of view, the only thing of importance " about the relation " father is that it defines this set of ordered which are such that x
Speaking generally, we say
couples.
The " extension " couples
(x, y)
of a relation
is
is
the father of
:
the class of those ordered
which are such that x has the relation
in question
to y.
We
can now go a step further in the process of abstraction, Given any relation,
and consider what we mean by " structure."
we
if it is
can,
For the sake extension
by Cy
is
a sufficiently simple one, construct a
the following couples
dy e ale five terms,
" a .
map
>
.
map
:
ab y
aCy
ady
no matter what. " of this relation
of
it.
which the
of definiteness, let us take a relation of be, ce, dcy
de y where
We may make
by taking
a
five points
on a plane and connecting them by arrows, as
in the
revealed
"
by
structure It is
accompanying the
map
figure.
What
what we
call
is
the
" of the relation.
clear that the
relation does not
terms that
is
make
"
structure
"
of the
depend upon the particular up the field of the relation.
The field may be changed without changing the structure, and the structure may be changed without changing the field for
61
Similarity of Relations
example, if we were to add the couple ae in the above illustration we should alter the structure but not the field. Two relations have the same " structure," we shall say, when the same map will do for both or, what comes to the same thing, when either can be a map for the other (since every relation can be its own
And that, as a moment's reflection shows, is the very same thing as what we have called " likeness." That is to say, two relations have the same structure when they have likeness, Thus what we i./. when they have the same relation-number. " relation-number " is the defined as the same thing as is very " structure " a word intended the word which, obscurely by
map).
important as it is, is never terms by those who use it.
(so far as
we know)
defined in precise
There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised.
For example, it is often said that space and time are subjective, but they have objective counterparts ; or that phenomena are subjective, but are caused
have differences
by things
which must
in themselves,
corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little inter se
about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts
would form a world having the same structure as the phenomenal world, and allowing us to infer from phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. If the phenomenal world has three
dimensions, so must the world behind
phenomena ; if the pheno must the other be and so on. every proposition having a communicable significance
menal world
is
Euclidean, so
;
In short, must be true of both worlds or of neither
must
lie
:
the only difference
in just that essence of individuality
which always eludes
words and bafHes description, but which, is
irrelevant to science.
Now
for that very reason, the only purpose that philosophers
62
Introduction
to
Mathematical Philosophy
have in view in condemning phenomena
is
themselves and others that the real world
in order to persuade is
very different from
the world of appearance. We can all sympathise with their wish to prove such a very desirable proposition, but we cannot con gratulate
do not
them on
their success.
It
is
true that
many
of
them
phenomena, and these Those who do assert counter
assert objective counterparts to
escape from the above argument. parts are, as a rule, very reticent on the subject, probably because they feel instinctively that, if pursued, it will bring about too
much
of a rapprochement
between the
real
and the phenomenal
they were to pursue the topic, they could hardly avoid the conclusions which we have been suggesting. In such ways,
world.
If
as well as in
number
is
many
others, the notion of structure or relation-
important.
CHAPTER
VII
RATIONAL, REAL, AND COMPLEX NUMBERS
WE
have now seen how to define cardinal numbers, and also relation-numbers, of which what are commonly called ordinal
numbers are a particular species. of these kinds of number may be
It will
be found that each
infinite just as well as finite.
capable, as it stands, of the more familiar exten sions of the idea of number, namely, the extensions to negative,
But neither
is
and complex numbers.
fractional,
irrational,
chapter we
shall briefly
In the present
supply logical definitions of these various
extensions.
One
of the mistakes that
definitions in this region
of
is
have delayed the discovery of correct the
common
number included the previous
idea that each extension
sorts as special cases.
thought that, in dealing with positive
and negative
It
was
integers, the
positive integers might be identified with the original signless integers. Again it was thought that a fraction whose denominator is
I
may
be identified with the natural number which
numerator. root of
2,
And
is
its
the irrational numbers, such as the square
were supposed to find their place among rational frac than some of them and less than the others,
tions, as being greater
so that rational and irrational
numbers could be taken together
as one class,
" called real numbers."
number was
further
And when
the idea of
extended so as to include " complex "
i.e. numbers I, involving the square root of that real numbers could be regarded as those thought
numbers,
complex numbers
in
which the imaginary part 63
(i.e.
it
was
among
the part
Introduction
64
to
Mathematical Philosophy
which was a multiple of the square root of All i) was zero. these suppositions were erroneous, and must be discarded, as we shall find,
correct definitions are to be given.
if
Let us begin with positive and negative integers. It is obvious on a moment's consideration that +1 and I must both be relations,
and
in fact
must be each
obvious and sufficient definition
is
other's
that
converses.
-f-i is
The
the relation of
m
and I is the relation of n to n-f-l. Generally, if inductive will be the relation of n-\-m to n number, -\-m any and will be the relation of n to n-\-m. Accord (for any n),
tt-f I to n, is
m
ing to this
n
long as
definition,
is
-\-m
a relation which
is
number number. But
a cardinal
inductive cardinal
-\-m
capable of being identified with
m
m
is
a class of classes. as
m
Indeed, -f
is
(finite or infinite)
y
is
one-one so
m
and
is
an
under no circumstances
which
is
not a relation, but
every bit as distinct from
m
is.
Fractions are more interesting than positive or negative integers.
We need fractions for many purposes, but perhaps most obviously measurement. My friend and collaborator Dr
for purposes of
A. N. Whitehead has developed a theory of fractions specially
adapted for their application to measurement, which But if all that is needed in Principia Mathematical
is
set forth
is
to define
objects having the required purely mathematical properties, this purpose can be achieved by a simpler method, which we shall
here adopt. relation
We
shall define the fraction
m/n
as being that
which holds between two inductive numbers x y when t
This definition enables us to prove that m/n is a oneor n is zero. And of course n/m one relation, provided neither
xn=ym.
m
is
the converse relation to m/n. From the above definition it
is
clear that the fraction
m/i
is
that relation between two integers x and y which consists in the x=my. This relation, like the relation -f-w, is by no means capable of being identified with the inductive cardinal fact that
number
m
%
because a relation and a class of classes are objects 1
Vol.
iii.
* 300
ff.,
especially 303.
Rational) Real, 1 of utterly different kinds.
same
relation,
and Complex Numbers be seen that o/
It will
is
whatever inductive number n may be;
65 always the
it is,
in short,
the relation of o to any other inductive cardinal. We may call this the zero of rational numbers ; it is not, of course, identical
with the cardinal number
o. Conversely, the relation ra/o is number the whatever inductive same, may be. There always We may call is not any inductive cardinal to correspond to m/o. " the is an instance of the sort of It of rationals." it infinity
m
mathematics, and that is represented sort from the true Cantorian different a totally by in our next chapter. The in infinite, which we shall consider for its definition or use, any finity of rationals does not demand,
infinite that is traditional in
"
This
oo ."
is
It is not, in actual fact, a
infinite classes or infinite integers.
very important notion, and we could dispense with it altogether The Cantorian infinite, on if there were any object in doing so. is of the greatest and most fundamental impor the understanding of it opens the way to whole new realms
the other hand,
tance of
;
mathematics and philosophy. be observed that zero and
It will
are not one-one.
There
is
Zero
m/n
one-many, and
is less
Given two
than p/q
if
mq
is
less
difficulty in proving that the relation serial,
"
ratios,
many-one. and less among
p/q,
less
we
shall
There
than pn.
we omit
zero
and
infinity
any smallest or largest ratio ; other than zero and infinity,
term and
from our it is
than," so defined,
infinity
m/2n
is
if
smaller and
though neither is zero or infinity, so that m/n
is
no is
In
the largest.
series, there is
obvious that is
say
is
so that the ratios form a series in order of magnitude.
this series, zero is the smallest If
mjn and
ratios
among
infinity is
not any difficulty in defining greater
ratios (or fractions).
that
is
infinity, alone
no longer
m/n any ratio 2m/n is larger, is
neither the smallest
Of course in practice we shall continue to speak of a fraction as (say) So greater or less than i, meaning greater or less than the ratio i/i. long as it is understood that the ratio i/i and the cardinal number i are 1
different, it is
not necessary to be always pedantic in emphasising the
difference.
5
66
Introduction
nor the largest
ratio,
omitted) there
is
arbitrarily.
In
Mathematical Philosophy
to
and therefore (when zero and
no smallest or
like
infinity are
m/n was chosen manner we can prove that however nearly largest, since
equal two fractions may be, there are always other fractions between them. For, let m/n and p/q be two fractions, of which the greater.
Then
is easy to see (or to prove) that (m+p)/(n-}-q) will be greater than m/n and less than p/q. Thus the series of ratios is one in which no two terms are consecutive,
p/q
is
it
but there are always other terms between any two. Since there are other terms between these others, and so on ad infinitum, it is obvious that there are an infinite number of ratios between
any two, however nearly equal these two may
be. 1
A
series
having the property that there are always other terms between " any two, so that no two are consecutive, is called compact." " " magnitude form a compact series. Such series have many important properties, and it is important to observe that ratios afford an instance of a compact series
Thus the
ratios in order of
generated purely logically, without any appeal to space or time or any other empirical datum. Positive and negative ratios can be defined in a way analogous that in which we defined positive and negative integers. Having first defined the sum of two ratios m/n and p/q as to
(mq+pn)/nq, we define where m/n is any ratio
-{-p/q as the relation of m/n-\-p/q to
and
is
m/n,
of course the converse of
p/q not the only possible way of defining positive and negative ratios, but it is a way which, for our purpose, has the merit of being an obvious adaptation of the way we adopted in -\-p/q-
This
;
is
the case of integers.
We
come now
to a
more
interesting extension of the idea of
" real " numbers, irrationals. In Chapter I. we had occasion to mention " incommensurables " and their disthe extension to what are called
number, which are the kind that embrace i.e.
1 Strictly speaking, this statement, as well as those following to the end " axiom of infinity," which of the paragraph, involves what is called the will be discussed in a later chapter.
Rational, Reaty
and Complex Numbers
67
It was through them, i.e. through covery by Pythagoras. geometry, that irrational numbers were first thought of. A
square of which the side is one inch long will have a diagonal of is the square root of 2 inches. But, as the ancients discovered, there is no fraction of which the square is 2.
which the length
This proposition is proved in the tenth book of Euclid, which is one of those books that schoolboys supposed to be fortunately lost in the days is
when Euclid was
extraordinarily simple.
of 2, so that ra 2 /ft 2 =2,
m
and therefore
an odd number for
p
is
if
half of m.
will
an unending
The proof
mjn be the square root Thus m 2 is an even number,
m2
i.e.
2n 2
.
must be an even number, because the square of odd. Now if m is even, m* must divide by 4,
m
Thus we shall have 4 2 =2 2 where Hence 2p 2 =n 2 and therefore n/p will also be the if 2. But then we can repeat the argument also be the square root of 2, and so on, through 2
=^.p
2
.
,
9
square root of
n=2q, pjq
used as a text-book.
is
then
m=2p,
still
If possible, let
:
numbers that are each
series of
half of its predecessor.
we divide a number by 2, and then halve the half, and so on, we must reach an odd number after a finite number of Or we may put the argument even more steps. simply by assuming that the m/n we start with is in its lowest terms in that case, m and n cannot both be even yet we have But
this is impossible
;
if
;
;
seen that, fraction
if
m
2
2
/n
2,
they must
m/n whose square
Thus no
be.
Thus there cannot be any
is 2.
fraction will express exactly the length of the diagonal whose side is one inch long. This seems like a
of a square
challenge thrown out
arithmetician
may
by nature to
arithmetic.
However the
boast (as Pythagoras did) about the power
numbers, nature seems able to baffle him by exhibiting lengths which no numbers can estimate in terms of the unit. But the
of
problem did not remain in this geometrical form. As soon as algebra was invented, the same problem arose as regards the solution of equations, though here it took on a wider form, since
it
also involved
complex numbers. can be found which approach nearer
It is clear that fractions
68
Introduction
and nearer to having ascending than
less
less
series
to
Mathematical Philosophy
their square equal to 2.
of fractions
all
of
We
can form an
which have their squares
but differing from 2 in their later members by than any assigned amount. That is to say, suppose I assign 2,
some small amount found that
all
in advance, say one-billionth, it will be the terms of our series after a certain one, say the
have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have tenth,
reached sooner or later a term in the after
by
series, say the twentieth, terms would have had squares differing from 2 than this still smaller amount. If we set to work to
which
less
all
extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-and-so
We can places, exactly fulfils the above conditions. equally well form a descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts
many
as
we come
to later terms of the series,
and
differing,
sooner or
than any assigned amount. In this way we seem later, by to be drawing a cordon round the square root of 2, and it may less
seem
difficult
to believe that it
is
it
can permanently escape
not by this method that
we
us.
shall actually
Nevertheless, reach the square root of 2. If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that, among those whose
squares are not less than
2, all
have
their squares greater than 2.
maximum to the ratios whose square is less and no minimum to those whose square is greater than 2.
There
is
no
than
2,
There
no lower limit short of zero to the difference between the numbers whose square is a little less than 2 and the numbers whose square is a little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one is
and there is no minimum to the other. Between these two classes, where V2 ought to be, there is nothing. Thus our class,
and Complex Numbers
Rational, Real,
69
cordon, though we have drawn it as tight as possible, has been 2. drawn in the wrong place, and has not caught
v
The above method of dividing all the terms of a series into two classes, of which the one wholly precedes the other, was 1 brought into prominence by Dedekind, and is therefore called " a Dedekind cut." With respect to what happens at the point of
there are four possibilities and a
section,
maximum
to the lower section
:
(i)
there
minimum
be a
may
to the upper
maximum to the one and no minimum be no maximum to the one, but a minimum to the other, (4) there may be neither a maximum to the one nor a minimum to the other. Of these four cases, the
section, (2) there
be a
may
to the other, (3) there
illustrated
first is
terms
may
by any
series in
which there are consecutive
in the series of integers, for instance, a lower section
:
must end with some number n and the upper section must then begin with n+i. The second case will be illustrated in the series of ratios
up
to
than
and including
The
I.
section
from
I
seen,
is
all
upward is less
square
is
than
2,
illustrated
and
I, I
we put
greater than
We may in series
if
is
than
(including
illustrated
as our lower section all ratios
and in our upper section
I,
third case
ratios less
square
we take
if
if
for our
itself).
case, as
in our lower section
and in our upper section
greater
for our lower
upper section
The fourth
all ratios
we have
all ratios
whose
ratios
whose
all
2.
neglect the
first
of our four cases, since it only arises
where there are consecutive terms.
our four cases,
all ratios
we take
we say
that the
maximum
In the second of
of the lower section
the lower limit of the upper section, or of any set of terms chosen out of the upper section in such a way that no term of
is
the upper section is before all of them. In the third of our four cases, we say that the minimum of the upper section is the
upper limit
of the lower section, or of
out of the lower section in such a section
is
after all of
1
Stetigkeit
und
them.
way
any
set of terms chosen
that no term of the lower
In the fourth case,
irrationale Zahlen,
2nd
we say
that
edition, Brunswick, 1892.
Introduction
70 there
"
"
a
is
gap
Mathematical Philosophy
to
neither the upper section nor the lower has In this case, we may also say that we
:
a limit or a last term. "
have an have
irrational section," since sections of the series of ratios
"
"
when they correspond
gaps
to irrationals.
What
delayed the true theory of irrationals was a mistaken " of series of " The there must be limits ratios. " " is of the utmost and before limit notion of importance, belief that
proceeding further it will be well to define it. A term x is said to be an " upper limit " of a class a with
P if (i) a has no maximum in P, (2) every a which belongs to the field of P precedes x, (3) every the field of P which precedes x precedes some member
respect to a relation
member member
of of
" we mean " has the relation P to.") precedes " maximum " This presupposes the following definition of a
of a.
(By
"
A term x is to a relation
said to be a
P
if
x
is
"
maximum "
member of a and of P to any other member
a
not have the relation
:
a with respect the field of P and does
of a class
of a.
These definitions do not demand that the terms to which
For example, given they are applied should be quantitative. a series of moments of time arranged by earlier and later, their " maximum " be the last of the moments ; but if (if any) will are and earlier, their " maximum " (if later arranged by they any) will be the first of the moments. The " minimum " of a class with respect to P is its maximum " " lower limit with with respect to the converse of P ; and the respect to P is the upper limit with respect to the converse of P. The notions of limit and maximum do not essentially demand
that the relation in respect to which they are defined should be serial, but they have few important applications except to cases is
when the
relation
often important
is
is serial
or quasi-serial.
"
the notion
upper
A
limit or
notion which
maximum,"
"
give the name upper boundary." Thus the " of a " set of terms chosen out of a series is upper boundary their last member if they have one, but, if not, it is the first
to which
term
we may
after all of
them,
if
there
is
such a term.
If there is neither
Rational) Real,
and Complex Numbers
71
a maximum nor a limit, there is no upper boundary. The " " lower boundary is the lower limit or minimum. Reverting to the four kinds of Dedekind section, we see that in the case of the first three kinds each section has a
boundary
(upper or lower as the case may be), while in the fourth kind It is also clear that, whenever the neither has a boundary. lower section has an upper boundary, the upper section has a lower boundary. In the second and third cases, the two boundaries are identical ; in the first, they are consecutive
terms of the
series.
" Dedekindian " when every section has a boundary, upper or lower as the case may be. We have seen that the series of ratios in order of magnitude
A
is
series is called
not Dedekindian.
From
the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving that there was
no rational limit to the ratios whose square is less than 2, they " " an irrational limit, which allowed themselves to postulate the Dedekind gap. Dedekind, in the above-mentioned work, set up the axiom that the gap must always be filled, i.e. that every section must have a boundary. It is for this reason
was to
fill
that series where his axiom
But there are an
infinite
is
verified are called
number
of series for 4
"
Dedekindian."
which
it is
not
verified.
"
"
what we want has many advan postulating tages ; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil.
The method
of
an irrational Dedekind cut in some way " repre " an irrational. In order to make use of this, which to sents It is clear that
begin with
way
is
no more than a vague feeling, we must find some from it a precise definition ; and in order to do
of eliciting
we must disabuse our minds of the notion that an irrational must be the limit of a set of ratios. Just as ratios whose de
this,
nominator
is
i
are not identical with integers, so those rational
Introduction
72
Mathematical Philosophy
to
numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios.
We have to of
define a
which some
will
new kind
numbers be rational and some "
"
of
called
"
real
irrational.
numbers," Those that
same kind of way in which the ratio n/i corresponds to the integer n but they are not the same as ratios. In order to decide what they are to be, are rational
to ratios, in the
correspond
;
let
us observe that an irrational
and a cut
represented by an irrational lower section. Let us confine
is
its
is
cut, represented by ourselves to cuts in which the lower section has no
maximum
;
lower section a " segment." Then those segments that correspond to ratios are those that consist of all ratios less than the ratio they correspond to, which is in this case
we
will call the
boundary ; while those that represent irrationals are those have no boundary. Segments, both those that have boundaries and those that do not, are such that, of any two their
that
pertaining to one series, one must be part of the other ; hence they can all be arranged in a series by the relation of whole and
A
series in which there are Dedekind gaps, i.e. in which there are segments that have no boundary, will give rise to more segments than it has terms, since each term will define a segment
part.
having that term for boundary, and then the segments without boundaries will be extra.
We
are
irrational
A
now
in a position to define a real
number and an
number.
" real number "
a segment of the series of ratios in order
is
of magnitude.
An
" irrational number "
is
a segment of the series of ratios
which has no boundary.
A
" rational real number "
is
a segment of the series of ratios
which has a boundary.
Thus a
rational real
certain ratio, to that ratio.
and
The
proper fractions.
number
it is
real
consists of all ratios less than a
the rational real
number
I,
number corresponding
for instance,
is
the class of
Rational, Real,
and Complex Numb en
73
we naturally supposed that an irrational the of a set of ratios, the truth is that it is the limit limit must be of the corresponding set of rational real numbers in the series In the cases in which
segments ordered by whole and part.
of
the upper limit of
For example, ^/^
is
those segments of the series of ratios that
all
correspond to ratios whose square is less than 2. More simply still, \/2 is the segment consisting of all those ratios whose square is less than 2. It is easy to prove that the series of segments of any series Dedekindian. For, given any set of segments, their boundary will be their logical sum, i.e. the class of all those terms that
is
belong to at least one segment of the
set. 1
" con definition of real numbers is an example of " " had another we struction as against of which postulation," example in the definition of cardinal numbers. The great
The above
advantage of this method is that it requires no new assumptions, but enables us to proceed deductively from the original apparatus of logic.
There
\L
and
is
no
difficulty in defining addition
numbers
for real v,
as
above defined.
and multiplication real numbers
Given two
each being a class of ratios, take any
any member
member
of
JJL
and
and add them together according to the rule for the addition of ratios. Form the class of all such sums obtainable by varying the selected members of p and v. This gives a
new
class is a
sum. of follows
The
p
of v
class of ratios,
it is
easy to prove that this
new
segment of the series of ratios. We define it as the and v. We may state the definition more shortly as
:
arithmetical
arithmetical
sums
other chosen in 1
and
For a
all
sum of two real numbers is member of the one and
of a
the class of the a
member
of the
possible ways.
treatment of the subject of segments and Dedekindian For a fuller Principia Mathematical, vol. ii. * 210-214.
fuller
relations, see
treatment of real numbers, see ibid., vol. Mathematics, chaps, xxxiii. and xxxiv.
iii.
* 310
ff.,
and Principles of
Introduction
74
We
to
Mathematical Philosophy
can define the arithmetical product of two real numbers
in exactly the same way, by multiplying a member of the one by a member of the other in all possible ways. The class of ratios
thus generated
is
defined as the product of the two real numbers. series of ratios is to be defined as
such definitions, the (In excluding o and infinity.) all
There
is
no
and negative
difficulty in
real
extending our definitions to positive
numbers and
their addition
and multiplication.
remains to give the definition of complex numbers. Complex numbers, though capable of a geometrical interpreta It
not demanded by geometry in the same imperative way " which irrationals are demanded. A " complex number means a number involving the square root of a negative number, whether
tion, are
in
Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. Using the letter i for the square
integral, fractional, or real.
I, any number involving the square root of a negative number can be expressed in the form x-\-yi, where x and y are The part yi is called the " imaginary " part of this number, real.
root of
" real " real " x being the (The reason for the phrase part. " " ima is that they are contrasted with such as are numbers time a have been for numbers long habitually Complex ginary.") used by mathematicians, in spite of the absence of any precise It has been simply assumed that they would obey definition. the usual arithmetical rules, and on this assumption their employ They are required less for profitable.
ment has been found
geometry than for algebra and analysis. We desire, for example, to be able to say that every quadratic equation has two roots, and every cubic equation has three, and so on. But if we are 2 confined to real numbers, such an equation as # -|-i=o has no has only one. Every roots, and such an equation as
x^io
number has
first presented itself as needed for negative numbers were needed in order that subtraction might be always possible, since otherwise a b would be meaningless if a were less than b ; fractions were needed
generalisation of
some simple problem
:
Rational) Real,
might be always possible
in order that division
numbers are needed
and Complex Numbers ;
75
and complex and solu
in order that extraction of roots
tion of equations may be always possible. But extensions of are not created by the mere need for them : they are
number
created by the definition, and it is to the definition of complex numbers that we must now turn our attention.
A complex number may be regarded and defined as simply an ordered couple of real numbers. Here, as elsewhere, many definitions are possible. All that is necessary is that the defini tions
adopted
In the case of
shall lead to certain properties.
complex numbers, they are defined as ordered couples of real we secure at once some of the properties required, numbers, if
namely, that two real numbers are required to determine a com plex number, and that among these we can distinguish a first
and a second, and that two complex numbers are only
when
the
first real
in the one is equal to the the second to the second. What
involved in the other, and needed further can be secured by defining the rules of addition
first is
identical
number involved
We
and multiplication.
are to have
Thus we
shall define that, given
numbers,
(#, y)
and
(#',
y'), their
y+y')> and their product
By
these definitions
shall
we
is
two ordered couples
sum
is
to be the couple
to be the couple (xx
of real r
(x+x
,
f
yy', xy'-\-x'y).
our ordered couples For example, take the
shall secure that
have the properties we
desire.
product of the two couples (o, y) and (o, y'). This will, by the above rule, be the couple ( yy', o). Thus the square of the couple (o, i) will be the couple ( I, o). Now those couples in
which the second term
o are those which, according to the usual nomenclature, have their imaginary part zero ; in the notation is
x-\- yi,
they are x+oi, which
as
is
it is natural to write simply x. Just natural (but erroneous) to identify ratios whose de nominator is unity with integers, so it is natural (but erroneous) it
Introduction
j6
to
Mathematical Philosophy
complex numbers whose imaginary part is zero with numbers. Although this is an error in theory, it is a con " " " " venience in practice ; x-}-oi may be replaced simply by x " " " " and o-\-yi by yi," provided we remember that the x " is not really a real number, but a special case of a complex number. And when y is I, " yi" may of course be replaced by " *." Thus to identify
real
the couple
(o,
l) is I.
represented by *, and the couple (1, o) is Now our rules of multiplication make the
represented by square of (o, l) equal to is
what we desired
(1,
the square of
o), i.e.
to secure.
Thus our
i is
i.
This
definitions serve all
necessary purposes.
easy to give a geometrical interpretation of complex numbers in the geometry of the plane. This subject was agree It is
ably expounded by W. K. Clifford in his Common Sense of the Exact Sciences, a book of great merit, but written before the importance of purely logical definitions had been realised.
Complex numbers of a higher order, though much less useful and important than those what we have been defining, have certain uses that are not without importance in geometry, as may be seen, for example, in Dr Whitehead's Universal Algebra.
The
definition of
complex numbers
obvious extension of the definition
complex number
of order
we have
n
is
obtained by an
We
given.
define a
n as a one-many relation whose domain numbers and whose converse domain from I to n. 1 This is what would ordi
of order
consists of certain real consists of the integers
narily be indicated
by the notation
(x l9
x 2 #3 ,
,
.
.
.
x n), where the
denote correlation with the integers used as suffixes, and the correlation is one-many, not necessarily one-one, because xr
suffixes
and xa may be equal when definition,
and
s are not equal.
with a suitable rule of multiplication,
purposes for
We
r
which complex numbers
will serve all
of higher orders are needed.
have now completed our review
of those extensions of
number which do not involve infinity. The application to infinite collections must be our next topic. 1
Cf Principles of Mathematics, .
The above
360, p. 379.
of
number
CHAPTER
VIII
INFINITE CARDINAL
NUMBERS
THE
definition of cardinal numbers which we gave in Chapter II. was applied in Chapter III. to finite numbers, i.e. to the ordinary " inductive natural numbers. To these we gave the name
numbers," because we found that they are to be defined as numbers which obey mathematical induction starting from o.
But we have not yet considered collections which do not have an number of terms, nor have we inquired whether such This is an collections can be said to have a number at all.
inductive
ancient problem, which has been solved in our own day, chiefly by Georg Cantor. In the present chapter we shall attempt to explain the theory of transfinite or infinite cardinal numbers as it
results
from a combination of his discoveries with those of
Frege on the logical theory of numbers. It cannot be said to be certain that there are in fact any infinite collections in the world.
we
the
"
axiom
The assumption that
there are
is
what
Although various ways suggest themselves by which we might hope to prove this axiom, there is reason to fear that they are all fallacious, and that there is no conclusive logical reason for believing it to be true. At the same call
time, there
is
of infinity."
certainly
and we are therefore
no logical reason against infinite
collections,
justified, in logic, in investigating
the hypo
thesis that there are such collections. The practical form of this hypothesis, for our present purposes, is the assumption that, if n is any inductive number, n is not equal to w-j-i. Various
subtleties arise in identifying this 77
form of our assumption with
Introduction
78
Mathematical Philosophy
to
the form that asserts the existence of infinite collections
;
but
we will leave these out of account until, in a later chapter, we come to consider the axiom of infinity on its own account. For the present we shall merely assume that, if n is an inductive number, n
not equal to n-\-i. This is involved in Peano's that no two inductive numbers have the same suc assumption cessor ; for, if n=n-}-i, then n I and n have the same successor,
namely
n.
is
Thus we are assuming nothing that was not involved
in Peano's primitive propositions.
Let us now consider the collection of the inductive numbers This
themselves.
is
a perfectly well-defined class.
In the
first
place, a cardinal number is a set of classes which are all similar to each other and are not similar to anything except each other. We then define as the " inductive numbers " those among
cardinals which belong to the posterity of o with respect to the relation of n to w-f-i, *
possessed by o and by the successors of possessors, meaning by the "successor" of n the number n-\-\. Thus the class of " inductive numbers " is perfectly definite. By our general definition of cardinal numbers, the number of terms in the class of inductive
numbers
is
to be defined as
are similar to the class of inductive classes is the
number
"
all
numbers
of the inductive
those classes that
"
i.e.
this set of
numbers according to our
definitions.
Now it is easy to see that this number is not one of the inductive If n is any inductive number, the number of numbers from o to n (both included) is n-\-i ; therefore the total number of inductive numbers is greater than n, no matter which of the
numbers.
inductive numbers n
numbers but term
may
be.
If
we arrange
the inductive
in a series in order of magnitude, this series has
;
if
n
is
an inductive number, every
series
no
whose
last field
has n terms has a last term, as it is easy to prove. Such differences Thus the number of inductive might be multiplied a
numbers ing
all
is
a
new number,
different
inductive properties.
It
from
may
them, not possess happen that o has a certain all
of
Cardinal Numbers
Infinite
79
yet that this new The difficulties that so long delayed the theory of infinite numbers were largely due to the fact that some, at least, of the inductive properties were wrongly judged property, and that if n has number does not have it.
it
so has
w+i, and
must belong to all numbers ; indeed it was thought that they could not be denied without contradiction. The first step in understanding infinite numbers consists in realising the to be such as
mistakenness of this view.
The most noteworthy and astonishing difference between an number and this new number is that this new number
inductive is
or
I
unchanged by adding
any
of a
necessarily
number
making
a
or subtracting
number
tion of
what he
I is
transfinite
calls
some
of
The
of as
fact of being
used by Cantor for the defini " but for cardinal numbers ;
appear as we proceed, it is cardinal number as one which does
which
better to define an infinite
or doubling or halving
which we think
larger or smaller.
not altered by the addition of " various reasons,
I
of other operations
will
not possess all inductive properties, i.e. simply as one which is not an inductive number. Nevertheless, the property of being unchanged by the addition of I is a very important one, and we
must dwell on
To say
it
for a time.
that a class has a
number which
is
not altered by the
the same thing as to say that, if we take a term x which does not belong to the class, we can find a one-one relation addition of
I is
whose domain
the class and whose converse domain
is
is
obtained
by adding x to the class. For in that case, the class is similar to the sum of itself and the term x, i.e. to a class having one extra term
;
so that
term, so that also
have
if
has the same number as a class with one extra
it
n
nn
domains consist
is I,
this i.e.
of the
consist of just one
number, n=n-\-\. In this case, we shall there will be one-one relations whose
whole
class
and whose converse domains
term short of the whole
class.
It
can be shown
that the cases in which this happens are the same as the apparently more general cases in which some part (short of the whole) can be
put into one-one relation with the whole.
When this can be done,
8o
Introduction
by which
the correlator
whole
Mathematical Philosophy
it is
done
class into a part of itself
be called " reflexive." " A " reflexive
Thus
may
be said to "
for this reason,
;
reflect
"
such classes
the will
:
one which "
class is
"
of itself.
A
to
similar to a proper part
is
a part short of the whole.) (A proper part " " reflexive cardinal number is the cardinal number of a is
reflexive class.
We One
have now to consider of the
most
illustration of the
of it
property of reflexiveness. " " reflexion is Royce's he imagines it decided to make a map this
striking instances of a
map
:
A
England upon a part of the surface of England. map, if is accurate, has a perfect one-one correspondence with its
original
;
thus our map, which
is
part,
is
in one-one relation
with
the whole, and must contain the same number of points as the whole, which must therefore be a reflexive number. Royce is
map, if it is correct, must contain must in turn contain a map of the map which map map, This point is interesting, of the map, and so on ad infinitum. but need not occupy us at this moment. In fact, we shall do
interested in the fact that the of the
a
well to pass from picturesque illustrations to such as are
and
for this
completely than consider the number-series definite,
The
purpose we cannot do
more better
itself.
n to
w-f-i, confined to inductive numbers, is one-one, has the whole of the inductive numbers for its domain, and all except o for its converse domain. Thus the whole class
relation of
of inductive
numbers
is
what the same
similar to
class
becomes
" reflexive " class according Consequently " " reflexive to the definition, and the number of its terms is a number. Again, the relation of n to 2n, confined to inductive
when we omit o.
numbers,
is
it is
a
one-one, has the whole of the inductive numbers for
its domain, and the even inductive numbers alone for its converse domain. Hence the total number of inductive numbers is the
same as the number of even inductive numbers. This property was used by Leibniz (and many others) as a proof that infinite numbers are impossible it was thought self-contradictory that ;
Cardinal Numbers
Infinite
" the part should be equal to the whole."
But
81 this
one of those
is
phrases that depend for their plausibility upon an unperceived " " has many meanings, but if it is the word vagueness equal taken to mean what we have called " similar," there is no contra :
diction, since
an
can perfectly well have parts
infinite collection
itself. Those who regard this as impossible have, unconsciously as a rule, attributed to numbers in general pro perties which can only be proved by mathematical induction,
similar to
their familiarity makes us regard, mistakenly, the as true beyond region of the finite. Whenever we can " reflect " a class into a part of itself, the same relation will necessarily reflect that part into a smaller
and which only
and so on ad infinitum.
part,
we have just seen, numbers we can, by as
For example, we can
reflect,
the inductive numbers into the even
all
the same relation (that of n to 2n) reflect ; the even numbers into the multiples of 4, these into the multiples of 8, and so on. This is an abstract analogue to Royce's problem of the map. The even numbers are a " map " of all the inductive
numbers
;
the
of the
the multiples of 4 are a map of the map ; the multiples If we had of 8 are a map of the map of the map ; and so on. " " the same the to to relation of our w+l," applied process map would have consisted of all the inductive numbers except o ; the
map map
The
of the
map would have consisted of all from 2 onward, map of the map of all from 3 onward and so on.
chief use of
;
such illustrations
is
in order to
become
familiar
with the idea of reflexive arithmetical
classes, so that apparently paradoxical propositions can be readily translated into the
language of reflexions and is
much
classes, in
which the
air of
paradox
less.
be useful to give a definition of the number which is the inductive cardinals. For this purpose we will define the kind of series exemplified by the inductive cardinals
It will
that first
of
in order of magnitude.
"
The kind
of series
which
is
called a
"
has already been considered in Chapter I. It is a which can be generated by a relation of consecutiveness
progression series
:
6
82
Introduction
Mathematical Philosophy
to
every member of the series is to have a successor, but there is to be just one which has no predecessor, and every member of the series is to be in the posterity of this term with respect to the relation " immediate These characteristics predecessor."
be "
may
A
summed up "
progession
in the following definition is
* :
a one-one relation such that there
is
just
one term belonging to the domain but not to the converse domain, and the domain is identical with the posterity of this one term. It is easy to see that a progression, so defined, satisfies
Peano's
The term belonging to the domain but not to the "o" the term to which converse domain will be what he calls
five
axioms.
;
a term has the one-one relation will be the
term he
and the domain
;
calls
"
number."
"
successor
of the one-one relation will
Taking
his five
the following translations " " o is a number becomes (1)
"
of the
be what
axioms in turn, we have
:
which
is
not a
member
the domain."
is
" The member of the domain
of the converse
domain
is
a
member
of
equivalent to the existence of such a given in our definition. We will call this
This
member, which
member "
:
is
term." " becomes " The successor of (2) any number is a number " The term to which a given member of the domain has the rela the
first
:
This is tion in question is again a member of the domain." member as follows the of the : definition, every proved By
member of the posterity of the first term hence successor of a member of the domain must be a member of
domain the
is
a
;
the posterity of the first term (because the posterity of a term always contains its own successors, by the general definition of posterity),
and therefore a member
the definition the posterity of the
domain, because by term is the same as the
of the
first
domain. " No two numbers have the same successor." (3) only to say that the relation
is
one-many, which
it is
(being one-one). 1
Cf.
Pnncipia Mathematica,
vol.
ii.
#
123.
by
This
is
definition
Cardinal Numbers
Infinite
(4) first
83
" The " becomes not the successor of any number term is not a member of the converse domain," which is " o
is
:
again an immediate result of the definition. " (5) This is mathematical induction, and becomes Every member of the domain belongs to the posterity of the first term," :
which was part of our
Thus progressions
definition.
we have
as
defined
them have the
five
formal properties from which Peano deduces arithmetic. It is " " in the sense similar easy to show that two progessions are
We
defined for similarity of relations in Chapter VI. can, of which is serial from the one-one relation
course, derive a relation
by which we
define a progression
member
a
the method used
:
and the relation
explained in Chapter IV.,
of its proper posterity
is
is
that
that of a term to
with respect to the original
one-one relation.
Two
transitive
asymmetrical relations which generate pro
same reasons for which the cor The class of all such " in " serial number progressions is a
gressions are similar, for the
responding one-one relations are similar. transitive generators of
the sense of Chapter VI.; it is in fact the smallest of infinite numbers, the number to which Cantor has given the name
serial
by which he has made it famous. But we are concerned, for the moment, with cardinal numbers. Since two progressions are similar relations, it follows that their domains (or their fields, which are the same as their domains) are similar classes. The domains of progressions form a cardinal
o>,
number, since every progression
is
easily
shown
This cardinal number
numbers
it
;
is
which
class
suffix o, to distinguish it
which have other is N N terms
that a class has
a
of a
domain of a progression. the smallest of the infinite cardinal
the smallest of infinite cardinals
it is
domain
the one to which Cantor has appropriated the
infinite cardinals,
that
similar to the
to be itself the
is
Hebrew Aleph with the
To say
is
member
of
N
,
and
suffixes.
from
larger
Thus the name
of
.
is
the same thing as to say
this is the
same thing
as to say
Introduction
84
to
Mathematical Philosophy
members of the class can be arranged in a progression. obvious that any progression remains a progression if we omit a finite number of terms from it, or every other term, or that the It is
except every tenth term or every hundredth term. These of thinning out a progression do not make it cease to
all
methods
be a progression, and therefore do not diminish the number of In fact, any selection from a pro its terms, which remains N .
it
may
nw
a progression
is
gression
be distributed.
n nW
if it
Take
has no last term, however sparsely (say) inductive numbers of the form
Such numbers grow very rare the number series, and yet there are just
,
of
or
.
there are inductive
Conversely,
numbers
altogether,
we can add terms
their
in the higher parts as
namely, N
to the inductive
number.
many
of
them
as
.
numbers without
One example, ratios. must be many more ratios
for
Take, increasing might be inclined to think that there
than integers, since ratios whose denominator is I correspond to the integers, and seem to be only an infinitesimal proportion
But in actual fact the number of ratios (or fractions) is exactly the same as the number of inductive numbers, namely, This is easily seen by arranging ratios in a series on the N If the sum of numerator and denominator in following plan of ratios.
.
:
one
is less
the
sum
is
than in the other, put the one before the other ; if equal in the two, put first the one with the smaller
numerator.
This series later.
their
i,
1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5,
is
a progression,
and
Hence we can arrange number is therefore N
It is
all
all
.
.
.
ratios occur in it sooner or
ratios in a progression,
and
.
not the case, however, that
N terms. than N ;
is
This gives us the series
The number it is,
greater than
all infinite collections
of real numbers, for example,
in fact, 2^,
and
n even when n
it is is
is
have
greater
not hard to prove that 2 n
infinite.
The
easiest
way
of
proving this is to prove, first, that if a class has n members, it contains 2 n sub-classes in other words, that there are 2 n ways
Carainal Numbers
Infinite
of selecting
where we
some
of its
members and
select all or none)
(including the extreme cases secondly, that the number of
;
sub-classes contained in a class
is
85
always greater than the number
Of these two propositions, the first is familiar in the case of finite numbers, and is not hard to extend The proof of the second is so simple and to infinite numbers.
members
of
of the class.
so instructive that
In the
we
shall give it
first place, it is
of a given class (say a)
since each
is
number
of sub-classes
at least as great as the
member
members, have a correlation of
:
clear that the
constitutes a sub-class,
number
of
and we thus
members with some of the sub Hence it follows that, if the number of sub-classes is classes. not equal to the number of members, it must be greater. Now it is easy to prove that the number is not equal, by showing that, given any one-one relation whose domain is the members and whose converse domain is contained among the set of sub classes, there must be at least one sub-class not belonging to the converse domain. The proof is as follows x When a oneone correlation R is established between all the members of a and some of the sub-classes, it may happen that a given member all
the
:
x
correlated with a sub-class of which
is
again,
it
may happen
that x
is
it is
a
member
;
or,
correlated with a sub-class of
not a member. Let us form the whole class, )3 say, members x which are correlated with sub-classes of which they are not members. This is a sub-class of a, and it is not correlated with any member of a. For, taking first the members of ]3, each of them is (by the definition of )8) correlated with some sub-class of which it is not a member, and is therefore not correlated with j3. Taking next the terms which are not members of jS, each of them (by the definition of j3) is correlated with
which
it is
of those
some
sub-class of
which
not correlated with
is
with 1
)3.
Since
This proof
is
R
j8.
it
is a member, and therefore again Thus no member of a is correlated
was any one-one correlation
of all
members
taken from Cantor, with some simplifications
Jahresbericht der deutschen Mathematiker-Vereinigung,
i.
:
(1892), p. 77.
see
86
Introduction
Mathematical Philosophy
to
with some sub-classes, it follows that there is no correlation It does not matter to the of all members with all sub-classes. all that proof if j3 has no members happens in that case is that the sub-class which is shown to be omitted is the null-class. :
Hence in any case the number of sub-classes is not equal to the number of members, and therefore, by what was said earlier, it is greater. Combining this with the proposition that, if n is n of number the members, 2 is the number of sub-classes, we have n the theorem that 2 is always greater than n, even when n is infinite.
from
It follows
However
to the infinite cardinal numbers.
number n may infinite
n will be, 2
numbers
accustomed to
it.
no
this proposition that there is
be
still
somewhat
is
We have, N -fw=N 2
o
,
greater.
surprising
maximum
great an infinite
The arithmetic
of
one becomes
until
for example,
where n
is
any inductive number,
=o-
(This follows from the case of the ratios, for, since a ratio is determined by a pair of inductive numbers, it is easy to see that the number of ratios is the square of the number of inductive
numbers,
i.e. it is
(This follows
then
fact, as
2 ;
but we saw that
it is
also
N =N 0> where n is any inductive number. N 2=No by induction ; for if NO "=N O from N +i=N 2 = N .)
But In
N
,
2^0
we
shall see later,
>N
2^
is
.
a very important number,
"
"
continuity namely, the number of terms in a series which has in the sense in which this word is used by Cantor. Assuming
space and time to be continuous in this sense (as
do in analytical geometry and kinematics),
we commonly
this
number of points in space or of instants in time the number of points in any finite portion of
;
will
it will
be the also be
space, whether
Cardinal Numbers
Infinite
or volume.
line, area,
After
N
,
2^
is
87
the most important and
interesting of infinite cardinal numbers.
Although addition and multiplication are always possible with infinite cardinals, subtraction and division no longer give definite results, and cannot therefore be employed as they are
employed with
so long as the the other number
if
:
n=&
N
Take subtraction
in elementary arithmetic.
,
if
n
definite result.
is
number subtracted reflexive, it
is
goes well
;
Thus
remains unchanged.
so far, subtraction gives a perfectly when we subtract N from
finite;
But
is finite, all
to begin
otherwise
it is
This is we may then get any result, from o up to N take inductive seen From the numbers, examples. easily by the of N terms collections following away
itself;
.
,
:
(1) All the inductive (2)
numbers
All the inductive
the numbers from o to n
remainder, zero.
numbers from n onwards remainder, I, numbering n terms in all.
odd numbers remainder, all the even numbers, N terms. numbering and All these are different ways of subtracting N from N (3)
All the
,
all
give different results.
As regards division, very similar results follow from the fact N is unchanged when multiplied by 2 or 3 or any finite number n or by N It follows that N divided by N may have that
.
any value from I up to N From the ambiguity of subtraction and division .
it
results
that negative numbers and ratios cannot be extended to infinite
Addition, multiplication, and exponentiation proceed subtraction, satisfactorily, but the inverse operations
numbers. quite
division,
and extraction
that depend upon
The
of roots
are ambiguous,
and the notions
fail when infinite numbers are concerned. by which we defined finitude was mathe i.e. we defined a number as finite when it
them
characteristic
matical induction,
obeys mathematical induction starting from o, and a class as when its number is finite. This definition yields the sort
finite
of result that a definition
ought to
yield,
namely, that the
finite
88
Introduction
to
Mathematical Philosophy
numbers are those that occur in the ordinary number-series But in the present chapter, the infinite num o, i, 2, 3, ... bers
we have
they have
also
discussed have not merely been non-inductive been reflexive. Cantor used reflexiveness as the :
definition of the infinite,
and
believes that
equivalent to non-inductiveness ; that is to say, he believes that every class and every cardinal is either inductive or reflexive. This may be true,
it
is
and may very possibly be capable of proof but the proofs by Cantor and others (including the present ;
hitherto offered
author in former days) are fallacious, for reasons which will be " explained when we come to consider the multiplicative axiom." At present, it is not known whether there are classes and cardinals
which are neither
n were such a cardinal, we should not have nn-\-i but n would not be one " natural of the numbers," and would be lacking in some of the inductive properties. All known infinite classes and cardinals reflexive
nor inductive.
If
y
but for the present it is well to preserve an open mind as to whether there are instances, hitherto unknown, of are reflexive
classes
;
and cardinals which are neither
reflexive nor inductive.
Meanwhile, we adopt
the following definitions A. finite class or cardinal is one which is inductive. An infinite class or cardinal is one which is not inductive.
All refiexive classes at present
We shall
whether
:
and cardinals are all infinite
infinite
classes
;
but
it is
not
and cardinals are
return to this subject in Chapter XII.
known
reflexive.
CHAPTER IX INFINITE SERIES
"
AN is
an
infinite series
"
infinite class.
one kind of
AND ORDINALS which the
may
be defined as a
We
have already had occasion to consider
infinite series,
series of
namely, progressions.
field
In this chapter
we shall consider the subject more generally. The most noteworthy characteristic of an infinite series is that its serial number can be altered by merely re-arranging In this respect there is a certain oppositeness between cardinal and serial numbers. It is possible to keep the cardinal
its
terms.
number to
it
of a reflexive class
;
in spite of adding terms
unchanged
on the other hand,
it
is
possible to change the serial
number of a series without adding or taking away any terms, by mere re-arrangement. At the same time, in the case of any infinite series it is also possible, as
without altering the the
way
in
serial
number
with cardinals, to add terms :
everything depends upon
which they are added.
In order to
make matters
Let us
examples.
first
clear, it will be best to begin with consider various different kinds of series
which can be made out various plans.
We
of the inductive
numbers arranged on
start with the series
3,
which, as finite
we have already
serial
seen, represents the smallest of in
numbers, the sort that Cantor
proceed to thin out this series 89
calls
co.
Let us
by repeatedly performing the
Introduction
90
Mathematical Philosophy
to
operation of removing to the end the first even number that occurs. We thus obtain in succession the various series :
w
i, 3, 4> 5,
6
i, 3, 5>
>
w+ 2
i, 3> 5, 7>
and so on. possible,
in
2 >4>
>
2 > 4> 6 >
'
>
we imagine
this process carried on as long as the reach series finally If
we
3,5,
i,
2>
>
n+ l
-
.
7,
.
which we have
.
2n+i,
first all
2,4,6,8,
.
..
2n,
the odd numbers and then
all
the even
numbers.
The
serial
w ~l~3>
numbers
2aj
of these various series are
Each
-
of these
numbers
is
of its predecessors, in the following sense
One
number
"
w+i, co+2, greater" than any
:
"
"
said to be
than another if greater any series having the first number contains a part having the second number, but no series having the second number contains a part having the first number. If
serial
we compare
the two series
i, 2, 3, 4,
.
3,4,5,
.
i,
we
is
see that the first
.
.
n,
.
.
+i,
,
.
.
2,
similar to the part of the second which
is
omits the last term, namely, the number 2, but the second not similar to any part of the first. (This is obvious, but
is is
demonstrated.) Thus the second series has a greater i.e. serial number than the first, according to the definition than But if we add a term at the is o>. CD+I beginning greater easily
of a progression instead of the end,
Thus
I
-\-a>o}.
Thus
i-fo>
is
we
still
have a progression.
not equal to co+i.
characteristic of relation-arithmetic generally
:
two relation-numbers, the general
p+v
to is
v-\-p,.
The
rule
is
that
if
p,
case of finite ordinals, in which there
This
and
is is
is
v are
not equal equality,
quite exceptional.
The
series
we
finally
reached just now consisted of first all the all the even numbers, and its serial
odd numbers and then
and Ordinals
Series
Infinite
91
number is 2o>. This number is greater than o> or a)-{-n where n is finite. It is to be observed that, in accordance with the 9
general definition of order, each of these arrangements of integers is
to be regarded as resulting
from some
definite relation.
E.g.
the qne which merely removes 2 to the end will be defined by " x and the following relation y are finite integers, and either :
y
and x
2
is
not
is
2,
one which puts first all ones will be defined by
x
is
odd and y
are even." in future
a
is
We
than y." The the odd numbers and then all the even
or neither
" :
even or x shall
is
and x
2
is less
x and y are finite integers, and either than y and both are odd or both
is less
not trouble, as a
rule, to give these formulae
but the fact that they could be given
;
is essential.
The number which we have called 2o>, namely, the number of series consisting of two progressions, is sometimes called a> .2. depends upon the order of the
Multiplication, like addition, factors
which
:
a progression of couples gives a series such as
is itself
a series which
a progression ; but a couple of progressions gives is twice as long as a It is therefore progression.
necessary to distinguish between 2cu and to 2. Usage is variable ; we shall use 2o> for a couple of progressions and a> 2 for a pro gression of couples, and this decision of course governs our .
.
"
general interpretation of
numbers
sum
of
a
" :
a
" .
j3
will
relations each
a
" .
)3
when a and
have to stand having
j3
are relation-
for a suitably constructed
terms.
jS
We can proceed indefinitely with the process of thinning out the inductive numbers. For example, we can place first the odd numbers, then their doubles, then the doubles of these, and so on.
We
thus obtain the series
3 5t 7
;
6 I0 H>
2>
8, 24, 40, 56,
of
which the number
Any
is o>
;
>
,
2 ,
since
one of the progressions in
it is
this
.
.
4>
I2 > 20 > 28,
.
.
;
.,
a progression of progressions new series can of course be .
Introduction
92
Mathematical Philosophy
to
thinned out as we thinned out our original progression. 3 4 proceed to o> , o> ,
.
.
co
w ,
and so on
we can always go further. The series of all the ordinals
;
however
far
We
we have
can
gone,
that can be obtained in this way,
that can be obtained by thinning out a progression, is longer than any series that can be obtained by re-arranging
all
i.e.
itself
the terms of a progression. (This is not difficult to prove.) cardinal number of the class of such ordinals can be shown
The
N it is the number which Cantor calls ordinal number of the series of all ordinals that can The Nj. be made out of an N taken in order of magnitude, is called o) v Thus a series whose ordinal number is coj has a field whose cardinal number is Nj. We can proceed from co x and Nx to co 2 and N2 by a process exactly analogous to that by which we advanced from w and N And there is nothing to prevent us from advancing to o>! and M x this way to new cardinals and new ordinals. in It indefinitely 2^ is to of the cardinals known whether in the not is any equal It is not even known whether it is comparable series of Alephs. for aught we know, it may be neither with them in magnitude to be greater than
;
,
.
;
equal to nor greater nor less than question
we
is
any one
of the Alephs.
This
connected with the multiplicative axiom, of which
shall treat later.
All the series
have
we have been
been what
is
called
considering so far in this chapter "well-ordered." well-ordered
A
one which has a beginning, and has consecutive terms, has a term next after any selection of its terms, provided and
series is
there are any terms after the selection. This excludes, on the one hand, compact series, in which there are terms between
any two, and on the other hand series which have no beginning, or in which there are subordinate parts having no beginning.
The
series of negative integers in order of
magnitude, having not well-ordered; but taken in the reverse order, beginning with I, it is well-ordered,
no beginning, but ending with being in fact a progression.
The
I,
is
definition
is
:
Series
Infinite
A
" well-ordered "
series
and Ordinals
is
93
one in which every sub-class
(except, of course, the null-class) has a first term. An " ordinal " number means the relation-number of a well-
ordered
series.
It is thus a species of serial
number.
Among well-ordered series, a generalised form of mathematical " induction applies. transfinitely property may be said to be
A
"
when
hereditary terms in a series, if,
it
belongs to a certain selection of the belongs to their immediate successor pro it
vided they have one.
In a well-ordered
series,
a transfinitely
hereditary property belonging to the first term of the series belongs to the whole series. This makes it possible to prove many propositions concerning well-ordered series which are not true of
all series.
numbers in series which are not well-ordered, and even to arrange them in compact series. For example, we can adopt the following plan consider It is easy to arrange the inductive
:
(exclusive), arranged in order a These form compact series ; between any magnitude. two there are always an infinite number of others. Now omit
the decimals from
*i
(inclusive) to
I
of
the dot at the beginning of each, and
we have
a
compact
series
such as divide by 10. If we wish to include those that divide by 10, there is no difficulty ; consisting of all finite integers except
instead of starting with
*i,
we
will include all
decimals less than
but when we remove the dot, we will transfer to the right any that occur at the beginning of our decimal. Omitting these, and returning to the ones that have no o's at the beginning, I,
o's
we can
arrangement of our integers as Of two integers that do not begin with the same digit, the one that begins with the smaller digit comes first. Of two follows
state the rule for the
:
that do begin with the same digit, but differ at the second digit, the one with the smaller second digit comes first, but first of all the one with no second digit ; and so on. Generally, if two integers agree as regards the first n digits, but not as regards
the
(n-f-i)**,
that one comes
digit or a smaller
first
which has either no (n+i) th
one than the other.
This rule of arrangement,
Introduction
94
Mathematical Philosophy
to
as the reader can easily convince himself, gives rise to a
compact
containing all the integers not divisible by 10 ; and, we saw, there is no difficulty about including those
series
as
that are divisible is
it
In
possible
fact,
by
It follows
10.
construct
to
we have already
compact
from series
this
example that
having N
seen that there are
N
terms.
ratios,
and
magnitude form a compact series ; thus we have here another example. We shall resume this topic ratios
order
in
of
in the next chapter.
Of the usual formal laws
of addition, multiplication,
and ex
ponentiation, all are obeyed by transfinite cardinals, but only some are obeyed by transfinite ordinals, and those that are obeyed " usual the by them are obeyed by all relation-numbers.
formal laws " I.
II.
III.
By
we mean
the following
:
The commutative law a+jS=j8+a and aX0=j8xa. The associative law :
:
(a+jS)+y=a+(j3-hy) The distributive law
and
(aXjS)Xy=aX (xy).
:
When
the commutative law does not hold, the above form law must be distinguished from
of the distributive
As we
shall see immediately,
one form
may
be true and the
other false.
IV.
The laws
of exponentiation
:
All these laws hold for cardinals, whether finite or infinite, {QI finite ordinals. But when we come to infinite ordinals,
and
or indeed to relation-numbers in general, some hold and some not. The commutative law does not hold ; the associative
do
law does hold
;
the distributive law (adopting the convention
Series
Infinite
we have adopted above
and Ordinals
95
as regards the order of the factors in a
product) holds in the form
but not in the form
the exponential laws
a?
.
hold, but not the
still
which
law
obviously connected with the commutative law for
is
multiplication.
The
definitions
of
multiplication
and exponentiation that
above propositions are somewhat complicated. The reader who wishes to know what they are and how the
assumed
are
in the
above laws are proved must consult the second volume of Principia Mathematics * 172-176. Ordinal transfinite arithmetic was developed by Cantor at an
earlier stage
than cardinal transfinite arithmetic, because
has various technical mathematical uses which led him to
But from the point it
is
of
it it.
view of the philosophy of mathematics less fundamental than the theory of
important and
less
Cardinals are essentially simpler than a curious historical accident that they first ordinals, as an abstraction from the latter, and only gradually appeared came to be studied on their own account. This does not apply transfinite
cardinals.
and
it is
to Frege's work, in which cardinals, finite and transfinite, were treated in complete independence of ordinals ; but it was
Cantor's
work that made the world aware
of the subject, while
Frege's remained almost unknown, probably in the main on account of the difficulty of his symbolism. And mathematicians,
other people, have more difficulty in understanding and " in the " using notions which are comparatively logical simple sense than in manipulating more complex notions which are
like
Mathematical Philosophy
96
Introduction
more akin
to their ordinary practice.
to
For these reasons,
it
was
only gradually that the true importance of cardinals in mathe matical philosophy was recognised. The importance of ordinals, though by no means small, is distinctly less than that of cardinals,
and
is
very largely merged in that of the more general conception
of relation-numbers.
CHAPTER X AND CONTINUITY
LIMITS
"
"
THE
limit is one of which the importance in conception of a has been mathematics found continually greater than had been thought. The whole of the differential and integral calculus,
indeed practically everything in higher mathematics, depends upon limits. Formerly, it was supposed that infinitesimals were involved in the foundations of these subjects, but Weierstrass showed that this is an error wherever infinitesimals were thought :
to occur,
what
really occurs
zero for their lower limit.
was an
a set of finite quantities having
is
It
used to be thought that
essentially quantitative notion,
" limit "
namely, the notion of a
quantity to which others approached nearer and nearer, so that among those others there would be some differing by less than any " limit " is a assigned quantity. But in fact the notion of purely ordinal notion, not involving quantity at all (except by accident series concerned happens to be given quantitative).
A
when the
point on a line may be the limit of a set of points on the line, without its being necessary to bring in co-ordinates or measure
ment
or anything quantitative.
The
cardinal
number N
limit (in the order of magnitude) of the cardinal 3,
is
numbers
the
I,
2,
...,...,
and a
finite
although the numerical difference between N O cardinal is constant and infinite from a quantitative :
numbers
no nearer to N as they grow What makes the limit of the finite numbers is the N O larger. fact that, in the series, it comes immediately after them, which point of view, finite
is
an ordinal
fact,
get
not a quantitative fact. 97
7
Introduction
98
Mathematical Philosophy
to
There are various forms of the notion of " limit," of in creasing complexity. The simplest and most fundamental form, been already defined, but we will here repeat the definitions which lead to it, in a general form in which they do not demand that the relation concerned
from which the
rest are derived, has
The definitions are as follows The " minima " of a class a with respect to a relation P are those members of a and the field of P (if any) to which no member of a has the relation P. The " maxima " with respect to P are the minima with respect shall be serial.
:
to the converse of P.
The " sequents " of a class a with respect to a relation P are " " " " of successors the minima of the successors of a, and the a are those members of the field of P to which every member of the common part of a and the field of P has the relation P. The " precedents " with respect to P are the sequents with respect to the converse of P. The " upper limits " of a with respect to P are the sequents provided a has no maximum ; but if a has a maximum, it has no
upper
limits.
The " lower
"
with respect to P are the upper limits with to the converse of P. respect Whenever P has connexity, a class can have at most one limits
maximum, one minimum, one sequent, etc. Thus, we are concerned with in practice, we can speak of
"
the limit
"
any).
(if
When P
is
definition of
"
in the cases
"
boundary
a serial relation, we can greatly simplify the above a limit. can, in that case, define first the
We
of a class a,
i.e. its
limits or
maximum, and then
proceed to distinguish the case where the boundary is the limit from the case where it is a maximum. For this purpose it is best to use the notion of
segment." " " segment of P defined by a class a as speak of the those terms that have the relation P to some one or more of
We all
the
"
will
members
of a.
This will be a segment in the sense defined
and
Limits
99
Continuity
in the sense there denned ; indeed^ every segment the segment defined by some class a. If P is serial, the segment defined by a consists of all the terms that precede
Chapter VII.
in is
some term or other be
of a.
maximum, every member and the whole
a,
by
of
a
of
a precedes some other member
of
therefore included in the segment defined
is
Take, for example, the class consisting of the fractions
a.
i i.e.
a has a maximum, the segment will maximum. But if a has no
If
the predecessors of the
all
i,
of all fractions of the
I,
form
if, for different finite values
I
2"
This series of fractions has no
of n.
that the segment which in order of magnitude)
maximum, and
it
defines (in the
is
the class of
whole
all
clear
it is
series of fractions
proper fractions.
Or,
again, consider the prime numbers, considered as a selection from the cardinals (finite and infinite) in order of magnitude. In this
case the segment defined consists of
all finite
integers.
"
"
Assuming that P is serial, the boundary of a class a will be the term x (if it exists) whose predecessors are the segment defined by a. " maximum " of a is a boundary which is a member of a. " An upper limit" of a is a boundary which is not a member of cu
A
has no boundary, it has neither maximum nor limit. the case of an " irrational " Dedekind cut, or of what is " called a gap." Thus the " upper limit " of a set of terms a with respect to a If a class
This
is
series
P
but
such that every earlier term comes before some of the a's. " " may define all the upper limiting-points of a set of
is
We terms
j3
is
that term x
(if it
exists)
as all those that are the
chosen out of
j8.
We
which comes
after all the a's,
upper limits of sets of terms have to distinguish upper
shall, of course,
limiting-points from lower limiting-points. example, the series of ordinal numbers
If
we
consider, for
:
I, 2, 3,
...
CO,
CO-f
I,
.
.
.
2CO,
2CO-H, ...
3^0,
...
CD
2 ,
...
CO
3 ,
...,
ioo
Introduction
Mathematical Philosophy
to
the upper limiting-points of the field of have no immediate predecessors, i.e. &J
I, CO, 2CO, 3&>>
2
The upper limiting-points I, co
On
2 ,
to>
>
this series are those that
2
-\-O),
.
.
,
2CO 2,
of the field of this
2co 2 ,
...
co
3 ,
co
3
+co 2
*
.
new .
.
.
CO 3
.
.
series will
be
.
and indeed every wellhas no lower limiting-points, because there are no terms except the last that have no immediate successors. But the other hand, the series of ordinals
ordered series
if
we
consider such a series as the series of ratios, every member both an upper and a lower limiting-point for
of this series is
suitably chosen sets.
and
we
If
select out of it the
rationals) will
limiting-points.
have
The
consider the series of real numbers,
rational real numbers, this set (the
the real numbers as upper and lower " first limiting-points of a set are called its
all
derivative," and the limiting-points of the called the second derivative, and so on.
With regard
to limits,
we may
first
derivative are
distinguish various grades of
what may be called " continuity " in a series. The word " con " had been used for a long time, but had remained without tinuity any precise
Each
definition until the time of
of these
two men gave a
but Cantor's definition
is
Dedekind and Cantor.
precise significance to the term,
narrower than Dedekind's
:
a series
which has Cantorian continuity must have Dedekindian con tinuity, but the converse does not hold.
The
first
a precise it
definition that
meaning
But
this
what we have
"
compactness," i.e. in the between any two terms of the series there are others. would be an inadequate definition, because of the
as consisting in
fact that
would naturally occur to a man seeking would be to define
for the continuity of series
called
"
" We in series such as the series of ratios. gaps saw in Chapter VII. that there are innumerable ways in which existence of
the series of ratios can be divided into two parts, of which one wholly precedes the other, and of which the first has no last term,
Limits while the second has no
contrary to the
first
and
101
Continuity
Such a state
term.
vague feeling we have
as to
of affairs
seems
what should character
"
continuity," and, what is more, it shows that the series of ratios is not the sort of series that is needed for many mathematical
ise
purposes. say that
Take geometry,
when two
for
example
:
we wish
to be able to
each other they have a the of in but if series common, points on a line were similar point " " to the series of ratios, the two lines might cross in a gap and straight lines cross
have no point in common. others might be given to
This
is
a crude example, but
show that compactness
is
many
inadequate as
a mathematical definition of continuity. It was the needs of geometry, as much as anything, that led " Dedekindian " It will be re to the definition of continuity.
membered that we
defined a series as Dedekindian
sub-class of the field has a boundary.
(It is
when every
sufficient to
assume
always an upper boundary, or that there is always a lower boundary. If one of these is assumed, the other can be
that there
is
deduced.) That is to say, a series is Dedekindian when there are no gaps. The absence of gaps may arise either through
terms having successors, or through the existence of limits in the absence of maxima. Thus a finite series or a well-ordered series is
Dedekindian, and so is the series of real numbers. The former Dedekindian series is excluded by assuming that our
sort of
series is compact ; in that case our series must have a property which may, for many purposes, be fittingly called continuity. Thus we are led to the definition " A series has " Dedekindian when it is Dedekindian :
continuity
and compact. But this definition is still too wide for many purposes. for example, that
we
Suppose,
desire to be able to assign such properties
make it certain that every point can be specified by means of co-ordinates which are real numbers this is not insured by Dedekindian continuity alone. We want to geometrical space as shall
:
to be sure that every point which cannot be specified by rational co-ordinates can be specified as the limit of a progression of points
IO2
Introduction
whose co-ordinates are
to
Mathematical Philosophy
rational,
and
this is a further property
which our definition does not enable us to deduce.
We
are thus led to a closer investigation of series with respect
This investigation was made by Cantor and formed the basis of his definition of continuity, although, in its simplest form, this definition somewhat conceals the considerations which to limits.
have given
some
rise to it.
We
shall, therefore, first travel
through
of Cantor's conceptions in this subject before giving his
definition of continuity.
Cantor defines a
series as
"
" perfect
when
all its
points are
But this limiting-points and all its limiting-points belong to it. definition does not express quite accurately what he means. There is no correction required so far as concerns the property points are to be limiting-points ; this is a property belonging to compact series, and to no others if all points are to be upper limiting- or all lower limiting-points. But if it is only
that
all its
assumed that they are limiting-points one way, without specify ing which, there will be other series that will have the property in question
for
example, the series of decimals in which a decimal is distinguished from the corresponding
ending in a recurring 9
terminating decimal and placed immediately before it. Such a very nearly compact, but has exceptional terms which
series is
are consecutive,
and
cessor, while the
second has no immediate successor.
of
which the
first
has no immediate prede
Apart from
series, the series in which every point is a limiting-point are compact series ; and this holds without qualification if it is specified that every point is to be an upper limiting-point (or
such
that every point
is
to be a lower limiting-point).
Although Cantor does not explicitly consider the matter, we
must distinguish
different kinds of limiting-points according to
the nature of the smallest sub-series by which they can be defined. Cantor assumes that they are to be defined by progressions, or
When regressions (which are the converses of progressions). our series is the limit of a progression or regres of member every
by
sion,
Cantor
calls
our series
"
condensed in
itself
"
(insichdicht).
Limits and Continuity
We
come now
to the second property
to be defined, namely, the property
" being
closed
"
as consisting in the fact that
belong to series
case,
is
it.
But
this only
all
by which perfection was
which Cantor
This, as
(abgescblosseri).
103
calls that of
we saw, was
first
defined
the limiting-points of a series
has any effective significance some other larger series (as
given as contained in
if
our
is
the
with a selection of real numbers), and limiting-points
e.g.,
are taken in relation to the larger series. Otherwise, if a series is considered simply on its own account, it cannot fail to contain its
limiting-points.
he says
;
different,
What Cantor means
is
not exactly what
indeed, on other occasions he says something rather which is what he means. What he really means is that
every subordinate series which is of the sort that might be ex pected to have a limit does have a limit within the given series ; i.e. every subordinate series which has no maximum has a limit, every subordinate series has a boundary. But Cantor does not state this for every subordinate series, but only for progres sions and regressions. (It is not clear how far he recognises that i.e.
this is a limitation.)
want
is
the following
A series
is
Thus,
finally,
we
find that the definition
we
:
said to be
"
closed
" (abgescblossen)
when every pro
gression or regression contained in the series has a limit in the series.
We then have the further definition A series is " perfect " when it is condensed :
when every term
in itself
and
closed,
the limit of a progression or regression, and every progression or regression contained in the series has a limit in the series. i.e.
is
is
In seeking a definition of continuity, what Cantor has in mind the search for a definition which shall apply to the series of
numbers and to any series similar to that, but to no others. this purpose we have to add a further property. Among the real numbers some are rational, some are irrational ; although real
For the
number
of irrationals is greater than the number of rationals, there are rationals between any two real numbers, however yet
Introduction
IO4
Mathematical Philosophy
to
may differ. The number of rationals, as we saw, This gives a further property which suffices to characterise continuity completely, namely, the property of containing a class the two
little is >S
.
N
of
members
in such a
that some of this class occur
way
between any two terms of our series, however near together. This property, added to perfection, suffices to define a class of series
class
which are
we say that
similar
and are
in fact a serial
Cantor defines as that of continuous
We may
A
all
slightly
his
simplify
number.
This
series.
definition.
To
begin
with,
:
" median " of a series class
members
of it are to
is
a sub-class of the
field
such
be found between any two terms of
the series.
Thus the numbers.
median
class in the series of real
It is obvious that there
cannot be median classes
rationals are a
except in compact series. We then find that Cantor's definition following
A
series is
" continuous " when
contains a median class having
To avoid continuity." tinuity,
equivalent to the
is
:
confusion, It will
Dedekindian,
(2) it
N terms.
" Cantorian speak of this kind as be seen that it implies Dedekindian con
we
shall
but the converse
Cantorian continuity
(i) it is
are
is
not the case.
All series having
but not
all series having Dedekindian continuity. The notions of limit and continuity which we have been defining must not be confounded with the notions of the limit of a function
for
similar,
approaches to a given argument, or the continuity of a function These are different
in the neighbourhood of a given argument.
notions, very important, but derivative from the above
The continuity
of
motion
motion
and more
is
(if continuous) complicated. an instance of the continuity of a function ; on the other hand, the continuity of space and time (if they are continuous) is an
is
instance of the continuity of series, or (to speak more cautiously) of a kind of continuity which can, by sufficient mathematical
Limits and Continuity
105
view manipulation, be reduced to the continuity of series. In of the fundamental importance of motion in applied mathe matics, as well as for other reasons, it will be well to deal with the notions of limits and continuity as applied to functions ; but this subject will be best reserved for a
briefly
separate chapter.
The
namely, those of closely to the
the
which we have been considering, Dedekind and Cantor, do not correspond very
definitions of continuity
mind
vague idea which
of the
man
is
associated with the
word
in the street or the philosopher.
in
They
conceive continuity rather as absence of separateness, the sort of general obliteration of distinctions which characterises a thick
A
fog gives an impression of vastness without definite It is this sort of thing that a meta multiplicity or division.
fog.
means by " continuity," declaring it, very truly, be characteristic of his mental life and of that of children
physician to
and animals.
The general idea vaguely indicated by the word " continuity " when so employed, or by the word " flux," is one which is certainly quite different from that which we have been defining. Take, for example, the series of real numbers. Each is what it is, quite definitely and uncompromisingly by imperceptible degrees into another unit,
and
its
does not pass over a hard, separate
;
it
;
it is
distance from every other unit
is
finite,
though
made less than any given finite amount assigned advance. The question of the relation between the kind
it
can be
in of
among the real numbers and the kind ex what we see at a given time, is a difficult and by
continuity existing hibited, e.g.
intricate one.
It is not to
be maintained that the two kinds
are simply identical, but it may, I think, be very well main tained that the mathematical conception which we have been
considering in this chapter gives the abstract logical scheme to which it must be possible to bring empirical material by suitable " " continuous manipulation, if that material is to be called in any precisely definable sense. It would be quite impossible
106
Introduction
to
Mathematical Philosophy
to justify this thesis within the limits of the present volume. is interested may read an attempt to justify
The reader who
by the present author in the Our Knowledge of the External World. With these indications, we must leave this problem, interesting as it is, in order to return to topics more it
as regards time in particular
Monist
for 1914-5, as well as in parts of
closely connected with mathematics.
CHAPTER XI LIMITS
IN this chapter
we
AND CONTINUITY OF FUNCTIONS shall
be concerned with the definition of the
any) as the argument approaches a given " con value, and also with the definition of what is meant by a tinuous function." Both of these ideas are somewhat technical,
limit of a function
(if
and would hardly demand treatment
mere introduction
a
in
to mathematical philosophy but for the fact that, especially through the so-called infinitesimal calculus, wrong views upon
our present topics have become so firmly embedded in the minds of professional philosophers that a prolonged effort
for
is
their
It
and considerable
has been thought
uprooting. required ever since the time of Leibniz that the differential and integral
calculus
required
infinitesimal
quantities.
Mathematicians
(especially Weierstrass) proved that this is an error ; but errors incorporated, e.g. in what Hegel has to say about mathematics,
die hard,
such
men
and philosophers have tended to ignore the work
of
as Weierstrass.
Limits and continuity of functions, in works on ordinary mathematics, are defined in terms involving number. This is
not essential, as Dr Whitehead has shown. 1 We will, however, begin with the definitions in the text-books, and proceed after wards to show how these definitions can be generalised so as to
apply to series in general, and not only to such as are numerical or numerically measurable. Let us consider any ordinary mathematical function fx9 where 1
See Principia Mathematica, vol. 107
ii.
* 230-234.
io8
Introduction
to
Mathematical Philosophy
x and/* are both real numbers, and fx is one-valued i.e. when x is given, there is only one value that/* can have. We call x " " the argument," and/* the value for the argument *." When " a function is what we call continuous," the rough idea for which
we
are seeking a precise definition
is
that small differences in *
and if we make the * small enough, we can make the differences in below any assigned amount. We do not want, if a function
shall correspond to small differences in/*,
differences in
/* fall
to be continuous, that there shall be sudden jumps, so that, some value of *, any change, however small, will make a
is
for
change in/* which exceeds some assigned finite amount. The ordinary simple functions of mathematics have this property 2 3 it belongs, for example, to * , * , log *, sin *, and so on. :
.
.
.
But it is not at all difficult to define discontinuous functions. " the place of birth of Take, as a non-mathematical example, the youngest person living at time t" This is a function of t ; constant from the time of one person's birth to the time of the next birth, and then the value changes suddenly its
value
is
from one birthplace to the other. An analogous mathematical " example would be the integer next below *," where x is a real number. This function remains constant from one integer to the next, and then gives a sudden jump. The actual fact is though continuous functions are more familiar, they are the exceptions there are infinitely more discontinuous functions that,
:
than continuous ones.
Many
functions are discontinuous for one or several values of
the variable, but continuous for sin I/*.
example I to from
I
The function
all
other values.
Take
as
an
sin 6 passes through all values
every time that 6 passes from
77/2 to 77/2, or
from
generally from (2w 1)77/2 to (2n-\- 1)77/2, where n is any integer. Now if we consider I/* when * is very small, we see that as * diminishes I/* grows faster and faster, so that 77/2 to 377/2, or
more and more quickly through the
cycle of values from one multiple of 77/2 to another as * becomes smaller and smaller. I Consequently sin i/x passes more and more quickly from it
passes
and
Limits to
and back again,
i
any
Continuity of Functions
x grows smaller.
as
interval containing o, say the interval
is some very small number, number of oscillations in this
we take
fact, if
from
e to -fe
where
go through an infioite and we cannot diminish
sin i/x will
e
the oscillations
In
109
interval,
by making the
Thus round
interval smaller.
about the argument o the function is discontinuous. It is easy to manufacture functions which are discontinuous in several
N
places, or in
in
places, or everywhere.
any book on the theory Proceeding
now
Examples
will
be found
of functions of a real variable.
to seek a precise definition of
what
is
meant
by saying that a function is continuous for a given argument, when argument and value are both real numbers, let us first "
" of a number x as all the numbers neighbourhood from x c to #-|-e, where e is some number which, in important It is clear that continuity at a given cases, will be very small. define a
point has to do with what happens in any neighbourhood of that point,
however small.
What we
desire
is
this
:
If
a
is
the argument for which
we wish
our function to be continuous, let us first define a neighbourhood (a say) containing the value /# which the function has for the
argument a
;
we
desire that,
if
we take
a sufficiently small
values for arguments throughout this neighbourhood shall be contained in the neighbourhood a,
neighbourhood containing
no matter how small we
we
a, all
may have made
decree that our function
not to
is
differ
a.
That
is
to say,
if
from/rf by more than
some very tiny amount, we can always
find a stretch of real
numbers, having a in the middle of
such that throughout
this stretch
fx
will
not differ
And
scribed tiny amount. tiny
amount we may
definition
select.
f
it,
rom fa by more than the pre
this
is
to remain true whatever
Hence we are
led to the following
:
The function f(x) ment a if, for every small as we please,
is
" said to be " continuous for the argu
positive
number
CT,
different
from
o,
but as
there exists a positive number e, different from o, such that, for all values of 8 which are numerically
no
Introduction
less 1
than
than
e,
to
Mathematical Philosophy
the difference
/(#+)/(#)
is
numerically
less
a.
In this definition, a
first
a neighbourhood of /(#),
defines
namely, the neighbourhood from/(tf) a to/(tf)-j-cr. The defini tion then proceeds to say that we can (by means of e) define a neighbourhood, namely, that from # e to a-\-e, such that, for
arguments within
this neighbourhood, the value of the function within the neighbourhood horn (a) a tof(a)+cr. If this can be done, however cr may be chosen, the function is " con " tinuous for the argument a. all
f
lies
So far we have not defined the " limit " of a function for a given argument. If we had done so, we could have defined the a function is continuous continuity of a function differently at a point where its value is the same as the limit of its value for :
approaches either from above or from below. But it is only " tame " function that has a definite limit as the exceptionally the argument approaches a given point. The general rule is that a function oscillates, and that, given any neighbourhood of a given argument,
however small, a whole stretch
of values
As
occur for arguments within this neighbourhood. the general rule, let us consider it first.
will is
Let us consider what
may happen
this
as the
argument approaches some value a from below. That is to say, we wish to consider what happens for arguments contained in the interval from a e to a, where e is some number which, in important cases, will
be very small.
The values
of the function for
arguments from a
e to a (a
excluded) will be a set of real numbers which will define a certain section of the set of real numbers, namely, the section consisting
numbers that are not greater than all the values for arguments from a e to a. Given any number in this section,
of those
there are values at least as great as this number for arguments e and #, i.e. for arguments that fall very little short
between a 1
A number is said to
e
and +e.
be " numerically
less
"
than e when
it lies
between
Limits of a
c
(if
is
and
Continuity of Functions
The
possible corresponding sections.
we
sections
approaches section
is
To say
a.
arguments between a not less than z.
all
" ultimate section " as the argument
will call the
to say that,
possible e's and all common part of all these
Let us take
very small).
in
that a
number
z belongs to the ultimate
however small we may make e, there are e and a for which the value of the function
is
We may i.e.
apply exactly the same process to upper sections,
to sections that go from
some point up
to the top, instead of
from the bottom up to some point. Here we take those numbers that are not less than all the values for arguments from a e to a
;
this defines
an upper section which
will
vary as
e varies.
Taking the common part of all such sections for all possible e's, we obtain the " ultimate upper section." To say that a number belongs to the ultimate upper section is to say that, however small we make e, there are arguments between a e and a for z
which the value If a
term
of the function
z belongs
ultimate upper section, " ultimate oscillation." sidering once
value
o.
We
is
we
say that
shall
We may
more the function shall
not greater than
z.
both to the ultimate section and to the it
belongs to the
matter by con x approaches the in with the above
illustrate the
sin
i/x as
assume, in order to
fit
approached from below. Let us begin with the " ultimate section." Between e and o, whatever e may be, the function will assume the value definitions, that this value
is
I for certain arguments, but will never assume any greater value. Hence the ultimate section consists of all real numbers, positive and negative, up to and including I i.e. it consists of all negative numbers together with o, together with the positive numbers up to and including I. ;
"
ultimate upper section " consists of all positive numbers together with o, together with the negative numbers Similarly the
down
to and including I. Thus the " ultimate oscillation " from I to i, both included.
consists of
all real
numbers
112
Introduction
Mathematical Philosophy
to
" " ultimate oscillation of say generally that the a function as the argument approaches a from below consists of all those numbers x which are such that, however near we
We may
come
to a y
we
shall
still
find values as great as
x and values as
small as x.
The ultimate or
many
oscillation
terms.
In the
may
first
contain no terms, or one term,
two cases the function has a
definite
If the ultimate oscillation limit for approaches from below. has one term, this is fairly obvious. It is equally true if it has none ; for it is not difficult to prove that, if the ultimate oscilla tion is null, the boundary of the ultimate section is the same as
that of the ultimate upper section, and limit of the function for approaches
ultimate oscillation has
many
may
be defined as the
But
from below.
terms, there
no
is
if
the
definite limit to
the function for approaches from below. In this case we can take the lower and upper boundaries of the ultimate oscillation
the lower boundary of the ultimate upper section and the upper boundary of the ultimate section) as the lower and upper " values for " limits of its ultimate approaches from below. " " ultimate Similarly we obtain lower and upper limits of the (i.e.
values for approaches from above. Thus we have, in the general case,/owr limits to a function for approaches to a given argument.
The limit
for a given
argument a only
exists
when
all
these four
are equal, and is then their common value. If it is also the value for the argument a, the function is continuous for this it is be taken as defining continuity equivalent to our former definition. We can define the limit of a function for a given argument
argument.
(if
it
This
exists)
and the four
may
:
without passing through the ultimate oscillation limits of the general case.
The
definition proceeds,
in that case, just as the earlier definition of continuity proceeded.
Let us define the limit for approaches from below. be a definite limit for approaches to a from below,
and
sufficient that, given
arguments
any small number
sufficiently near to a (but
both
less
cr,
If there is to it is
necessary
two values
than
for
a) will differ
and
Limits
Continuity of Functions
113
and our arguments then the difference both lie between a e and a (a excluded), between the values for these arguments will be less than cr. less
by
This
than
cr
;
i.e. if
to hold for
is
function has
e is sufficiently small,
any
a limit
however small
cr,
in that case the
;
approaches from below.
for
Similarly a limit for approaches from above. These two limits, even when both exist, need not be identical ; and if they are identical, they still need not be identical
we
when
define the case
there
is
It is only in this last case a. the function continuous for the argument a. " continuous " function is called (without qualification)
with the value for the argument
we
that
A when
call
it is
continuous for every argument.
method of reaching the definition the following " Let us say that a function ultimately converges into a " class a if there is some real number such that, for this argument Another
slightly different
of continuity
and is
" if
a
all
is
:
arguments greater than
member
converges into there
this,
the value of the function
we
say that a function " argument approaches x from below some argument y less than x such that throughout of the class a.
is
a
Similarly
shall
as the
the interval from y (included) to x (excluded) the function has values which are members of a. may now say that a
We
continuous for the argument
function
is
value fa,
if it satisfies
(1)
Given any
a, for
four conditions, namely
real
number
verges into the successors
approaches a from below
of
less
this
which
it
has the
:
than /#, the function con
number
as
the argument
;
real number greater than /, the function con into the verges predecessors of this number as the argument approaches a from below ; (2)
Given any
(3)
and
(4)
Similar conditions for approaches to a from above. of this form of definition is that it analyses
The advantages
the conditions of continuity into four, derived from considering and values arguments respectively greater or less than the
argument and value
for
which continuity
is
to be defined.
8
H4
Introduction
We may now
to
Mathematical Philosophy
generalise our definitions so as to apply to series
which are not numerical or known to be numerically measurable. The case of motion is a convenient one to bear in mind. There a story by H. G. Wells which will illustrate, from the case of motion, the difference between the limit of a function for a given argument and its value for the same argument. The hero of
is
who
the story,
possessed, without his knowledge, the power of was being attacked by a policeman, but on
realising his wishes,
"Go to
If f(t)
"
he found that the policeman disappeared. was the policeman's position at time t, and t the moment
ejaculating
of the ejaculation, the limit of
the policeman's positions as
t
from below would be in contact with the hero, But such occur whereas the value for the argument t was approached to
t
.
rences are supposed to be rare in the real world,
though without adequate evidence, that
all
and
it is
assumed, motions are continu
position at time t,f(t) " " the meaning of continuity involved in such statements which we now wish to define as ous,
is
i.e.
that, given
any body,
a continuous function of
t.
if /(*) is its
It
is
simply as possible.
The
definitions given for the case of functions
and value are
real
where argument
numbers can readily be adapted
more
for
general use.
Let serial,
P and Q be two though
should be is
so.
relations, which it is well to imagine not necessary to our definitions that they Let R be a one-many relation whose domain
it
is
contained in the
field of
P, while
R
Then
tained in the field of Q.
its is
converse domain
is
con
(in a generalised sense) a
whose arguments belong to the
field of Q, while its values belong to the field of P. Suppose, for example, that we let Q be the timeare dealing with a particle moving on a line
function,
:
R
P
the the series of points on our line from left to right, relation of the position of our particle on the line at time a to
series,
the time
a,
" the
R
of a
"
is its
position at time a.
This
may be borne in mind throughout our definitions. shall say that the function R is continuous for the argument
illustration
We
so that
and
Limits
Continuity of Functions
115
given any interval a on the P-series containing the value of the function for the argument #, there is an interval on the Q-series containing a not as an end-point and such that, through a
if,
out this interval, the function has values which are members " " of a. (We mean by an interval all the terms between any
x and y are two members of the
field of P, and x has " " P-interval x to y the relation P to y, we shall mean by the all terms z such that x has the relation P to x and z has the rela
two
tion
;
P
We mate
i.e. if
to y
together,
when
so stated, with x or y themselves.)
"
"
"
ulti and the ultimate section can easily define the " for " ultimate section oscillation." To define the
approaches to the argument a from below, take any argument y which precedes a (i.e. has the relation Q to a), take the values of the function for all
form the section of of the P-series
these values.
and take
their
P
arguments up to and including
defined
which are
Form all such common part
The ultimate upper
by these
earlier
section
values,
i.e.
than or identical
;
this will
all y's that precede a, be the ultimate section.
and the ultimate
oscillation are then
convergence
and the
no
difficulty
resulting alternative definition of continuity offers of any kind.
say that a function
and
sections for
defined exactly as in the previous case. The adaptation of the definition of
We
y,
members with some of
those
R
"
ultimately Q-convergent into a if there is a member y of the converse domain of and the field of Q such that the value of the function for the argument is
"
R
y and for any argument to which y has the relation Q is a member " We say that of a. Q-converges into a as the argument " a if there is a term approaches given argument a y having the relation Q to a and belonging to the converse domain of
R
R
and such that the value of the function for any argument Q-interval from y (inclusive) to a (exclusive) belongs to a. Of the four conditions that a function must to be continuous for the
argument
the value for the argument a
:
a,
the
first is,
fulfil
in the
in order
putting b for
1 1
6
Introduction
to
Mathematical Philosophy
P
Given any term having the relation
to
b,
R
Q-converges
into the successors of b (with respect to P) as the
argument
approaches a from below. The second condition converse
;
obtained by replacing P by its is the third and fourth are obtained from the first and
second by replacing Q by its converse. There is thus nothing, in the notions of the limit of a function or the continuity of a function, that essentially involves number. Both can be defined generally, and many propositions about them can be proved for any two series (one being the argumentseries
and the other the
definitions
value-series).
do not involve
It will
infinitesimals.
be seen that the
They involve
infinite
growing smaller without any limit short of but do not involve any intervals that are not finite. zero, they This is analogous to the fact that if a line an inch long be halved, classes of intervals,
then halved again, and so on indefinitely, we never reach infini after n bisections, the length of our bit is
tesimals in this 'way 2n be.
of
an inch
The
divisions
;
and
process
:
this is finite
of
whatever
successive
whose ordinal number
finite
bisection
is infinite,
number n may
does
since
not lead to
it is
essentially
a one-by-one process. Thus infinitesimals are not to be reached in this way. Confusions on such topics have had much to do
with the infinity
difficulties
which have been found in the discussion of
and continuity.
CHAPTER
XII
SELECTIONS AND THE MULTIPLICATIVE AXIOM IN this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic, and which is con venient, though not indispensable, in certain portions of mathe matics.
It is convenient, in the sense that
propositions,
many
interesting
seems natural to suppose true, cannot help ; but it is not indispensable, because
which
it
be proved without its even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form. Before enunciating the multiplicative axiom, we must first explain the theory of selections, and the definition of multi plication
when the number
of factors
may
be
infinite.
In defining the arithmetical operations, the only correct pro cedure is to construct an actual class (or relation, in the case relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but of
it
is
prove the existence of the number Take, as the simplest example, the case of addition.
essential in order to
defined.
Suppose we are given a cardinal number ^, and a terms. How shall we define ju.+/z ? For fji
has
We
a which
this
purpose having //, terms, and they must not can construct such classes from a in various ways,
we must have two overlap.
class
classes
which the following is perhaps the simplest Form first all the ordered couples whose first term is a class consisting of a single member of a, and whose second term is the null-class ; of
then, secondly, form
:
all
the ordered couples whose 117
first
term
is
n8
Introduction
Mathematical Philosophy
to
the null-class and whose second term
a class consisting of a
is
These two classes of couples have no single member of a. member in common, and the logical sum of the two classes will have /z-f/*- terms. Exactly analogously we can define p,-\-v, is the number of some class a and v is the number given that /z,
of
some class j3. Such definitions,
as a rule, are merely a question of a suitable in the case of multiplication, where the
technical device.
But
number
may
of factors
be
important problems
infinite,
arise out
of the definition.
Multiplication difficulty.
when the number
Given two
of factors
a and
classes
no
finite offers
which the
of
j8,
is
has
first
terms and the second v terms, we can define fix v as the number ju, of ordered couples that can be formed by choosing the first term out of a and the second out of finition does
It will
]3.
not require that a and
j3
be seen that
de
this
should not overlap
it
;
even remains adequate when a and jS are identical. For example, Then the class let a be the class whose members are x l9 # 2 #3 .
,
which
is
used to define the product
(*i, *i), (*i, *a)> (*i> *B)
(#3 > *s)
5
/x
X p, is
(**> *i)
the class of couples
(*2>
* 2)>
(*2> *a)
:
(*3> *i),
>
(*s> *a)
This definition remains applicable when \i or v or both are infinite, and it can be extended step by step to three or four or
any
finite
number
No
of factors.
this definition, except that
it
difficulty
arises
as regards
cannot be extended to an
infinite
number of factors. The problem of multiplication when the number of factors may be infinite arises in this way Suppose we have a class K consisting of classes suppose the number of terms in each of is these classes given. How shall we define the product of all If we can frame our definition generally, it these numbers ? :
;
will
be applicable whether K
is
finite or infinite.
It
is
to be
observed that the problem is to be able to deal with the case when K is infinite, not with the case when its members are. If
and
Selections
the Multiplicative
Axiom
119
K is not infinite, the method defined above is just as applicable when its members are infinite as when they are finite. It is the case when K is infinite, even though its members may be finite, that we have to find a way of dealing with. The following method of defining multiplication generally is due to Dr Whitehead. It is explained and treated at length in * 80 ff., and vol. ii. * 114. Principia Mathematics*, vol. i. Let us suppose to begin with that K is a class of classes no two of which overlap say the constituencies in a country where
there
is
no plural voting, each constituency being considered Let us now set to work to choose one term
as a class of voters.
out of each class to be
when they
representative, as constituencies
its
members
elect
do
assuming that by law who is a voter in that
of Parliament,
each constituency has to elect a man constituency. We thus arrive at a class of representatives, who make up our Parliament, one being selected out of each con
How many
stituency.
Parliament are there of its voters, it
can make
different
possible ways Each constituency can
and therefore
JLC
;
constituencies
thus is
if
The
choices.
are independent of
?
it is
finite,
of
choosing a
select
any one
there are p voters in a constituency, choices of the different constituencies
obvious that, when the total number number of possible Parliaments
the
obtained by multiplying together the numbers of voters in the various constituencies. When we do not know whether the
is
number of constituencies is finite or infinite, we may take the number of possible Parliaments as defining the product of the numbers of the separate constituencies. This is the method products are defined. We must now drop our and proceed to exact statements. Let K be a class of classes, and let us assume to begin with that no two members of ic overlap, i.e. that if a and j3 are two different
by which
infinite
illustration,
members other.
of K, then
We
sists of just
tion
"
no member of the one "
shall call a class a " selection
one term from each
from K
if
every
member
member of JJL
of
is
a
member
of the
from K when it con K ; i.e. p, is a " selec
belongs to some
member
I2O
Introduction
and
of K,
in
Mathematical Philosophy
a be any member of
if
The
common. "
to
class of all
"
K, /i
"
and a have exactly one term " from K we shall call
selections
The number of terms in the multiplicative class of /c, i.e. the number of possible selections from K, is defined as the product of the numbers of the members the
multiplicative class
This definition
of K.
of K.
equally applicable whether K
is
finite
is
or infinite.
we can be wholly
Before
must remove the
with these definitions, we of K are to
satisfied
no two members
restriction that
For this purpose, instead of defining first a class overlap. " a called selection," we will define first a relation which we will call
"
a
from K
A
selector."
from every member of
if,
representative of that there
R
definition
A
to
is
a
and
;
this is to
is
x
is
a
a selector from
a
A
a
be
all
that
R
member a
of
/c,
a and has the The formal does. of
K is a one-many relation, and such that, if x has the domain,
member /c,
term which has the relation of
given any
member
is
class of classes
for its converse
relation to a, then
R
picks out one term as the
it
/c,
" be called a " selector
:
" selector " from a
having K If
member,
will
i.e. if,
just one term x which
is
relation
R
relation
of a.
and a
is
a
member
of K,
and x
is
the
R to a, we call x the " representative "
in respect of the relation R.
" selection " from K
selector
;
will
now be
and the multiplicative
defined as the
class, as before, will
domain
of a
be the class
of selections.
But when the members
of
K overlap, there
may be more selectors
than selections, since a term x which belongs to two classes a and j8 may be selected once to represent a and once to represent j3, giving rise to different selectors in the two cases, but to the
same
For purposes of defining multiplication, it is the Thus we define selectors we require rather than the selections. " The product of the numbers of the members of a class of " is the number of selectors from /c. classes K selection.
:
We
can define exponentiation by an adaptation of the above
and
Selections
Axiom
the Multiplicative
121
We
plan.
from v
might, of course, define /A" as the number of selectors But there are classes, each of which has ju, terms.
objections
to
definition, derived
this
axiom
multiplicative
(of
which we
sarily involved if it is adopted.
construction
shall
from the fact that the speak shortly)
We adopt
is
unneces
instead the following
:
class having (JL terms, and j3 a class having v terms. Let y be a member of j3, and form the class of all ordered couples that have y for their second term and a member of a for
Let a be a
There will be p such couples for a given y, since a may be chosen for the first term, and a has /z we now form all the classes of this sort that result
their first term.
any member members.
of
If
from varying
we obtain
altogether v classes, since y may be has v members. These v classes are each
y,
any member of j8, and j8 them a class of couples, namely, all the couples that can be formed of a variable member of a and a fixed member of j8. We define \L V as the number of selectors from the class consisting of
of
these v classes.
Or we may equally well
selections, for, since
the
number
define
ju,"
as the
number of
our classes of couples are mutually exclusive,
of selectors
is
the
same
as the
number
of selections.
A selection from our class of classes will be a set of ordered couples, of
be exactly one having any given member of jS second term, and the first term may be any member of a.
which there
for its
Thus
ju,"
is
will
defined
by the
selectors
from a certain
set of v classes
one having a certain structure p, and a more manageable composition than is the case in general. terms, but the set
each having
The relevance
is
of this to the multiplicative
axiom
will
appear
shortly.
What two
numbers it
as the
member and
applies to exponentiation applies also to the product of
cardinals.
j8
We
might define "jz.Xi'" as the sum of the having JJL terms, but we prefer to define
of v classes each
number of
has v
formed consisting of a a followed by a member of j5, where a has ^ terms terms. This definition, also, is designed to evade the of ordered couples to be
necessity of assuming the multiplicative axiom.
122
Introduction
to
Mathematical Philosophy
we can prove the usual formal laws of But there is one thing we multiplication and exponentiation. cannot prove we cannot prove that a product is only zero when one of its factors is zero. We can prove this when the number of factors is finite, but not when it is infinite. In other words, we cannot prove that, given a class of classes none of which is or that, given a class null, there must be selectors from them With our
definitions,
:
;
of
mutually exclusive classes, there
must be
at least one class
term out of each of the given classes. These things cannot be proved ; and although, at first sight, they seem
consisting of one
obviously true, yet reflection brings gradually increasing doubt, we become content to register the assumption and
until at last
consequences, as we register the axiom of parallels, without assuming that we can know whether it is true or false. The
its
assumption, loosely worded,
is
when we should expect them. of stating it precisely.
is
that selectors and selections exist
There are
We may
many equivalent ways begin with the following :
" Given any class of mutually exclusive classes, of which none null, there is at least one class which has exactly one term in
common
with each of the given classes." we will call the "
1
This proposition
We and is
multiplicative axiom." will first give various equivalent forms of the proposition, then consider certain ways in which its truth or falsehood
of interest to
mathematics.
multiplicative axiom is equivalent to the proposition that a product is only zero when at least one of its factors is zero ;
The
that, if any number of cardinal numbers be multiplied together, the result cannot be o unless one of the numbers concerned is o. i.e.
The
axiom
equivalent to the proposition that, be any relation, and K any class contained in the converse if domain of R, then there is at least one one-many relation implying multiplicative
is
R
R
and having K
for its converse
domain.
multiplicative axiom is equivalent to the assumption that a be any class, and K all the sub-classes of a with the exception
The
if
1
See Principia Mathematica, vol.
i.
* 88.
Also vol.
iii.
* 257-258.
and
Selections
the Multiplicative
of the null-class, then there is
is
Axiom
at least one selector
the form in which the axiom was
first
"
the learned world by Zermelo, in his
123
from
This
K.
brought to the notice of Beweis, dass jede
Menge
1
Zermelo regards the axiom as an wohlgeordnet werden kann." be confessed that, until he made It must truth. unquestionable mathematicians had used
it explicit,
it
without a qualm
would seem that they had done so unconsciously.
And
;
but
it
the credit
having made it explicit is entirely independent of the question whether it is true or false. The multiplicative axiom has been shown by Zermelo, in the
due to Zermelo
for
above-mentioned proof, to be equivalent to the proposition that every class can be well-ordered, i.e. can be arranged in a series in
which every sub-class has a class).
The
full
term (except, of course, the nullis difficult, but it is not
first
proof of this proposition
the general principle upon which it proceeds. It " Zermelo's axiom," i.e. it assumes uses the form which we call difficult to see
any class a, there is at least one one-many relation R whose converse domain consists of all existent sub-classes of a that, given
such that, if x has the relation R to f , then x is a member of f Such a relation picks out a " representative "
and which
is
.
from each sub-class
of course, it will often
;
sub-classes
have the same representative.
in effect,
to count off the
of
R
and
is
We
x r Then a except x 1
of a, one
by
one,
does,
by means
the representative take the representative of the class consisting
transfinite induction.
of a; call it
of all of
members
happen that two
What Zermelo
;
x2
call it
.
put
first
must be
It
different
from xl9
because every representative is a member of its class, and x is shut out from this class. Proceed similarly to take away X 2 , and
#3 be the representative of what is obtain a progression x^ X 2 xm
let
left.
In this
way we
first
assuming that a is not finite. We then take away the whole progression let #w be the In this way we can go on representative of what is left of a. ,
.
.
.
.
.
.,
;
until nothing
is left.
The
successive representatives will form a
1 Mathematische Annalen, vol. speak of it as Zermelo's axiom.
lix.
pp. 514-6.
In this form we shall
Introduction
124
to
Mathematical Philosophy
well-ordered series containing all the members of a. (The above a hint of the lines of of the This course, only is, general proof.) " is called Zermelo's theorem." proposition The multiplicative axiom is also equivalent to the assumption
that of any two cardinals which are not equal, one must be the If the axiom is false, there will be cardinals p and v greater.
We
neither less than, equal to, nor greater than v. have seen that Nj and 2 No possibly form an instance of such a pair. Many other forms of the axiom might be given, but the above
such that
ju-
is
most important of the forms known at present. As to the truth or falsehood of the axiom in any of its forms, nothing are the
known at present. The propositions that depend upon the axiom, without being known to be equivalent to it, are numerous and important. Take
is
first
the connection of addition and multiplication. We naturally sum of v mutually exclusive classes, each having
think that the jit
When
terms, must have p,Xv terms.
But when
proved.
v
is infinite, it
v
this
is finite,
can be
cannot be proved without the
multiplicative axiom, except where, owing to some special cir cumstance, the existence of certain selectors can be proved. The
Suppose way the multiplicative axiom enters in is as follows we have two sets of v mutually exclusive classes, each having ^ terms, and we wish to prove that the sum of one set has as many terms as the sum of the other. In order to prove this, we must :
Now, since there are in each case some one-one relation between the two sets of but what we want is a one-one relation between their Let us consider some one-one relation S between the Then if K and A are the two sets of classes, and a is some
establish a one-one relation. v classes, there classes
;
terms. classes.
member
is
of K, there will be a
correlate of
a
with respect to S.
and are therefore relations of a and
member
Now
similar.
There
The
trouble
jS.
j3
A which
of
a and
j3
are, accordingly, is
of A,
we have
/x
be the terms,
one-one cor
that there are so many.
order to obtain a one-one correlation of the
sum
will
each have
to pick out one selection
sum
from a
of
In
K with the
set of classes
Selections of correlators,
a with
of
j3.
and
the Multiplicative
Axiom
125
one class of the set being all the one-one correlators If K and A are infinite, we cannot in general know
we can know that the multi cannot establish the usual we Hence plicative and multiplication. addition kind of connection between that such a selection exists, unless
axiom
is
true.
This fact has various curious consequences. To begin with, *s N othat N 2 x commonly inferred from
=
=N
we know
this that the
sum
N
of
^
classes each
have N members, but
itself
having N
do not know that the number of terms in such a
N
members must
this inference is fallacious, since
sum
is
we
N XN
This has a bearing upon the theory prove that an ordinal which " has NO predecessors must be one of what Cantor calls the second class," i.e. such that a series having this ordinal number will have
nor consequently that
it is
N terms
.
It is easy to
of transfinite ordinals.
It is also easy to see that,
in its field.
progression of ordinals of the their limit
terms.
It
second
form at most the sum of N is
thence
inferred
if
we take any
the predecessors of classes each having N
class,
fallaciously,
unless
the multi
that the predecessors of the limit are N " second in number, and therefore that the limit is a number of the class." That is to say, it is supposed to be proved that any pro
axiom
plicative
is
true
gression of ordinals of the second class has a limit which
is again the corol with This an ordinal of the second proposition, lary that a} (the smallest ordinal of the third class) is not the
class.
limit of
any progression,
is
involved in most of the recognised In view of the way in
theory of ordinals of the second class.
which the multiplicative axiom is involved, the proposition and corollary cannot be regarded as proved. They may be true,
its
or they may not. All that can be said at present is that we do not know. Thus the greater part of the theory of ordinals of the second class must be regarded as unproved.
Another
know of N that
that pairs
it is
illustration
may
help to
make
the point clearer. might suppose that the
2XN =N Hence we must have N terms. But .
sometimes the
case,
this,
We sum
though we can prove
cannot be proved to happen always
126
Introduction
unless
we assume the
by the
millionaire
to
Mathematical Philosophy
multiplicative axiom. This is illustrated a pair of socks whenever he bought
who bought
a pair of boots, and never at any other time, and
who had such
a passion for buying both that at last he had N
pairs of boots
The problem is How many boots had pairs of socks. and how many socks ? One would naturally suppose that he had twice as many boots and twice as many socks as he had pairs of each, and that therefore he had N of each, since that and N O
:
he,
number
is not increased by doubling. But this is an instance of the difficulty, already noted, of connecting the sum of v classes
each having p terms with fjiXv. Sometimes this can be done, sometimes it cannot. In our case it can be done with the boots,
but not with the socks, except by some very
The reason
for the difference
tinguish right and
one out of each all
the
left
suggests
is
this
Among
artificial device.
boots
and therefore we can make a
left,
we can
dis
selection of
namely, we can choose all the right boots or but with socks no such principle of selection
pair,
boots
;
and we cannot be
itself,
:
sure, unless
we assume
the
multiplicative axiom, that there is any class consisting of one sock out of each pair. Hence the problem.
We may
put the matter in another way. To prove that a has N terms, it is necessary and sufficient to find some way of arranging its terms in a progression. There is no difficulty in this with the boots. The pairs are given as forming an NO , doing class
and therefore take the
left
of the pair all
Within each
as the field of a progression.
boot
pair,
and the
first
right second, keeping the order in this way we obtain a progression of
unchanged But with the socks we ;
the boots.
shall
have to choose arbi
with each pair, which to put first ; and an infinite number of arbitrary choices is an impossibility. Unless we can find a trarily,
rule for selecting,
that a selection
i.e.
is
a relation which
is
principle of selection. :
there will
we do not know
Of course,
in the
we always can find some For example, take the centres of mass be points p in space such that, with any
case of objects in space, like socks, of the socks
a selector,
even theoretically possible.
Selections
and
the Multiplicative
Axiom
127
mass of the two socks are not both at exactly same distance from the p ; thus we can choose, from each pair, that sock which has its centre of mass nearer to p. But there is no theoretical reason why a method of selection such as this should always be possible, and the case of the socks, with a little goodwill on the part of the reader, may serve to show how a pair, the centres of
might be impossible. be observed that, if it were impossible to select one out each pair of socks, it would follow that the socks could not be
selection
It is to
of
arranged in a progression, and therefore that there were not N of them. This case illustrates that, if fj, is an infinite number,
one set of
p
pairs
may
not contain the same number of terms as
; for, given N pairs of boots, there are but we cannot be sure of this in the case of boots, certainly the socks unless we assume the multiplicative axiom or fall back
another set of
p,
pairs
N
upon some
fortuitous geometrical
method
of selection such as
the above.
Another
axiom will
is
important problem involving the multiplicative the relation of reflexiveness to non-inductiveness. It
be remembered that in Chapter VIII. we pointed out that a number must be non-inductive, but that the converse
reflexive
(so far as is known at present) can only be proved if we assume the multiplicative axiom. The way in which this comes about is
as follows
:
It is easy to
prove that a reflexive
class is
one which contains
N terms. (The class may, of course, itself Thus we have to prove, if we can, that, given
sub-classes having
have N terms.)
any non-inductive
class, it is possible to
out of
Now
its
terms.
there
is
no
choose a progression difficulty in showing that
a non-inductive class must contain more terms than any inductive class, or, what comes to the same thing, that if a is a non-induc
and v is any inductive number, there are sub-classes a that have v terms. Thus we can form sets of finite sub classes of a First one class having no terms, then classes having I term (as many as there are members of a), then classes having tive class of
:
128
Introduction
2 terms,
Mathematical Philosophy
to
We
and so on.
thus get a progression of sets of sub
each set consisting of all those that have a certain given number of terms. So far we have not used the multiplica
classes, finite
tive axiom,
but we have only proved that the number of
tions of sub-classes of
a
a reflexive number, 2* is the
is
number of members of a, so that ^ classes of a and 2 2 is the number of the
i.e.
that,
number
collec
p
if
collections of sub-classes, *
not inductive, 2 2f must be reflexive. from what we set out to prove.
But
is
then, provided JLC this is a long way
is
of sub
In order to advance beyond this point, we must employ the multiplicative axiom. From each set of sub-classes let us choose out one, omitting the sub-class consisting of the nullclass alone. That is to say, we select one sub-class containing
one term,
04,
say
;
one containing two terms, a 2 say ,
;
one con
(We can do this if the multipli taining three, a 3 say and so on. cative axiom is assumed otherwise, we do not know whether ;
,
;
we can always do a i a 2> as>
f
collections of sub-classes
We now know
goal. if
is
ju,
We
have now a progression not.) sub-classes of a, instead of a progression of or
it
thus
;
we
are one step nearer to our
assuming the multiplicative axiom, a non-inductive number, 2* must be a reflexive number. that,
The next step is to notice that, although we cannot be sure that new members of a come in at any one specified stage in the we can be sure that new members progression a x a 2 a 3 ,
,
,
.
.
.
Let us illustrate. keep on coming in from time to time. class c^, which consists of one term, is a new beginning;
The let
the one term be xv or
may
may
The
not contain x1
;
class
a 2 consisting
if it
it
,
does,
of
two terms,
introduces one
new
must introduce two new terms, say possible that a3 consists of xl9 #2 xst and so introduces no new terms, but in that case a 4 must introduce a v contain, at a new term. The first v classes a ly a 2 a3 term
;
#2 xz ,
.
and
if it
does not,
In this case
it
it is
,
,
.
.
.
+" terms, j/(v+i)/2 terms; there were no repetitions in the v classes, to go on with only repetitions from the
the very most, 1+2+3+ thus it would be possible, first
,
i.e.
if
Selections
and
Axiom
the Multiplicative
129
th v(v+i)/2 class. But by that time the old terms would no longer be sufficiently numerous to form a next class
class to the
with the right number of members, i.e. v(i/-|-i)/2-[-i, therefore new terms must come in at this point if not sooner. It all follows that, if we omit from our progression 04, a 2 , a3 , .
,
,
those classes that are composed entirely of members that have occurred in previous classes, we shall still have a progression.
Let our new progression be called fi l9 j8 2 j8 3 (We shall have a>i=pi and a 2 =j3 2 because a x and a 2 must introduce new .
.
,
.
.
,
We may or may not have a3 =j83
terms. will
p^
a,,
where v
them, say
Now
contains
jS^,
,
but, speaking generally,
some number greater than p
is
are some of the a's.)
j8's
of
be
these
jS's
;
are such that
the
i.e.
any one
members which have not occurred
in
Let y^ be the part of /^ which consists j8's. Thus we get a new progression y l9 y 2 y3 if a 2 does not (Again y5 will be identical with j8j and with c^ contain the one member of al9 we shall have y 2 =j3 2 =a 2 but if
any
of
of the previous
new members.
,
,
.
.
.
;
,
a 2 does contain
this
member
This
a 2 .)
of
gression
a
and
i.e. if
;
member
xl
of y s ,
will consist of the other
new progression of y's consists of mutually Hence a selection from them will be a pro
exclusive classes.
is
one member, y 2
is
the
member
and so on
;
of y l9 x 2 is a then xl9 # 2 #3 , .
,
.
member .
is
of y a ,
xs
a progression,
a sub-class of a.
Assuming the multiplicative axiom, such a selection can be made. Thus by twice using this axiom we can prove that, if the axiom is true, every non-inductive cardinal must be reflexive. This could also be deduced from is
Zermelo's theorem, that, if the axiom is true, every class can be well ordered ; for a well-ordered series must have either a finite or a reflexive
There
number
of
terms in
its field.
one advantage in the above direct argument, as deduction from Zermelo's theorem, that the above against does not demand the universal truth of the multi argument is
plicative axiom, but only its truth as applied to a set of It
may happen
for larger
that the axiom holds for
numbers
of classes.
For
N
classes,
N
classes.
though not
this reason it is better,
9
when
Introduction
130 it
is
to
Mathematical Philosophy
to content ourselves
possible,
assumption. The assumption made ment is that a product of N factors the factors
" N "
is
We may
zero.
with the more restricted
above direct argu never zero unless one of
in the is
state this assumption in the
form
:
a multipliable number," where a number v is defined as " when a product of v factors is never zero unless multipliable is
We
can prove that a finite number is always multipliable, but we cannot prove that any infinite number
one of the factors
The
is so.
that
all
is
zero.
multiplicative
cardinal
axiom
is
equivalent to the assumption But in order to
numbers are multipliable.
identify the reflexive with the non-inductive, or to deal with the
problem of the boots and socks, or to show that any progression of numbers of the second class is of the second class, we only
much
need the very
smaller assumption that
N
is
multipliable.
not improbable that there is much to be discovered in regard to the topics discussed in the present chapter. Cases the be found where which seem to involve may propositions It is
It is conceivable multiplicative axiom can be proved without it. that the multiplicative axiom in its general form may be shown
to be false.
From
the best hope
:
this point of view, Zermelo's theorem offers the continuum or some still more dense series
might be proved to be incapable of having its terms well ordered, which would prove the multiplicative axiom false, in virtue of Zermelo's theorem. results has
obscurity.
But so
far,
no method
of obtaining such
been discovered, and the subject remains wrapped in
CHAPTER
XIII
THE AXIOM OF INFINITY AND LOGICAL TYPES
THE axiom as follows
"
of infinity is
an assumption which
may
n be any inductive cardinal number, there of individuals having n terms."
If
class
be enunciated
:
If this is true, it follows, of course,
is
that there are
at least one
many
classes
having n terms, and that the total number of individuals in the world is not an inductive number. For, by the axiom, there is at least one class having n-f- 1 terms, from which it follows that there are many classes of n terms and that n is not the number of individuals in the world. Since n is any of individuals
inductive number,
number.
it
follows that the
number
of individuals
must
(if our axiom be true) exceed any inductive In view of what we found in the preceding chapter,
in the world
about the possibility of cardinals which are neither inductive nor reflexive, we cannot infer from our axiom that there are at
N individuals, unless we assume the multiplicative axiom. But we do know that there are at least N classes of classes, since the inductive cardinals are classes of classes, and form a progression if our axiom is true. The way in which the need for this axiom arises may be explained as follows One of Peano's assumptions is that no two inductive cardinals have the same successor, i.e. that we shall not have ra-f !=-{- 1 unless least
:
m
In Chapter VIII. we had occasion to use what is virtually the same as the above assumption of Peano's, namely, that, if n is an inductive cardinal,
m=n,
if
and n are inductive cardinals.
Introduction
132 n
Mathematical Philosophy
to
not equal to w-f-i. It might be thought that this could be proved. We can prove that, if a is an inductive class, and n
is
is
the
number
of
This proposition
members is
of a, then
is
is
not equal to
+i.
proved by induction, and might be
easily
thought to imply the other. might be no such class as
n
n
But
in fact
What
it
does not, since there
If does imply is this class an inductive cardinal such that there is at least one
having n members, then n
a.
is
it
:
not equal to n-\-i.
The axiom
of
infinity assures us (whether truly or falsely) that there are classes having n members, and thus enables us to assert that n is not
equal to +i. But without this axiom we should be left with the possibility that n and n-\-i might both be the null-class.
Let us
illustrate this
possibility
by an example
:
Suppose
there were exactly nine individuals in the world. (As to what " is meant by the word individual," I must ask the reader to
be patient.) Then the inductive cardinals from o up to 9 would be such as we expect, but 10 (defined as 9+ 1 ) would be the It will be remembered that n-\-i may be defined as null-class. follows
tt-j- I
:
is
the collection of
term x such that, when x of
all
those classes which have a
taken away, there remains a class
Now applying this definition, we 9+1 is a class consisting of no
n terms.
case supposed, the null-class.
9+w,
is
unless
n
The same will be true Thus 10 and zero.
is
of all
see that, in the classes,
i.e.
it is
or generally of subsequent inductive
9+2,
all be identical, since they will all be the null-class. In such a case the inductive cardinals will not form a progression,
cardinals will
nor will
and 10
it
will
be true that no two have the same successor, for 9 both be succeeded by the null-class (10 being itself
the null-class). It is in order to prevent such arithmetical catastrophes that we require the axiom of infinity.
As a matter metic of
of fact, so long as
we
are content with the arith
and do not introduce
finite integers,
either infinite
integers or infinite classes or series of finite integers or ratios, it is
possible to obtain
infinity.
That
is
all
to say,
desired results without the
we can
axiom
of
deal with the addition, multi-
The Axiom of
and Logical Types
Infinity
133
and exponentiation of finite integers and of ratios, but we cannot deal with infinite integers or with irrationals.
plication,
Thus the theory
How
fails us.
of the transfinite
of real numbers come about must now be
and the theory
these various results
explained.
Assuming that the number of individuals in the world is n, number of classes of individuals will be 2 n This is in virtue of the general proposition mentioned in Chapter VIII. that the the
.
number n is 2
of classes contained in a class
Now
.
2n
of classes in the
which has n members
is always greater than n. Hence the number world is greater than the number of individuals.
now, we suppose the number of individuals to be 9 i.e. 512. just now, the number of classes will be 2 If,
,
9, as
we
Thus
did
we
if
take our numbers as being applied to the counting of classes instead of to the counting of individuals, our arithmetic will
be normal until we reach 512 the first number to be null be 513. And if we advance to classes of classes we shall do :
better
:
the
number
of
them
will
be 2 512 , a number which
large as to stagger imagination, since
it
has about 153
will still
is
so
digits.
And
if we advance to classes of classes of classes, we shall obtain number represented by 2 raised to a power which has about the number of digits in this number will be about 153 digits
a
;
three times io 152
.
to write out this
In a time of paper shortage
number, and
if
we want
it is
undesirable
larger ones
we can
obtain them by travelling further along the logical hierarchy. In this way any assigned inductive cardinal can be made to find its place
among numbers which
are not null, merely
by
1 travelling along the hierarchy for a sufficient distance. As regards ratios, we have a very similar state of affairs.
If a ratio p,/v is to have the expected properties, there must be enough objects of whatever sort is being counted to insure that the null-class does not suddenly obtrude itself. But this
can be insured, for any given ratio
JJL/V,
without the axiom of
On this subject see Principia Mathematica, vol. ii. * 120 ff. corresponding problems as regards ratio, see ibid., vol. iii. * 303 1
On ff.
the
Introduction
134 infinity,
by merely
to
Mathematical Philosophy
travelling
up the hierarchy a
sufficient distance.
we cannot succeed by counting individuals, we can try counting if we still do not succeed, we can classes of individuals try If
;
of classes,
classes
and so on.
Ultimately, however few indi
may be in the world, we shall reach a stage where many more than /x objects, whatever inductive number
viduals there there are
Even
be.
p may
there were no individuals at
if
all,
would
this
be true, for there would then be one
class, namely, the null(namely, the null-class of classes and the class whose only member is the null-class of individuals), 4 classes of classes of classes, 16 at the next stage, 65,536 at the next
still
class, 2 classes of classes
stage,
and so on.
infinity
Thus no such assumption as the axiom of required in order to reach any given ratio or any given
is
inductive cardinal. It is
when we wish
to deal with the whole class or series of
inductive cardinals or of ratios that the
need the whole
axiom
is
required.
We
class of inductive cardinals in order to establish
the existence of
N
and the whole
,
series in order
to establish
the existence of progressions for these results, it is necessary that we should be able to make a single class or series in which :
We
no inductive cardinal
is null.
in order of
in order to define real
need the whole
series of ratios
numbers
as segments not give the desired result unless the series compact, which it cannot be if the total number of
magnitude
:
this definition will
of ratios is
ratios, at the stage concerned, is finite.
would be natural to suppose as I supposed myself in former days that, by means of constructions such as we have been It
considering, the said
:
axiom
of infinity could be proved.
It
Let us assume that the number of individuals
n may be o without
spoiling our
argument
;
then
if
may
is n,
be
where
we form
the
set of individuals, classes, classes of classes, etc., all
complete taken together, the number of terms in our whole set
which
is
N
.
Thus taking
all
will
be
kinds of objects together, and not
The Axiom of
and Logical Types
Infinity
135
confining ourselves to objects of any one type, we shall certainly obtain an infinite class, and shall therefore not need the axiom
So
of infinity.
it
might be
said.
before going into this argument, the that there is an air of hocus-pocus about it
Now,
first
thing to observe
something reminds one of the conjurer who brings things out of the hat. The man who has lent his hat is quite sure there wasn't a live rabbit in it
is
before, but
the reader,
he if
is
at a loss to say
how
he has a robust sense
:
the rabbit got there.
of reality, will feel
So
convinced
impossible to manufacture an infinite collection out of a finite collection of individuals, though he may be unable to that
it is
say where the flaw is in the above construction. It would be a mistake to lay too much stress on such feelings of hocus-pocus ; But they like other emotions, they may easily lead us astray. afford a
prima facie ground for scrutinising very closely any argument which arouses them. And when the above argument is scrutinised it will, in my opinion, be found to be fallacious, though the fallacy
is
a subtle one and
by no means easy
to avoid
consistently.
The
"
con the fallacy which may be called " " would fusion of types." To explain the subject of fully types require a whole volume ; moreover, it is the purpose of this book fallacy involved
is
to avoid those parts of the subjects which are still obscure and controversial, isolating, for the convenience of beginners, those parts which can be accepted as embodying mathematically ascer tained truths. Now the theory of types emphatically does not
belong to the finished and certain part of our subject this theory
of
is still
inchoate, confused,
some doctrine of types
the doctrine should take infinity
it is
;
and obscure.
:
much
of
But the need
doubtful than the precise form and in connection with the axiom of
is less
particular'y easy to see the necessity of
some such
doctrine.
This necessity results, for example, from the " contradiction of the greatest cardinal." We saw in Chapter VIII. that the number of classes contained in a given class
is
always greater than the
Introduction
136
number
members
of
to
Mathematical Philosophy and we inferred that there is But if we could, as we suggested
of the class,
no greatest cardinal number. a moment ago, add together into one
class the individuals, classes
of individuals, classes of classes of individuals, etc.,
obtain a class of which
The
its
sub-classes
class "consisting of all objects that
sort,
if
must,
Since
the greatest possible. of it, there cannot be
Hence we
When
more
can be counted, of whatever
number which members
sub-classes will be
all its
them than
of
we should
would be members.
there be such a class, have a cardinal
is
I
own
there are members.
arrive at a contradiction.
came upon this contradiction, in the year 1901, some flaw in Cantor's proof that there is
I first
attempted to discover
no greatest cardinal, which we gave in Chapter VIII.
Apply
ing this proof to the supposed class of all imaginable objects, I was led to a new and simpler contradiction, namely, the following
:
The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other " " such a thing as everything," then every " is something, and is a member of the class everything." But normally a class is not a member of itself. Mankind, for
words, " thing
there
if
Form now
not a man.
is
example,
is
which are not members
member are not
of itself or not
themselves,
its
There lib.
types
is
it
is,
Vol.
is, it is
:
one of those classes that
a
member
and that
contradictory.
no
Thus
of itself.
not, a
it is
This
is
of the
member
two hypo
of itself
each
a contradiction.
manufacturing similar contradictions The solution of such contradictions by the theory of is
difficulty in
Mathematical and also, more by the present author in the American Journal
set forth fully in Principia
briefly, in articles 1
If it
of themselves, i.e. it is not a member of itself. not one of those classes that are not members of
i.e. it is
that
theses
ad
?
members
If it is not, it is
implies
the assemblage of all classes is it a This is a class
of themselves.
i.,
Statement.
Introduction, chap,
ii.,
# 12
and
* 20;
vol
ii.,
Prefatory
The Axiom of
Infinity
and Logical Types
137
1 of Mathematics and in the Revue de Metaphysique et de Morale? For the present an outline of the solution must suffice.
The "
fallacy consists in the formation of
"
impure
As we
classes,
i.e.
what we may
call
which are not pure as to " type." chapter, classes are logical fictions, and
classes
shall see in a later
a statement which appears to be about a class will only be signi if it is capable of translation into a form in which no mention
ficant is
made
This places a limitation upon the ways in
of the class.
which what are nominally, though not can occur significantly a sentence or
really,
names
for classes
set of symbols in which such pseudo-names occur in wrong ways is not false, but strictly devoid of meaning. The supposition that a class is, or that it :
member
of itself is meaningless in just this way. And to generally, suppose that one class of individuals is a member, or is not a member, of another class of individuals
is
not, a
more will
be to suppose nonsense ; and to construct symbolically any whose members are not all of the same grade in the logical
class
hierarchy
is
to use symbols in a
way which makes them no
longer symbolise anything. Thus if there are n individuals in the world, and 2 n classes of
we cannot form
a new class, consisting of both and having w-f-2 n members. In this way the attempt to escape from the need for the axiom of infinity breaks down. I do not pretend to have explained the doctrine of types, or done more than indicate, in rough outline, why there
individuals,
individuals and classes
need of such a doctrine.
I have aimed only at saying just was required in order to show that we cannot 'prove the existence of infinite numbers and classes by such conjurer's methods as we have been examining. There remain, however,
is
much
so
as
certain other possible
methods which must be considered.
Various arguments professing to prove the existence of infinite classes are given in the Principles of Mathematics, 1
"
Mathematical Logic as based on the Theory of Types,"
1908, pp. 222-262.
"
339
Les paradoxes de
la logique," 1906, pp.
627-650.
(p. 357). vol. xxx.,
Introduction
138
to
Mathematical Philosophy
In so far as these arguments assume that, if n is an inductive cardinal, n is not equal to n-\-i, they have been already dealt
There is an argument, suggested by a passage in Plato's ParmfnidlSy to the effect that, if there is such a number as I, then I has being ; but I is not identical with being, and therefore with.
and being are two, and therefore there is such a number as 2, and 2 together with I and being gives a class of three terms, and I
This argument
so on.
is
fallacious, partly
"
"
because
being
is
not a term having any definite meaning, and still more because, if a definite meaning were invented for it, it would be found that
numbers do not have being "
logical fictions,'' as
we
they
shall see
are, in fact, what are called when we come to consider
the definition of classes.
The argument that the number of numbers from o to n (both inclusive) is n-\-i depends upon the assumption that up to and including n no number is equal to its successor, which, as we have if the axiom of It infinity is false. must be understood that the equation n=n-\-i, which might be true for a finite n\in exceeded the total number of individuals in the world, is quite different from the same equation as applied to a reflexive number. As applied to a reflexive number, it means that, given a class of n terms, this class is " similar " to that obtained by adding another term. But as applied to a number which is too great for the actual world, it merely means that there is no class of n individuals, and no class of n-\-\ indi it does not mean that, if we mount the viduals hierarchy of types sufficiently far to secure the existence of a class of n terms,
seen, will not be always true
;
we
shall
then find this class " similar " to one of
n-\- 1 terms, for inductive this will not be the case, quite independently of the truth or falsehood of the axiom of infinity.
if
n
is
There
is
an argument employed by both Bolzano
1
and Dede-
to prove the existence of reflexive classes. The argument, in brief, is this : An object is not identical with the idea of the
kind
2
1
1
Bolzano, Paradoxien des Unendlichen, 13. Dedekind, Was sind und was sollen die Zahlen ?
No. 66.
The Axiom of object, but there
is
Infinity
(at least in the
and Logical Types
139
realm of being) an idea of any
of it is one-one, and object. " idea ideas are only some among objects. Hence the relation " of constitutes a reflexion of the whole class of objects into a
The
relation of
an object to the idea
namely, into that part which consists of ideas. the class of objects and the class of ideas are both Accordingly, infinite. This argument is interesting, not only on its own part of
itself,
account, but because the mistakes in it (or what I judge to be mistakes) are of a kind which it is instructive to note. The
main
error consists in assuming that there
object.
It
is,
is
an idea of every
of course, exceedingly difficult to decide
meant by an " idea "
;
but
let
we know.
us assume that
what
is
We are
then to suppose that, starting (say) with Socrates, there is the idea of Socrates, and so on ad inf. Now it is plain that this is not the case in the sense that
all
existence in people's minds.
these ideas have actual empirical Beyond the third or fourth stage
they become mythical. If the argument is to be upheld, the " " ideas intended must be Platonic ideas laid up in heaven, for
But then it at once becomes certainly they are not on earth. doubtful whether there are such ideas. If we are to know that basis of some logical theory, proving to a thing that there should be an idea of it. necessary certainly cannot obtain this result empirically, or apply it,
there are,
that
We as
it
must be on the
it is
Dedekind does, to " meine Gedankenwelt "
the world of
my
thoughts.
we were concerned to examine fully the relation of idea and object, we should have to enter upon a number of psychological If
and
logical inquiries, which are not relevant But a few further points should be noted.
to our If
main purpose.
" idea "
is
to be
understood logically, it may be identical with the object, or it may stand for a description (in the sense to be explained in a
In the former case the argument fails, subsequent chapter). because it was essential to the proof of reflexiveness that object and idea should be distinct. In the second case the argument also fails, because the relation of object
and description
is
not
Introduction
140
to
Mathematical Philosophy
there are innumerable correct descriptions of any given " Socrates the master of (e.g) may be described as object. " as the or who drank the Plato," hemlock," or as philosopher
one-one
:
" the husband of Xantippe." If to take up the remaining " " is to be idea hypothesis interpreted psychologically, it must be maintained that there is not any one definite psychological entity which could be called the idea of the object
:
there are in
numerable beliefs and attitudes, each of which could be called an " idea of the object in the sense in which we might say my idea of Socrates is quite different from yours," but there is not any central entity (except Socrates himself) to bind together various
" ideas of Socrates," and thus there tion of idea as
we have
ideas (in
and object
as the
already noted,
is it
however extended a
is
not any such one-one rela
argument supposes. sense) of
of the things in the world.
Nor, of course,
true psychologically that there are
For
more than a tiny proportion
all
these reasons, the above
in favour of the logical existence of reflexive classes
argument must be rejected. It
might be thought that, whatever
may
be said of
logical
arguments, the empirical arguments derivable from space and time, the diversity of colours, etc., are quite sufficient to prove the actual existence of an infinite number of particulars. I do
We have no
reason except prejudice for believ ing in the infinite extent of space and time, at any rate in the sense in which space and time are physical facts, not mathematical
not believe
fictions.
this.
We
naturally regard space and time as continuous, or,
at least, as
compact
"
"
but this again is mainly prejudice. The in physics, whether true or false, illustrates
;
theory of quanta the fact that physics can never afford proof of continuity, though The senses are not it might quite possibly afford disproof. sufficiently exact to distinguish
between continuous motion and
in a cinema. rapid discrete succession, as anyone may discover world in which all motion consisted of a series of small finite
A
would be empirically indistinguishable from one in which motion was continuous. It would take up too much space to
jerks
The Axiom of
and Logical Types
Infinity
141
defend these theses adequately ; for the present I am merely suggesting them for the reader's consideration. If they are valid, it
follows that there
number
is
no empirical reason for believing the world to be infinite, and that there
of particulars in the
never can be
;
also that there is at present
no empirical reason
number to be finite, though it is theoretically conceivable that some day there might be evidence pointing, to believe the
though not conclusively, in that direction.
From
the fact that the infinite
is
not self-contradictory, but
is
logically, we must conclude that nothing can be known a priori as to whether the number of things in the world is finite or infinite. The conclusion is, therefore,
also not
demonstrable
to adopt a Leibnizian
worlds are
knowing
some
phraseology, that some of the possible
of infinity will be
false in others
we cannot
and we have no means of two kinds our actual world belongs.
infinite,
to which of these
The axiom and
finite,
;
whether
in some possible worlds true or false in this world,
true is
it
tell.
" " individual and chapter the synonyms have been used without It be would particular explanation. to them without a impossible explain longer disquisi adequately tion on the theory of types than would be appropriate to the
Throughout
"
this
"
we
present work, but a few words before
leave this topic
may
do something to diminish the obscurity which would otherwise envelop the meaning of these words. In an ordinary statement we can distinguish a verb, expressing an attribute or relation, from the substantives which express the " Caesar subject of the attribute or the terms of the relation. " " " lived ascribes an attribute to Caesar
;
Brutus
killed Caesar
expresses a relation between Brutus and Caesar. Using the word in a "subject" generalised sense, we may call both Brutus and Caesar subjects of this proposition
:
the fact that Brutus
matically subject and Caesar object is the same occurrence may be expressed killed
by Brutus," where
Caesar
is
is
gram
logically irrelevant, since
in the
words " Caesar was
the grammatical subject.
Introduction
142
to
Mathematical Philosophy
in the simpler sort of proposition we shall have an attribute " " or relation holding of or between one, two or more subjects in the extended sense. (A relation may have more than two
Thus
"
A gives B
"
is a relation of three terms.) Now often happens that, on a closer scrutiny, the apparent subjects are found to be not really subjects, but to be capable of analysis ; the only result of this, however, is that new subjects take their
terms
:
e.g.
to
C
it
places.
It also
made
happens that the verb " we
may
grammatically be a relation which
subject e.g. may say, Killing holds between Brutus and Caesar." But in such cases the :
is
misleading, and in a straightforward statement, the rules that should guide philosophical grammar, following Brutus and Cssar will appear as the subjects and killing
grammar
is
as the verb.
We are thus led to the conception of terms which, when they occur in propositions, can only occur as subjects, and never in any other way. This is part of the old scholastic definition of substance ; but persistence through time, which belonged to that notion, forms no part of the notion with which we are con We shall define " proper names " as those terms which can only occur as subjects in propositions (using " subject " in the extended sense just explained). We shall further define " " " " individuals or as the objects that can be particulars cerned.
named by proper names. directly, rather than they are symbolised
(It
by means ;
would be better
to define
them
symbols by which but in order to do that we should have of the kind of
to plunge deeper into metaphysics than is desirable here.) It that is, of course, possible that there is an endless regress : whatever appears as a particular is really, on closer scrutiny,
some kind of complex. If this be the case, the axiom of infinity must of course be true. But if it be not the case, it must be for possible theoretically analysis to reach ultimate " " and it is these that give the meaning of particulars subjects, " or It is to the number of these that the axiom individuals." of infinity is assumed to apply. If it is true of them, it is true a class or
The Axiom of of classes of them,
similarly
if it is
and
false of
Hence it is natural
Infinity
and Logical Types
classes of classes of
them,
it is
false
143
them, and so on
throughout
;
this hierarchy.
to enunciate the axiom concerning them rather than concerning any other stage in the hierarchy. But whether the axiom is true or false, there seems no known method of
discovering.
CHAPTER XIV INCOMPATIBILITY AND THE THEORY OF DEDUCTION
WE
have now explored, somewhat hastily it is true, that part mathematics which does not demand a
of the philosophy of critical
examination of the idea of
chapter, however,
we
In the preceding found ourselves confronted by problems class.
which make such an examination imperative. Before we can undertake it, we must consider certain other parts of the philos which we have hitherto ignored. In a synthetic treatment, the parts which we shall now be concerned with come first they are more fundamental than anything
ophy
of mathematics,
:
we have discussed hitherto. Three topics before we reach the theory of classes, namely that
:
will
concern us
(i)
the theory
Of prepositional functions, (3) descriptions. these, the third is not logically presupposed in the theory of of deduction,
(2)
but it is a simpler example of the kind of theory that needed in dealing with classes. It is the first topic, the theory of deduction, that will concern us in the present chapter. classes,
is
Mathematics
starting from certain a strict arrives, by process of deduction, at the premisses, various theorems which constitute it. It is true that, in the past, is
a deductive science
:
it
mathematical deductions were often greatly lacking in rigour true also that perfect rigour Nevertheless, in so far as rigour
it is
is
is
;
a scarcely attainable ideal. lacking in a mathematical
proof, the proof is defective ; it is no defence to urge that common sense shows the result to be correct, for if we were to rely upon that, it
would be better to dispense with argument altogether, 144
and
Incompatibility
the
Theory of Deduction
145
rather than bring fallacy to the rescue of common sense. No " appeal to common sense, or intuition," or anything except strict
deductive logic, ought to be needed in mathematics after the premisses have been laid down. Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required
an appeal to the
figure,
reasoning according
invented a theory of mathematical is never strictly
which the inference
to
but always requires the support of what is called The whole trend of modern mathematics, with increased pursuit of rigour, has been against this Kantian
logical,
"
intuition."
its
The
theory.
things in the mathematics of Kant's day which
cannot be proved, cannot be known for example, the axiom of What can be known, in mathematics and by mathe parallels. matical methods, else is to
wise
is
belong to
what can be deduced from pure logic. What human knowledge must be ascertained other
empirically, through the senses or through experience in
some form, but not a
priori.
The
positive grounds for this
are to be found in Principia Mathematica, passim ; a controversial defence of it is given in the Principles of Mathe thesis
We
matics.
cannot here do more than refer the reader to those
works, since the subject
we
is
too vast for hasty treatment.
assume that
Mean
deductive, and what is involved in deduction. In deduction, we have one or more propositions called pre
while,
shall
all
mathematics
is
proceed to inquire as to
misses,
from which we
For our purposes,
infer a proposition called the conclusion.
be convenient, when there are originally several premisses, to amalgamate them into a single proposition, so as to be able to speak of the premiss as well as of the con it will
Thus we may regard deduction as a process by which pass from knowledge of a certain proposition, the premiss, to knowledge of a certain other proposition, the conclusion.
clusion.
we
But we
shall not regard
it is correct,
i.e.
such a process as logical deduction unless is such a relation between premiss
unless there
and conclusion that we have a
right to believe the conclusion
10
Introduction
146
to
Mathematical Philosophy
we know
the premiss to be true. It is this relation that is chiefly of interest in the logical theory of deduction. In order to be able validly to infer the truth of a proposition,
if
we must know there
is
that (as
i.e.
that some other proposition is true, and that of the sort called "implication,"
between the two a relation
we
say) the premiss
"
" implies
the conclusion.
Or we may know that a
shall define this relation shortly.)
(We
certain
other proposition is false, and that there is a relation between " " 1 the two of the sort called disjunction," expressed by p or ^," so that the knowledge that the one is false allows us to infer that the other
true.
is
some
the falsehood of
Again, what we wish to infer This proposition, not its truth.
inferred from the truth of another proposition, provided
that the two are It
is false.
"
may
incompatible,"
proposition, in just the of the other i.e.
i.e.
also be inferred
that
if
one
is
in
be
we know
true, the other
from the falsehood
same circumstances
be
may may
of another
which the truth
might have been inferred from the truth of the one
from the falsehood
q implies p.
of
p we may
infer the falsehood of q,
All these four are cases of inference.
;
when
When
our
minds are fixed upon inference, it seems natural to take " impli " as the cation primitive fundamental relation, since this is the relation
which must hold between p and q
if
we
are to be able
But for technical to infer the truth of q from the truth of p. reasons this is not the best primitive idea to choose. Before proceeding to primitive ideas and definitions, let us consider further the various functions of propositions suggested by the
above-mentioned relations of propositions. The simplest of such functions is the negative, " not-^>." This is that function of p which is true when p is false, and false
when p
is
It is
true.
convenient to speak of the truth of a pro " "
2 truth-value i.e. truth is ; position, or its falsehood, as its " truth-value " of a true and the falsehood of a false proposition,
one.
Thus not-
has the opposite truth-value to p.
1
We shall use the letters p,
2
This term
is
due to Frege.
q, r, s, t
to denote variable propositions.
and
Incompatibility
Theory of Deduction
the
"
We may
take next disjunction, whose truth-value is truth when p
but
falsehood
is
when both p and
Next we may take has falsehood for
Take next
its
true
and
also
is
a function
when
is
q
true,
q are false.
" conjunction,
when p and
for its truth-value
This
p or
is
147
p and q"
q are both true
This has truth otherwise
;
it
truth-value.
incompatibility,
i.e.
"
p and q are not both true."
the negation of conjunction ; it is also the disjunction " Its truthof the negations of p and q, i.e. it is not-/) or not-y." value is truth when p is false and likewise when q is false ; its
This
is
is falsehood when p and q are both true. " " Last take implication, i.e. p implies q," or if p, then
truth-value
to infer the truth of q
pret
it
as
meaning
if
" :
we know Unless p
the truth of p. is
false, q
is
Thus we
true," or
"
inter either
" " is (The fact that implies capable of other meanings does not concern us ; this is the meaning which is convenient for us.) That is to say, " p implies q " is to mean " " is or its truth-value is to be truth if likewise
p
is
false or q is true."
not-/>
if
q
is
true,
q
:
and
is
to be falsehood
We have thus five functions: incompatibility, for
if
p
is
true
p and q
false,
is false.
negation, disjunction, conjunction,
and implication.
We
might have added others, and not-^," but the above not-p differs the other four in being from Negation "
example, joint falsehood,
five will suffice.
a function of one proposition, whereas the others are functions of two. But all five agree in this, that their truth-value depends
only upon that of the propositions which are their arguments. Given the truth or falsehood of p, or of p and q (as the case may
we
are given the truth or falsehood of the negation, disjunc function of tion, conjunction, incompatibility, or implication.
be),
A
" truth-function." propositions which has this property is called a The whole meaning of a truth-function is exhausted by the statement of the circumstances under which it is true or false. " Not-/)," for example, is simply that function of p which is true when p is false, and false when p is true there is no further :
Introduction
148
Mathematical Philosophy
to
" " meaning to be assigned to it. The same applies to p or q and the rest. It follows that two truth-functions which have the same truth-value for
values of the argument are indis For example, " p and q " is the negation of tinguishable. " " and vice versa ; thus either of these may be not-/) or not-^ the as defined negation of the other. There is no further meaning in a truth-function over it is
and above the conditions under which
true or false.
It is clear that the
pendent. is
all
above
five truth-functions are
not
all
We can define some of them in terms of others.
no great
difficulty in reducing the
number
to
two
;
inde
There the two
chosen in Principia Mathematica are negation and disjunction. " " ; Implication is then defined as not-/) or q incompatibility " " as as the or negation of incompati not-/> not-q ; conjunction But it has been shown by Sheffer * that we can be content bility.
with one primitive idea for all five, and by Nicod 2 that this enables us to reduce the primitive propositions required in the theory of deduction to two non-formal principles and one formal one.
For this purpose, we
may
take as our one indefinable either
incompatibility or joint falsehood. We will choose the former. Our primitive idea, now, is a certain truth-function called
"
incompatibility," which we will denote by p/q. Negation can be at once defined as the incompatibility of a proposition " " " with is defined as itself, i.e. Disjunction is not-/) />//>." the incompatibility of not-/) and not- Conjunction is the negation of incompatibility, i.e. it is (p/q) \
(p/q)'
Thus
all
our four other functions are defined in terms
of incompatibility. It is
obvious that there
functions, either
arguments. this subject 1 2
is
no limit to the manufacture
by introducing more arguments
What we
are concerned with
is
or
by repeating the connection of
with inference. Trans.
Am. Math.
Soc., vol. xiv. pp.
Proc. Camb. Phil. Soc., vol. xix.,
i.,
of truth-
481-488.
January 1917.
and
Incompatibility If
we know that p
to assert
q.
true and that
is
There
is
logical about inference
Theory of Deduction
the
:
p implies
q,
149
we can proceed
always unavoidably something psycho inference
new knowledge, and what
is
a
method by which we
arrive
not psychological about it is the relation which allows us to infer correctly ; but the actual passage at
is
from the assertion of p to the assertion of q is a psychological process, and we must not seek to represent it in purely logical terms.
In mathematical practice, when we
we have always
infer,
some expression containing variable propositions, say p and q which is known, in virtue of its form, to be true for all values we have also some other expression, part of the former, of p and q which is also known to be true for all values of p and q and in y
;
;
we
virtue of the principles of inference, of our original expression,
and
assert
are able to drop this part
what
This somewhat
is left.
may be made clearer by a few examples. Let us assume that we know the five formal principles of
abstract account
deduction enumerated in Principia Matbematica. (M. Nicod has reduced these to one, but as it is a complicated proposition,
we
begin with the
will
follows
" p or p implies p
" (1)
then p
is
These
five propositions
are as
i.e. if
either
p
is
true or
p
is
true,
true.
" (2)
five.)
:
q implies
when one
" p or q
i.e.
of its alternatives
is
the disjunction
"
p or
" q
is
true
true.
"
" p or q implies q or />." This would not be required we had a theoretically more perfect notation, since in the "
(3) if
conception of disjunction there is no order involved, so that " " and " q or p " should be identical. But since our p or q
symbols, in any convenient form, inevitably introduce an order, we need suitable assumptions for showing that the order is irrelevant. (4)
or
If either
"
p
or r
increase
its
"
p
is
true or
true.
"
q or r
"
is
true,
then either q
is
true
(The twist in this proposition serves to deductive power.) is
Introduction
150
Mathematica
to
Philosophy
then " p or q " implies " p or r." These are the formal principles of deduction employed in (5)
If q implies
r,
A
Principia Mathematica.
double use, and
it is
formal principle of deduction has a make this clear that we have
in order to
cited the above five propositions.
has a use as the premiss
It
an inference, and a use as establishing the fact that the pre miss implies the conclusion. In the schema of an inference we have a proposition p, and a proposition " p implies ," from which we infer q. Now when we are concerned with the princi of
ples of deduction, our apparatus of primitive propositions has " " of our inferences. to yield both the p and the p implies q
That
to say, our rules of deduction are to be used, not only as " rules, which is their use for establishing p implies q" but also is
as substantive premisses,
we wish
i.e.
as the
of our
p
schema.
Suppose,
p implies q, then if q r it follows that p here a relation of r. We have implies implies three propositions which state implications. Put for example,
pi=p
implies
Then we have take the
q,
to prove that
p 2 =q implies
to prove that
fifth of
if
and p 3 =p implies
r,
r.
p implies that p z implies p s
our above principles, substitute " "
Now
.
not-/> for p,
is and remember that not-/) or q by definition the same Thus our fifth principle yields q." implies p
as
"
:
"
If q implies r,
"
i.e.
then
'
'
p implies
q
'
implies
p 2 implies that p^ implies p 3 ."
'
p implies Call this
r,'
propo
sition A.
But the fourth
of our principles,
when we
substitute not-/),
not-, for p and q and remember the definition of implication, y
becomes " If p implies that q implies :
r,
then q implies that p implies
Writing p 2 in place of p, p in place of
becomes " If p z implies that p 1 implies
q,
and p 3
r."
in place of ry this
:
1>"
Call this B.
/> 3 ,
then p l implies that p 2 implies
Incompatibility
and
Theory of Deduction
the
1
5
1
Now we
proved by means of our fifth principle that " " p 2 implies that p^ implies p 3 which was what we called A.
Thus we have here an instance
A
since
"
the schema of inference, B represents the
of
represents the p of our scheme, and
p implies
Hence we
q."
"
pl
arrive at
q,
namely, "
implies that p z implies p 3
which was the proposition to be proved. In this proof, the adaptation of our fifth principle, which yields A, occurs as a substantive premiss
;
while the adaptation of our fourth principle,
which yields B, is used to give the form of the inference. The formal and material employments of premisses in the theory
and
of deduction are closely intertwined,
to keep
theory
The is
them separated, provided we
it is
not very important
realise that
they are in
distinct. earliest
one which
method
illustrated
is
new
from a premiss in the above deduction, but which
of arriving at
results
can hardly be called deduction. The primitive propositions, whatever they may be, are to be regarded as asserted for all possible values of the variable propositions p, q, r which occur
itself
in them.
We may therefore substitute for (say) p
any expression
"s implies t," always a proposition, e.g. not-p, By means of such substitutions we really obtain
whose value
is
and so on.
sets of special cases of
our original proposition, but from a prac
view we obtain what are virtually new propositions. The legitimacy of substitutions of this kind has to be insured by tical
point of
means
of a non-formal principle of inference. 1
We may now
state the one formal principle of inference to
which M. Nicod has reduced the purpose we
will first
show how
defined in terms of incompatibility.
p
|
(q/q)
five
certain
We
given above.
For
this
truth-functions can be
saw already that
means " p implies q"
1 No such principle is enunciated in Pnncipia Mathematics, or in M. Nicod's article mentioned above. But this would seem to be an omission,
Introduction
152
We now
observe that
p For
Mathematical Philosophy
to
|
means " p implies both
(q/r)
"
means
q
and r"
incompatible with the incom r," p implies that q and r are not incom " " patible," i.e. for, as p implies that q and r are both true we saw, the conjunction of q and r is the negation of their this expression
patibility of q
and
i.e.
p
is
"
incompatibility.
Observe next that
t \
(t/t)
means "
t
implies itself."
This
is
a
particular case of p (q/q). Let us write p for the negation of \
negation of p/s,
i.e. it
will
p ; thus p/s will mean the the conjunction of p and s. It
mean
follows that
is
false, i.e. s
states that
and q are both true ; in still simpler words, s jointly imply s and q jointly.
p and
P=p
Now, put
|
(q/r),
Q=(s/q)\p/s.
Then M. Nicod's
sole formal principle of deduction
is
Pk/Q, in other words,
P
implies both
TT
and Q.
He
employs in addition one non-formal principle belonging to the theory of types (which need not concern us), and one corresponding to the principle that, given p, and given that p implies q, we can assert q. This principle is :
From true, true, then q is true." (r/q) this apparatus the whole theory of deduction follows, except in so far as we are concerned with deduction from or to the " "If p
and p
is
|
is
existence or the universal truth of
which we There
is
?
if
I
am
prepositional functions,"
next chapter. not mistaken, a certain confusion in the
shall consider in the
Incompatibility
minds
of
some authors
and
be valid to infer q from
In order that
valid.
is
153
between propositions,
as to the relation,
which an inference
in virtue of
Theory of Deauction
the
may
it
only necessary that p should be " " true and that the proposition should be true. not-/> or q Whenever this is the case, it is clear that q must be true. But it is />,
"
inference will only in fact take place when the proposition not-/> " or q is known otherwise than through knowledge of not-/) or
"
"
is knowledge of q. Whenever p is false, not-/> or q but is useless for inference, which requires that p should be Whenever q is already known to be true, " not-/) or q "
course also since q
is
inferred.
known
to be true, but
is
true, true. is
of
again useless for inference,
already known, and therefore does not need to be In fact, inference only arises when " not-/) or q "
can be known without our knowing already which of the two alternatives it
is
that makes the disjunction true.
Now, the
circumstances under which this occurs are those in which certain relations of
that of r is
.
form
exist
between p and
r implies the negation of
if
Between "
r implies not-5
s,
"
For example, we know
q.
then s implies the negation " s
and "
a formal relation which enables us to
implies not-r
know that the
there
first
implies the second, without having first to know that the first is false or to know that the second is true. It is under such circum stances that the relation of implication
is
practically useful for
drawing inferences.
But this formal relation is only required in order that we may be able to know that either the premiss is false or the conclusion " " is true. It is the truth of
that is required for not-/) or q the validity of the inference ; what is required further is only required for the practical feasibility of the inference. Professor
Lewis
*
has especially studied the narrower, formal relation " may call formal deducibility." He urges that the wider relation, that expressed by " not-/> or q" should not be " called implication." That is, however, a matter of words. C.
I.
which we
1
See Mind, vol. xxi., 1912, pp. 522-531
240-247.
;
and
vol. xxiii.,
1914,
pp.
Introduction
154
to
Provided our use of words define
The
them.
Mathematical Philosophy is
consistent,
essential
of
it
matters
difference
little
how we
between the
point theory which I advocate and the theory advocated by Professor Lewis is this He maintains that, when one proposition q is " " from another p, the relation which we formally deducible between them one which he calls " strict implication," is perceive which is not the relation expressed by " not-p or q " but a narrower relation, holding only when there are certain formal connections :
between p and
q.
I
maintain that, whether or not there be
such a relation as he speaks of, it is in any case one that mathe matics does not need, and therefore one that, on general grounds
economy, ought not to be admitted into our apparatus of fundamental notions ; that, whenever the relation of " formal " holds between two propositions, it is the case that deducibility of
we can
see that either the first
nothing beyond premisses
;
and
is false
or the second true, and that
this fact is necessary to
that, finally, the reasons of detail
Lewis adduces against the view which in detail,
be admitted into our
and depend
I
which Professor
advocate can
for their plausibility
all
be met
upon a covert and
unconscious assumption of the point of view which I reject. I conclude, therefore, that there is no need to admit as a funda
mental notion any form of implication not expressible as a truth-function.
CHAPTER XV PROPOSITIONAL FUNCTIONS
WHEN,
in the preceding chapter,
we were
discussing propositions,
"
we did not attempt to give a definition of the word proposition." But although the word cannot be formally defined, it is necessary to say something as to its meaning, in order to avoid the very " are to confusion with prepositional functions," which
common
be the topic of the present chapter. We mean by a " proposition " primarily a form of words which " I say primarily," expresses what is either true or false. because I do not wish to exclude other than verbal symbols, or
they have a symbolic character. But I " should be limited to what may, proposition " to such further and in some sense, be called symbols symbols," " two and two
even mere thoughts " think the word
if
and falsehood. Thus " will be and " two and two are five propositions, " " Socrates is not a man." and so will " Socrates is a man and The statement " Whatever numbers a and b may be, " 2 " but the bare formula as give expression to truth
are four
"
:
a*+2ab+b
a?-\-2ab-\-b
we all
a proposition ; alone is not, since
is
2
"
it
asserts nothing definite unless
are further told, or led to suppose, that a and b are to possible values, or are to have such-and-such values.
have
The
former of these
is tacitly assumed, as a rule, in the enunciation mathematical formulae, which thus become propositions ; but if no such assumption were made, they would be " preposi is an tional functions." A " in
of
fact, prepositional function," expression containing one or more undetermined constituents,
Introduction
156
to
Mathematical Philosophy
such that, when values are assigned to these constituents, the expression becomes a proposition. In other words, it is a function
But
whose values are propositions. be used with caution.
A
this latter definition
descriptive function,
must
" the hardest
e.g.
proposition in A's mathematical treatise," will not be a proBut in positional function, although its values are propositions.
such a case the propositions are only described tional function, the values
human "
is
undetermined, is
must actually enunciate
propositions. " functions are x prepositional easy to give a prepositional function ; so long as x remains
of
Examples is
in a proposi-
:
:
it
neither true nor false, but
is
assigned to x
becomes a true or
it
when a value
false proposition.
Any
So long as is merely an
mathematical equation is a prepositional function. the variables have no definite value, the equation
expression awaiting determination in order to become a true or false proposition. it
If it is
an equation containing one variable,
becomes true when the variable
the equation, otherwise " " it will be true identity
of
it
made
is
becomes
when the
false
;
variable
equal to a root
but is
if
it
is
an
any number.
The equation
to a curve in a plane or to a surface in space is a true for values of the co-ordinates belong function, propositional the or surface, false for other values. to on curve ing points
" all A is B " are proExpressions of traditional logic such as A and B have to be determined as definite positional functions :
such expressions become true or false. The notion of " cases " or " instances " depends upon proConsider, for example, the kind of process positional functions. and let us take what is called " classes before
generalisation," suggested by some very primitive example, say, " lightning is followed by " " instances of this, i.e. a thunder." We have a number of
number
of propositions such as
and
followed
is
" instances " of " If x
" this
by thunder." What They are instances
is
are of
a flash of lightning
these
the
occurrences
propositional followed by thunder." process of generalisation (with whose validity we are fortun-
function
The
?
:
:
is
a flash of lightning, x
is
Prepositional Functions
157
ately not concerned) consists in passing from a number of such instances to the universal truth of the prepositional function : " If x is a flash of lightning, x is followed by thunder." It will
be found that, in an analogous way, prepositional functions are always involved whenever we talk of instances or cases or examples.
We "
do not need to ask, or attempt to answer, the question " is a function A function :
What
standing
?
prepositional prepositional alone may be taken to be a mere schema, a
all
mere
an empty receptacle for meaning, not something already We are concerned with prepositional functions, significant. shell,
broadly speaking, in two ways " true in all cases " and "
:
first,
as involved in the notions
some cases " secondly, as the theory of classes and relations. The second of we will postpone to a later chapter the first must true in
involved in these topics
;
;
occupy us now.
When we all
cases,"
say that something " it is clear that the
A
a proposition. is
proposition
an end of the matter.
" Socrates
is
a
man
are propositions, " in being true
"
and all
or it
"
" " or true in always true " involved cannot be something is
is
"
Napoleon died at St Helena." would be meaningless to speak This phrase
cases."
prepositional functions. that is often said when
by B.
of their
only applicable to Take, for example, the sort of thing causation is being discussed. (We are
its logical analysis.)
instance, followed
These
is
net concerned with the truth or falsehood of what only with
and there
just true or false,
There are no instances or cases of
Now
We if
is
said,
but
A is, in every " instances " of A,
are told that
there are
A
must be some general concept of which it is significant to say " x is " #3 is A," and so on, where x l9 x2 x3 are #! is A," 2 A," which are not This identical one with another. particulars "
,
applies,
e.g.,
to our previous case of lightning.
We
say that
But the separate lightning (A) is followed by thunder (B). flashes are particulars, not identical, but sharing the common property of being lightning.
The only way
of
expressing a
Introduction
158
common of a
true
to
Mathematical Philosophy
property generally
is
to say that a
common
property
number of objects is a prepositional function which becomes when any one of these objects is taken as the value of the
" " the objects are of the instances truth of the prepositional function for a prepositional function, though it cannot itself be true or false, is true in certain instances " " " and false in certain unless it is true In this case
variable.
all
others,
false."
When,
or
always
to return to our example,
every instance followed by B, we if x is an A, it is followed by a B
mean
we say
that,
that
whatever x
always
A
is
may
in be,
that is, we are asserting that ; " a certain propositional function is always true." " " " Sentences involving such words as all," a," every," " " some " the," require propositional functions for their inter The way in which propositional functions occur pretation.
can be explained by means of two of the above words, namely, " " and " all some."
There
are, in the last analysis,
only two things that can be one is to assert that it is
done with a propositional function
:
true in all cases, the other to assert that
it is
true in at least one
some cases (as we shall say, assuming that there is no necessary implication of a plurality of cases). All the
case, or in
to be
other uses of propositional functions can be reduced to these two. When we say that a propositional function is true " in all cases," or
"
" always
tion),
fa," then
(f>a
For example, is
human
shall also say,
that
function, and a
"
we
(as
we mean
is is
"
if
or not
;
all
its
without any temporal sugges " " If
values are true.
fa
is
the
the right sort of object to be an argument to to be true, however a may have been chosen. " a is human, a is mortal is true whether a in fact, every proposition of this
form
is
true.
Thus the propositional function " if x is human, x is mortal " " true " in all cases." is Or, again, the state always true," or " " no unicorns is the same as the statement are there ment " the propositional function cases."
The
assertions
" positions, e.g.
'
p
or q
'
*
x
is
not a unicorn
'
is
true in
all
the preceding chapter about pro " * are really assertions implies q or p,
in
Prepositional Functions
159
that certain prepositional functions are true in
all
cases.
We
do
not assert the above principle, for example, as being true only of this or that particular p or q, but as being true of any p or q
concerning which it can be made significantly. The condition that a function is to be significant for a given argument is the same as the condition that
either true or false.
it
shall
have a value
The study
for that
argument,
of the conditions of significance
belongs to the doctrine of types, which we shall not pursue beyond the sketch given in the preceding chapter.
Not only the
principles of deduction, but all the primitive
propositions of logic, consist of assertions that certain preposi tional functions are always true. If this were not the case, they
would have to mention particular things or concepts Socrates, or redness, or east and west, or what not, and clearly it is not the province of logic to make assertions which are true concerning one such thing or concept but not concerning another. It is part of the definition of logic (but not the whole of that all its propositions are completely general,
its definition)
consist of the assertion that
they all some propositional function con
taining no constant terms
always true.
our
final
is
We
i.e.
shall return in
chapter to the discussion of propositional functions
containing no constant terms. For the present we will proceed to the other thing that is to be done with a propositional function,
" namely, the assertion that it is sometimes true," i.e. true in at least one instance. When we say " there are men," that means that the pro" x is a man " is sometimes true. When we positional function " some men are Greeks," that means that the propositional say " function x is a man and a Greek " is sometimes true. When we " cannibals still exist in Africa," that means that the prosay " x is a cannibal now in Africa " is sometimes positional function To say " there are at least true, i.e. is true for some values of x. " n individuals in the world is to say that the propositional function " a is a class of individuals
number n "
is
sometimes
true, or, as
and a member
of the cardinal
we may say,
true for certain
is
160
Introduction
Mathematical Philosophy
to
This form of expression is more convenient when it which is the variable constituent which
values of a.
necessary to indicate
is
we
are taking as the argument to our prepositional function. For example, the above prepositional function, which we may " shorten to a, is a class of n individuals," contains two variables,
The axiom of infinity, in the language of prepositional " The prepositional function if n is an inductive functions, is number, it is true for some values of a that a is a class of n indi a and
n.
*
:
viduals
'
true for
is
all
possible values of
subordinate function, " a
is
is
a is
said to be, in respect of a, sometimes true
that this happens respect of
,
if
n
Here there
a class of n individuals," which
."
;
and the assertion
an inductive number
is
is
said to be, in
always true.
The statement that a function fa of the statement that not- fa
is
is always true is the negation sometimes true, and the state
ment that fa is sometimes true is the negation of the state ment that Tiot-fa is always true. Thus the statement " all
men
are mortals
"
the negation of the statement that the man " is sometimes true. And the " statement there are unicorns " is the negation of the state ment that the function " x is not a unicorn " is always true. 1 We say that fa is " never true " or " always false " if not-fa is " always true. We can, if we choose, take one of the pair always," " sometimes " as a primitive idea, and define the other by means Thus if we choose " sometimes " as of the one and negation. ' " ' is always true is to our primitive idea, we can define (f>x " * 2 for But that notis sometimes true.' it is false mean fa
function
"x
is
is
an immortal
:
reasons connected with the theory of types it seems more correct " " and " sometimes " as primitive ideas, to take both always and define by their means the negation of propositions in which
they occur. That is to say, assuming that we have already 1 The method of deduction is given in Principia Mathematica, vol. 2
*
i.
For
9.
linguistic reasons, to
avoid suggesting either the plural or the " " yx is not always false rather
singular, it is often convenient to say
than
"
cpx
sometimes
" or "
is
sometimes true."
161
Prepositional Functions
defined (or adopted as a primitive idea) the negation of pro " The positions of the type to which x belongs, we define ' ' ' * negation of x always is not-0# sometimes ; and the nega :
'
tion of
sometimes
(j>x
we can
re-define
'
*
is
not-<# always.'
'
In like manner
disjunction and the other truth-functions,
applied to propositions containing apparent variables, in terms of the definitions and primitive ideas for propositions
as
.
containing no apparent variables.
Propositions containing no
"
apparent variables are called elementary propositions." From these we can mount up step by step, using such methods as have just been indicated, to the theory of truth-functions as applied to propositions containing one, two, three . . variables, or any .
number up to n, where n is any assigned finite number. The forms which are taken as simplest in traditional formal logic are really far from being so, and all involve the assertion of all values or some values of a compound prepositional function. " Take, to begin with,
S
all
is
P."
We
will
take
it
that S
is
by a prepositional function x, and P by a prepositional " is human " function i/jx. E.g., if S is men, <j)X will be if P is x ; defined
mortals,
"
all
S
t/jx
P
is
be " there means " '
will
"
:
to be observed that
"
all
is
Then
a time at which x dies." '
x implies i/jx always true." It is " S is P does not apply only to those
is
terms that actually are S's ; it says something equally about terms which are not S's. Suppose we come across an x of which
we do not know whether "
P
"
it is
an S or not
;
still,
our statement
x, namely, that if x is an S, then x is a P. And this is every bit as true when x is not an S as when x is an S. If it were not equally true in both cases, the all
S
is
reductio ad
tells
us something about
absurdum would not be a valid method
essence of this
method
;
for the
consists in using implications in cases
We
where
(as it afterwards turns out) the hypothesis is false. may the matter another way. In order to understand " all S is P," put it is not necessary to be able to enumerate what terms are S's ;
provided we being a P,
know what is meant by being an S and what by we can understand completely what is actually affirmed II
1
62
Introduction
"
Mathematical Philosophy
to
P," however little we may know of actual instances This shows that it is not merely the actual terms that of either. " all is are S's that are relevant in the statement S P," but all the
by
S
all
is
terms concerning which the supposition that they are S's is significant, i.e. all the terms that are S's, together with all the terms that are not S's i.e. the whole of the appropriate logical " type." What applies to statements about all applies also to " There are statements about
men," e.g., means that Here all values of x
some.
"
x
is
(i.e.
human "
is
true for some values of x.
"
values for which
all
x
"
human
is
significant,
and not only those that
true or false) are relevant,
human.
is
(This becomes obvious
we
if
whether
in fact are
how we
consider
could
prove such a statement to be false.) Every assertion about " all " or " some " thus involves not only the arguments that make a certain function true, but all that make it significant, i.e. all
for
which
it
has a value at
all,
whether true or
false.
We may now proceed with our interpretation of the traditional forms of the old-fashioned formal logic. We assume that S fa is true, and P is those for which fa a later chapter, all classes are derived we shall see in (As Then from way prepositional functions.)
is
those terms x for which
is
true.
in this
:
" All S
"
means " fa implies fa is always true." " is P means " fa and fa is sometimes true." " P means " fa implies not-fa is always true." " Some S is not P " means " fa and not-fa is sometimes is
P
'
" Some S " No S is
'
'
*
'
'
'
'
true." It will
be observed that the propositional functions which are all or some values are not fa and fa them
here asserted for selves, x.
but truth-functions of fa and fa for the same argument
The
intended (j>a
ing
and all
easiest is
ipa,
"
where a
men "
way
to conceive of the sort of thing that
to start not from is
some constant.
are mortal
If
Socrates
"
is
:
we
is
fa and fa in general, but from will
Suppose we are consider
begin with
human, Socrates
is
mortal,"
Prepositional Functions
163
"
"
as replaced by a variable x Socrates regard " The object to be secured is that, occurs. wherever Socrates although x remains a variable, without any definite value, yet " " " " it is to have the same value in fa as in fa when we are " " asserting that fa implies fa is always true. This requires with a function whose values are such as shall start that we " rather than with two separate functions fa cf>a implies $a" and fa ; for if we start with two separate functions we can
and then we
will
"
never secure that the
while remaining undetermined, shall
x,
have the same value in both. "
"
when we fa always implies iftx " is mean Propositions always true. fa " are called " formal " of the form fa always implies fa " this name is given equally if there are several ; implications For
brevity we say " that (j>x implies
variables.
The above
show how
definitions
forms are such propositions as tional logic begins.
"
far
removed from the simplest is P," with which tradi
S
all
It is typical of the lack of analysis
involved
"
that traditional logic treats "all S is P as a proposition of " " all " the same form as it treats men are mortal " x is P e.g., as of the
same form
seen, the first
second
is
is
"
as
of the
It will
mortal."
is
As we have just
fa"
while the
form "
fa" The emphatic separation of these which was effected by Peano and Frege, was a very
of the
two forms, advance
vital
Socrates
form " fa always implies
in symbolic be seen that "
logic. all
S
is
P
"
and " no S
is
P
" do not
by the substitution of not-iffx for fa, and that the same applies to " some S is P " and " some S is really differ in form, except
not P."
It
should also be observed that the traditional rules
of conversion are faulty,
if
we adopt
the view, which
is
the only
" all S is P" technically tolerable one, that such propositions as " " do not involve the existence of S's, i.e. do not require that there should be terms which are S's.
The above
definitions
lead to the result that, if fa is always false, i.e. if there are no " " all S is P and no S is P " will both be true, whatS's, then
Introduction
164
P may
ever
be.
to
Mathematical Philosophy
For, according
"
to the definition in the last
"
means " not- fa or fa" which is chapter, fa implies fa always true if not-<# is always true. At the first moment, this result
but a
might lead the reader to desire different definitions, practical experience soon shows that any different
little
would be inconvenient and would conceal the important The proposition " fa always implies fa, and fa
definitions ideas.
sometimes true "
is
is
essentially composite,
and
it
would be
very awkward to give this as the definition of "all S is P," " for then we should have no language left for fa always implies fa," which is needed a hundred times for once that the other is " " needed. But, with our definitions, all S is P does not imply " some S is P," since the first allows the non-existence of S and the second does not; thus conversion per accidens becomes invalid,
Darapti fails if
and some moods " All
:
there
is
M
is S, all
of the
M is P,
syllogism are fallacious, e.g. therefore some S is P," which
no M.
" has several forms, one of which will occupy us in the next chapter ; but the fundamental form " some is that which is derived immediately from the notion of " (< a function times true." We say that an argument a satisfies
The notion
fa
if
a
is
of
true
;
"
existence
this is the
same sense
in
true,
"
there are #'s for which
arguments satisfying fa
exist"
This
of
an
Now if fa is sometimes
equation are said to satisfy the equation.
we may say
which the roots
it is
is
we may say mean
true, or
the fundamental
" existence." Other meanings are either derived ing of the word from this, or embody mere confusion of thought. We may " " " men x is a man is some exist," meaning that correctly say " Men times true. But if we make a pseudo-syllogism exist, Socrates is a man, therefore Socrates exists," we are talking " " " Socrates is not, like men," merely an un nonsense, since :
determined argument to a given prepositional function. The " Men are fallacy is closely analogous to that of the argument numerous, Socrates is a man, therefore Socrates is numerous." :
In this case
it is
obvious that the conclusion
is
nonsensical, but
Prepositional Functions
165
not obvious, for reasons which will For the present let us " merely note the fact that, though it is correct to say men exist," it is incorrect, or rather meaningless, to ascribe existence to a " terms given particular x who happens to be a man. Generally, in the case of existence
appear more
satisfying exists
"
it is
fully in the next chapter.
fa exist" means "fa
(where a
is
is
sometimes true"; but
devoid of significance.
"a
a mere noise or shape, It will be found that by bearing in mind
a term satisfying fa)
is
simple fallacy we can solve many ancient philosophical puzzles concerning the meaning of existence. this
Another
set of notions as to
which philosophy has allowed
to fall into hopeless confusions through not sufficiently separating propositions and prepositional functions are the " " notions of modality : necessary, possible, and impossible.
itself
(Sometimes contingent or assertoric is used instead of possible) traditional view was that, among true propositions, some were necessary, while others were merely contingent or assertoric ;
The
among false propositions some were impossible, namely, those whose contradictories were necessary, while others merely
while
happened not to be
any
clear account of
In the case of prepositional functions, the three " " If obvious. fa is an undetermined value of a
of necessity. fold division
fact, however, there was never what was added to truth by the conception
In
true.
is
certain prepositional function, is
if it is
be necessary if the function sometimes true, and impossible if
it will
always true, possible never true. This sort of situation arises in regard to prob ability, for example. Suppose a ball x is drawn from a bag
it is
which contains a number " if
x
is
white
none,
it is
"
of balls
:
if
all
the balls are white,
some are white, it is possible ; necessary impossible. Here all that is known about x is that is
;
if
" x was a a certain prepositional function, namely, ball in the bag." This is a situation which is general in prob and not uncommon in practical life e.g. when ability problems
it
satisfies
a person calls of whom we know nothing except that he brings a letter of introduction from our friend so-and-so. In all such
1
66
Introduction
to
Mathematical Philosophy
the prepositional function is relevant. For clear thinking, in many very diverse directions, the habit of keeping prepositional functions sharply separated from propositions is of the utmost importance, and cases,
the
as in regard to modality in general,
failure
philosophy.
to
do
so in
the past has been a disgrace to
CHAPTER XVI DESCRIPTIONS
WE
dealt in the preceding chapter with the words all and some ; in this chapter we shall consider the word the in the singular,
and
in the next chapter
we
shall consider the
word
the in the
plural. may be thought excessive to devote two chapters to one word, but to the philosophical mathematician it is a
It
of very great importance like Browning's Grammarian with the enclitic Se, I would give the doctrine of this word if I " and not merely in a prison. were " dead from the waist down " had
word
We
:
have already
to mention descriptive " " " the father of x or the sine functions," i.e. such expressions as " of x" These are to be defined by first defining descriptions." A " description " may be of two sorts, definite and indefinite (or
ambiguous).
An
occasion
indefinite description
is
a phrase of the
form " a so-and-so," and a definite description is a phrase of " " the so-and-so the form Let us begin with (in the singular). the former.
"
Who
"
did you meet
?
"I met
We
indefinite description."
a man."
" That
a very
is
are therefore not departing from
Our question is What do I really met a man " ? Let us assume, for the moment, that my assertion is true, and that in fact I met Jones. " It is clear that what I assert is not I may I met Jones." say " " I met a but it was not in that man, case, though I lie, Jones
usage in our terminology. when I assert " assert
:
I
;
I
do not contradict myself, as
I
should do
167
if
when
I
say
I
met a
1
68
man
Introduction I really
mean
Mathematical Philosophy
to
that
met Jones.
I
It is clear also that the
person to whom I am speaking can understand what I say, even he is a foreigner and has never heard of Jones. But we may go further not only Jones, but no actual man,
if
:
enters into
my
statement.
This becomes obvious
when the
state
is false, since then there is no more reason why Jones should be supposed to enter into the proposition than why any one else should. Indeed the statement would remain significant,
ment
though at
it
"
all.
could not possibly be true, even if there were no " " I met a " I met a unicorn or sea-serpent if
be a unicorn or a sea-serpent,
i.e.
we know what
perfectly significant assertion,
fabulous monsters.
Thus
it is
it
man is
a
would be to
what is the definition of these only what we may call the concept
" that enters into the proposition. In the case of unicorn," for example, there is only the concept there is not also, some where among the shades, something unreal which may be called :
" a unicorn."
to say
"
I
Therefore, since
met a unicorn,"
it is
it
is
significant (though false)
clear that this proposition, rightly
" a unicorn," though analysed, does not contain a constituent " it does contain the concept unicorn." The question of " unreality," which confronts us at this a very important one. Misled by grammar, the great majority of those logicians who have dealt with this question have dealt with it on mistaken lines. They have regarded point,
is
grammatical form as a surer guide in analysis than, in fact, it is. And they have not known what differences in gram matical form are important. " I met Jones " and " I met a man " would count traditionally as propositions of the same form, but in actual fact they are of quite different forms the first an names actual person, Jones ; while the second involves a " The prepositional function, and becomes, when made explicit ' I met x and x is human function is sometimes true." (It :
:
'
be remembered that we adopted the convention of using " sometimes " as not implying more than once.) This proposi " I tion is obviously not of the form met x," which accounts
will
1
Descriptions
"
for the existence of the proposition
of the fact that there
For want
is
I
met a unicorn "
69
in spite
" no such thing as a unicorn."
of the
apparatus of prepositional functions, many that there are logicians have been driven to the conclusion unreal objects. It is argued, e.g. by Meinong, 1 that we can " " the the round square," golden mountain," speak about and so on ; we can make true propositions of which these are the subjects ; hence they must have some kind of logical being, since otherwise the propositions in which they occur would be it seems to me, there is a failure which ought to be preserved even in the most abstract studies. Logic, I should maintain, must no
meaningless.
In such theories,
of that feeling for reality
more admit a unicorn than zoology can
;
for logic is concerned
with the real world just as truly as zoology, though with
its
more abstract and general features. To say that unicorns have an existence in heraldry, or in literature, or in imagination, a most pitiful and paltry evasion. What exists in heraldry is not an animal, rrlade of flesh and blood, moving and breathing of its own initiative. What exists is a picture, or a description is
in words.
Similarly,
exists in his
own
imagination, just
to maintain that Hamlet,
for
example,
world, namely, in the world of Shakespeare's as
truly
as
(say)
Napoleon existed in the
ordinary world, is to say something deliberately confusing, or else confused to a degree which is scarcely credible. There is " " the real one world world, only Shakespeare's imagination :
is
part of
are real.
But
it, and the thoughts that he had in writing Hamlet So are the thoughts that we have in reading the play.
of the
very essence of fiction that only the thoughts,
feelings, etc., in
Shakespeare and his readers are real, and that an objective Hamlet. When
it is
there
is
not, in addition to them,
you have taken account of all the feelings roused by Napoleon in writers and readers of history, you have not touched the actual
man
;
him.
but in the case If
1
of
Hamlet you have come
to the
end
of
no one thought about Hamlet, there would be nothing
Untersuchungen zur Gegenstandstheorie und Psychologic, 1904.
Introduction
170 left of
him
;
if
Mathematical Philosophy
to
no one had thought about Napoleon, he would it that some one did. The sense of reality is
have soon seen to
and whoever juggles with it by pretending that has another kind of reality is doing a disservice to Hamlet thought. A robust sense of reality is very necessary in framing a correct analysis of propositions about unicorns, golden moun vital in logic,
tains,
round squares, and other such pseudo-objects.
In obedience to the feeling of
reality,
we
shall insist that,
"
" unreal the analysis of propositions, nothing is to be admitted. But, after all, if there is nothing unreal, how, it
in
be asked, could we admit anything unreal ? The reply that, in dealing with propositions, we are dealing in the first
may is
instance with symbols, and
we
attribute significance to groups significance, we shall fall into the
if
symbols which have no
of
error of admitting unrealities, in the only sense in
namely,
possible,
"
as
objects
described.
In
the
which
this is
proposition
met a unicorn," the whole four words together make a signi " " unicorn by itself is significant, ficant proposition, and the word I
same sense
word " man."
But the two words " a unicorn " do not form a subordinate group having a meaning Thus if we falsely attribute meaning to these two of its own. " a unicorn," and with words, we find ourselves saddled with
in just the
as the
the problem how there can be such a thing in a world where " " unicorn is an indefinite there are no unicorns. descrip
A
which describes nothing. It is not an indefinite description which describes something unreal. Such a proposition as " " is " " x a description, x is unreal only has meaning when tion
in that case the proposition will be true ; a description which describes nothing. But whether " x " describes the description something or describes nothing, definite or indefinite
if
" x"
is
of the proposition in which it " a unicorn " occurs ; like just now, it is not a subordinate group its own. of All this results from the fact that, a meaning having " " " " " when " x is a description, x is unreal or x does not exist
it is
is
in
any case not a constituent
not nonsense, but
is
always significant and sometimes true.
171
Descriptions
We may now
proceed to define generally the meaning of Suppose propositions which contain ambiguous descriptions. we wish to make some statement about " a so-and-so," where
"so-and-so's" are those objects that have a certain property <,
i.e.
true. t/)X
those objects x for which the prepositional function (fax is " a " " a man as our instance of so-and-so," (E.g. if we take
be " x is human.")
will
"
Let us
now wish
to assert the property " a so-and-so " has wish to assert that
of a so-and-so," i.e. we that property which x has when i/jx is true. (E.g. in the case " I met " I met a the proposition be Now will of man," ifix #.") " " a that so-and-so has the property ift is not a proposition of " " " would have to be a so-and-so the form 0#." If it were,
ifj
x for a suitable x ; and although (in a sense) this be true in some cases, it is certainly not true in such a case may " as a unicorn." It is just this fact, that the statement that a
identical with
so-and-so has the property ijj is not of the form ifrx, which makes " " to be, in a certain clearly definable a so-and-so it possible for
"
unreal."
sense,
The
definition
is
as follows
:
The statement that " an object having the property " the property
^
has
ift
means
:
" The joint assertion of
So far as
x
and
i/ix
logic goes, this is the
is
not always false."
same proposition
as
might
be expressed by " some <'s are ^'s " ; but rhetorically there is a difference, because in the one case there is a suggestion of singularity, is
and
in the other case of plurality.
not the important point.
This, however,
The important point
is
that,
when
" " a so-and-so rightly analysed, propositions verbally about are found to contain no constituent represented by this phrase.
And there
is why such propositions can be significant even when no such thing as a so-and-so.
that is
The
definition of existence, as applied to
tions, results
chapter.
We
from what was said say that
"
men
exist
at the
"
or
ambiguous descrip end of the preceding
" a
man
exists
"
if
the
Introduction
172
to
Mathematical Philosophy
"
x is human prepositional function " a so-and-so " exists if " x generally true.
We may
" Socrates
is
put
man
a
this in
"
is
" is
is
sometimes true
so-and-so
other language.
"
is
;
and
sometimes
The proposition
no doubt equivalent to " Socrates
is
not the very same proposition. The is of " " Socrates is human expresses the relation of subject and " " Socrates is a man predicate ; the is of expresses identity.
human," but
it
is
It is a disgrace to the human race that it has chosen to employ " is " the same word a for these two entirely different ideas
disgrace which a symbolic logical language of course remedies. " " Socrates is a man is identity in identity between an
The
object
named
qualifications
(accepting
explained
" Socrates " as a name, subject to and an later) object ambiguously
" " exist when object ambiguously described will at least one such proposition is true, i.e. when there is at least
An
described.
one true proposition of the form " x is a so-and-so," where " x " is a name. It is characteristic of ambiguous (as opposed to definite) descriptions that there may be any number of true propositions of the above form Socrates is a man, Plato is a " a man exists " follows from man, etc. Thus Socrates, or Plato, or
anyone
else.
With
definite descriptions,
on the other
" x is the hand, the corresponding form of proposition, namely, " " x " is a so-and-so name), can only be true for one (where value of x at most. This brings us to the subject of definite
which are to be defined in a way analogous to that employed for ambiguous descriptions, but rather more
descriptions,
complicated.
We
come now
to the
main subject word the
of the present chapter,
the singular). One " " a so-and-so very important point about the definition of " " the so-and-so the definition to be sought ; applies equally to is a definition of propositions in which this phrase occurs, not a " a In the case of definition of the phrase itself in isolation. no one could suppose that so-and-so," this is fairly obvious
namely, the definition of the
(in
:
" a
man " was
a definite object, which could be defined by
itself.
173
Descriptions Socrates
is
a man, Plato
cannot infer that " a
is
man
a man, Aristotle
"
is
means the same
a man, but as
"
we
Socrates
"
" means and also the same as " Plato means and also the same " " as Aristotle means, since these three names have different meanings.
men "
Nevertheless,
in the world, there
This
'
man, and not only so, but it is the a man,' the quintes entity that is just an indefinite man without being any
is
sential
when we have enumerated all the is nothing left of which we can say,
a
body
in particular."
there
is
It is of course quite clear that
in the world is definite
:
if it is
a
man
it is
whatever
one definite
man and not any other. Thus there cannot be such an entity " " as a man to be found in the world, as opposed to specific man. And accordingly it is natural that we do not define " a man " itself, but only the propositions in which it occurs. In the case of " the so-and-so " this at first sight less obvious.
We may
is
equally true, though
demonstrate that
this
must
be the case, by a consideration of the difference between a name " and a Take the Scott is the proposition,
definite description.
author of Waverley" We have here a name, " Scott," and a " the author of Waverley" which are asserted to description,
apply to the same person. The distinction between a all other symbols may be explained as follows
name and
:
A name
a simple symbol whose meaning
something that i.e. something of the kind that, in " " " Chapter XIII., we defined as an individual or a particular." " " And a simple symbol is one which has no parts that are " " Scott is a symbols. Thus simple symbol, because, though it has parts (namely, separate letters), these parts are not symbols. On the other hand, " the author of Waverley " is not a simple symbol, because the separate words that compose the phrase is
is
can only occur as subject,
are parts which are symbols. If, as may be the case, whatever seems to be an " individual " is really capable of further analysis, we shall have to content ourselves with what may be called
" relative individuals," which
will
be terms that, throughout
the context in question, are never analysed and never occur
Introduction
174
Mathematical Philosophy
to
otherwise than as subjects. And in that case we shall have " relative names." correspondingly to content ourselves with
From
the standpoint of our present problem, namely, the defini
tion of descriptions, this problem, whether these are absolute names or only relative names, may be ignored, since it con
cerns different stages in the hierarchy of " types," whereas we have to compare such couples as " Scott " and " the author of
Waverley" which both apply to the same object, and do not problem of types. We may, therefore, for the moment, treat names as capable of being absolute ; nothing that we shall
raise the
have to say
may
be a
We is is
will
depend upon
shortened by
little
assumption, but the wording
have, then, two things to compare
a name, which a simple symbol, directly designating an individual which
its
meaning, and having
dependently of the
which consists fixed,
"
this
it.
meanings
meaning
(i)
meaning in its own right, in words ; (2) a description,
this
of all other
whose meanings are already whatever is to be taken as the
of several words,
and from which "
:
results
of the description.
A
proposition containing a description what that proposition becomes when a
not identical with
is
name
is
substituted,
name names
the same object as the description " Scott is the " is describes. author of Waverley obviously a " " different proposition from : the first is a fact Scott is Scott
even
if
the
And if we put " the author of Waverley"
in literary history, the second a trivial truism.
anyone other than Scott in place of
our proposition would become false, and would therefore certainly no longer be the same proposition. But, it may be said, our " is proposition
is
essentially of the
same form
as (say)
Scott
which two names are said to apply to the same The reply is that, if " Scott is Sir Walter " really means
Sir Walter," in
person. " the person
named
e
Scott
'
is
the person
named
then the names are being used as descriptions
:
i.e.
'
Sir Walter,'
'
the individual,
instead of being named, is being described as the person having that name. This is a way in which names are frequently used
175
Descriptions
and there will, as a rule, be nothing in the phraseology show whether they are being used in this way or as names. When a name is used directly, merely to indicate what we are in practice,
to
speaking about, it is no part of the fact asserted, or of the falsehood it is our assertion happens to be false merely part of the our we which thought. What we want express symbolism by if
:
to express
is
something which might
(for
example) be translated
it is something for which the actual ; words are a vehicle, but of which they are no part. On the other " the hand, when we make a proposition about person called " " " ' enters into what we are Scott actual name
into a foreign language
the
Scott,'
asserting,
and not merely into the language used
in
making the
Our proposition will now be a different one if we ' " the But so long as person called Sir Walter.' " we are using names as names, whether we say " Scott or whether " " we say Sir Walter is as irrelevant to what we are asserting Thus so long as names as whether we speak English or French. " Scott is Sir Walter " is the same trivial are used as names, " Scott is Scott." This completes the proof that proposition as " " is not the same Scott is the author of Waverley proposition " the author of results from name as for substituting a Waverley" assertion.
'
substitute
no matter what name
may
be substituted.
When we
use a variable, and speak of a propositional function, the (/>x say, process of applying general statements about x to cases will consist in substituting a name for the letter particular
"
x"
is a function which has individuals for its " " Suppose, for example, that <j>x is always true ; " let it be, say, the law of identity," x=x. Then we may sub "x" stitute for any name we choose, and we shall obtain a true " for the that
assuming that ^
arguments.
moment Socrates," Assuming " " and Aristotle are names (a very rash assumption), Plato," we can infer from the law of identity that Socrates is Socrates, Plato is Plato, and Aristotle is Aristotle. But we shall commit proposition.
"
a fallacy if we attempt to infer, without further premisses, that the author of Waverley is the author of Waverley. This results
Introduction
176
to
Mathematical Philosophy
from what we have just proved, that, if we substitute a name for " " the author of in a proposition, the proposition Waverley we obtain is a different one. That is to say, applying the result to our present case
proposition as
"
:
If
x"
is
propositions of the
"
a name,
the author of Waverley " "
no matter what name all
"
x=x "
is
not the same
the author of Waverley" Thus from the fact that
is
x may be. " are true we cannot form " x=x
infer,
without more ado, that the author of Waverley is the author of " the so-and-so Waverley. In fact, propositions of the form " is the so-and-so are not always true it is necessary that the so-and-so should exist (a term which will be explained shortly). :
It is false that the present King of France is the present King of France, or that the round square is the round square. When we substitute a description for a name, prepositional functions " " if the true become
which are
realise
we
may
always
describes nothing.
There
is
false,
no mystery
description
we when
in this as soon as
(what was proved in the preceding paragraph) that the result is not a value of the
substitute a description
propositional function in question. are now in a position to define propositions in which a The only thing that distinguishes definite description occurs.
We
" the so-and-so " from " a so-and-so " is the implication of " of London," inhabitant the cannot of We speak uniqueness. because inhabiting London is an attribute which is not unique. We cannot speak about " the present King of France," because " the present King of there is none ; but we can speak about " the so-and-so " Thus about always England." propositions " a so-and-so," imply the corresponding propositions about with the addendum that there is not more than one so-and-so. " " could as Scott is the author of
Such a proposition
Waverley been had never written, or if several Waverley it ; and no more could any other proposition had written people resulting from a propositional function x by the substitution not be true
if
" the " " for x." We may say that ( " means " the value of x for which x wrote author of Waverley
of
"
the author of Waverley
177
Descriptions '
Thus the proposition " the author
true."
is
Waverley
Waverley was Scotch," " (1)
" (2)
"
example, involves
if
of
:
x wrote Waverley " is not always false ; " if x and y wrote Waverley, x and y are identical always true
(3)
for
is
;
x wrote Waverley, x was Scotch "
is
always true.
These three propositions, translated into ordinary language, state (1)
(2) (3)
:
at least one person wrote Waverley
;
most one person wrote Waverley whoever wrote Waverley was Scotch. at
;
" the author of Waverley was Scotch." Conversely, the three together (but no two of them) Hence the imply that the author of Waverley was Scotch. All these three are implied
by
three together may be taken as defining what is " the author of Waverley was Scotch." proposition
meant by the
We may
somewhat simplify these three propositions. The " and second together are equivalent to There is a term such that x wrote Waverley is true when x is c and is false
first
c
:
'
when x
is
'
not
In other words, " There
c ."
is
a term c such that
* 9 " always equivalent to x is c. (Two " " when both are true or both are propositions are equivalent We have here, to begin with, two functions of x, " x false.) wrote Waverley " and " x is r," and we form a function of c by *
x wrote Waverley
'
is
considering the equivalence of these two functions of x for all values of x ; we then proceed to assert that the resulting function " of c is sometimes true," i.e. that it is true for at least one value of c. (It obviously cannot be true for more than one value of c .) These two conditions together are defined as giving the meaning " of the author of Waverley exists." We may now define " the term satisfying the function x exists."
This
ticular case.
the function
is
" '
the general form of which the above
The author
of Waverley
x wrote Waverley' "
"
is
And
is a par " the term satisfying " the "
so-and-so 12
will
Introduction
178
Mathematical Philosophy
to
always involve reference to some prepositional function, namely, that which defines the property that makes a thing a so-and-so.
Our
definition
is
as follows
:
" " The term means satisfying the function fa exists " There is a term c such that fa is always equivalent to x is c? " the author of Waverley was Scotch," In order to define we have still to take account of the third of our three proposi " Whoever wrote tions, namely, Waverley was Scotch." This will be satisfied by merely adding that the c in question is to " is be Scotch. Thus " the author of Waverley was Scotch * " There is a term c such that (i) x wrote Waverley is always ' equivalent to x is cj (2) c is Scotch." " " the term is And generally satisfying x satisfies fa :
'
'
:
9
:
defined as meaning
" There '
x
:
a term c such that (i)
is
9
is c,
(2)
ific
is
<{>x
is
always equivalent to
true."
the definition of propositions in which descriptions occur. It is possible to have much knowledge concerning a term
This
is
"
know many
the sopropositions concerning and-so," without actually knowing what the so-and-so is, i.e. " without knowing any proposition of the form x is the so-and-so," described,
i.e.
where " x "
to
In a detective story propositions about " are accumulated, in the hope that ultimately they will suffice to demonstrate that it was A who did the deed. We may even go so far as to say that,
"
the
is
a name.
man who
in all such
did the deed
knowledge as can be expressed in words with the " this " and " that " and a few other words of
exception of
which the meaning varies on different occasions no names, in the strict sense, occur, but what seem like names are really descriptions.
We may
inquire
significantly
" Homer existed, which we could not do if " the so-and-so exists " is
"
whether Homer
were a name.
The
whether
significant, proposition " a " is a true or false ; but if a is the so-and-so (where name), " " It is only of descriptions a exists are meaningless. the words
179
Descriptions definite
asserted
to be a
"a "
name anything
name,
tion, like
"
is
be
significantly
must name something what not a name, and therefore, if intended
a name,
is
can
existence
that
indefinite
for, if
;
does not
or
is
it
:
a symbol devoid of meaning, whereas a descrip
the present King of France," does not become in on the ground that it
capable of occurring significantly merely
it is a complex symbol, derived from that of its constituent
describes nothing, the reason being that of
which the meaning
is
symbols. And so, when we ask whether Homer existed, we are " Homer " as an abbreviated we using the word description " author of the Iliad and the it the Odyssey" may replace by (say) :
The same like
considerations apply to almost
all
uses of
what look
proper names.
When
descriptions occur in propositions, " " distinguish what may be called primary
The
occurrences.
abstract distinction
is
it
is
necessary to
and " secondary "
as follows.
A
descrip the proposition in primary which it occurs results from substituting the description for " " x in some prepositional function (/>x ; a description has a tion
has
a
"
"
"
"
occurrence
when
when
occurrence
the result of substituting the of the proposition con <j>x gives only part An instance will make this clearer. Consider " the cerned. " the present King of present King of France is bald." Here " has a primary occurrence, and the proposition is false. France
secondary
description for
x in
Every proposition
in
which a description which describes nothing
has a primary occurrence present King of France we are first to take " x " "
is is
is
false.
not bald."
But now consider " the This
is
If
ambiguous. " the
bald," then substitute
present France for x" and then of the the occurrence result, King deny " " the present King of France of is secondary and our proposition but if we are to take " x is not bald " and substitute is true ; " the " " for x" then " the present present King of France " King of France has a primary occurrence and the proposition is
false.
Confusion of primary and secondary occurrences
ready source of fallacies where descriptions are concerned.
is
a
i8o
Introduction
to
Mathematical Philosophy
mathematics chiefly in the form of " the term having the relation R to descriptive functions, i.e. " " the " as we may say, on the analogy of the R of y y," or " " the father of To father of and similar is Descriptions occur
in
y
function of to
'
x
is
value of
say
phrases.
rich," for example, c
"
cj
is
c.
It
y
to say that the following prepositional
c is rich,
and
'
x begat y
'
is
always equivalent sometimes true," i.e. is true for at least one obviously cannot be true for more than one
:
"
is
"
value.
The theory chapter, of
is
of descriptions,
of the
knowledge.
briefly
outlined in the present
utmost importance both in
But
for
purposes
of
logic
and
mathematics,
in theory the more
philosophical parts of the theory are not essential, and have therefore been omitted in the above account, which has confined itself to
the barest mathematical requisites.
CHAPTER XVIi CLASSES IN the present chapter we shall be concerned with plural
:
so on.
saw
the in the
the inhabitants of London, the sons of rich men, and In other words, we shall be concerned with classes. We
in Chapter II. that a cardinal
class of classes,
and
number
is
in Chapter III. that the
defined as the class of
all
unit classes,
i.e.
to be defined as a
number
I is
of all that
one member, as we should say but for the vicious
to be
have
just
circle.
Of
classes,
to
that
when the number I is defined as the class " " unit classes must be defined so as not we know what is meant by " one " in fact, they
are defined
in a
way
course,
;
closely analogous to that used for descriptions,
A class
a
"
an a
*
x
is
is
function of if
there
of all unit
is
said to be a
namely
:
the prepositional function 9 " ' (regarded as a always equivalent to x is c
'
is
c) is
" unit "
assume
class
if
not always false, i.e., in more ordinary language, c such that x will be a member of a when x is c
a term
This gives us a definition of a unit class if we a class is in general. Hitherto we have, in already " class " as a dealing with arithmetic, treated primitive idea. But, for the reasons set forth in Chapter XIII., if for no others, we cannot accept " class " as a primitive idea. We must seek a definition on the same lines as the definition of descriptions,
but not otherwise.
know what
a definition
which
will assign a
meaning to propositions in whose verbal or symbolic expression words or symbols apparently representing classes occur, but which will assign a meaning that i.e.
altogether eliminates
all
mention of classes from a right analysis 181
1
82
of
Introduction
such propositions.
symbols
for
classes
objects called
to
Mathematical Philosophy
We are
" classes,"
tions, logical fictions, or (as
The theory
shall then
be able to say that the
mere conveniences, not representing and that classes are in fact, like descrip "
we
incomplete symbols." than the theory of descrip
say)
of classes is less complete
and there are reasons (which we
tions,
shall give
in
outline)
for regarding the definition of classes that will be suggested as
not finally satisfactory. Some further subtlety appears to be required ; but the reasons for regarding the definition which will be offered as being approximately correct and on the right lines
are overwhelming.
The
first
thing
is
to realise
classes
why
cannot be regarded
as part of the ultimate furniture of the world.
to explain precisely
what one means by
consequence which it implies If
we had
may
It is difficult
this statement,
be used to elucidate
its
but one
meaning.
a complete symbolic language, with a definition for
everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would repre " the ultimate furniture of sent symbolically what I mean by " " the world." I am maintaining that no symbols either for class in general or for particular classes
apparatus of undefined symbols.
would be included in
On
the other hand,
all
this
the
particular things there are in the world would have to have
names which would be included among undefined symbols. We might try to avoid this conclusion by the use of descriptions. Take (say) " the last thing Cassar saw before he died." This we might use it as (in one is a description of some particular of that particular. But perfectly legitimate sense) a definition " a " is a name for the same a in which if particular, proposition " a " occurs is not (as we saw in the preceding chapter) identical " a " we substitute with what this proposition becomes when for " the last thing Caesar saw before he died." If our language " a" or some other name for the same does not contain the name ;
particular,
we
shall
have no means
which we expressed by means
of
of expressing the proposition
" a " as opposed to the one that
Classes
we expressed by means
183
Thus
of the description.
descriptions
would not enable a perfect language to dispense with names
for
In this respect, we are maintaining, classes particulars. differ from particulars, and need not be represented by undefined
all
Our
symbols.
We
first
business
is
to give the reasons for this opinion.
have already seen that classes cannot be regarded as a on account of the contradiction about
species of individuals,
which are not members
classes
themselves
of
(explained in
Chapter XIIL), and because we can prove that the number
of
greater than the number of individuals. We cannot take classes in the pure extensional way as simply heaps or conglomerations. If we were to attempt to do that, we should find it impossible to understand how there can be such classes
is
members at all and cannot " " be regarded as a heap ; we should also find it very hard to understand how it comes about that a class which has only one member is not identical with that one member. I do not mean " to assert, or to deny, that there are such entities as heaps." As a mathematical logician, I am not called upon to have an a class as the null-class, which has no
opinion on this point.
All that I
am
maintaining
are such things as heaps, we cannot identify composed of their constituents.
We try
shall
to
come much nearer
identify
class, as
classes
we explained
positional function
and
in
Chapter
which
it
is
first is false.
For
if
if
there
classes
if
we
functions.
Every by some prothe members of the class is
defined
a class can be defined
by one
can equally well be defined by any
prepositional function, other which is true whenever the
ever the
II.,
true of
But
that,
to a satisfactory theory
with prepositional
false of other things.
is
them with the
first is
true and false
this reason the class
when
cannot be identi
with any one such prepositional function rather than with other and given a prepositional function, there are always any others which are true when it is true and false when it is many fied
" say that two prepositional functions are formally " when this happens. Two propositions are " equivaequivalent
false.
We
Introduction
184 lent
"
when both
functions
x,
equivalent to
Mathematical Philosophy
are true or both false
are
ifjx
to
"
two prepositional
;
" formally equivalent
when
is
always
It is the fact that there are other functions
iftx.
formally equivalent to a given function that makes it impossible to identify a class with a function ; for we wish classes to be such that no two distinct classes have exactly the same members, and therefore two formally equivalent functions will have to
determine the same
class.
When we have
decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or
and
aggregates,
also that they cannot be identified with pro-
becomes very difficult to see what they they are to be more than symbolic fictions. And if it
positional functions,
can be, if we can find any
way
of dealing with
them
as symbolic fictions,
we
increase the logical security of our position, since we avoid the need of assuming that there are classes without being com
make
the opposite assumption that there are no classes. abstain from both assumptions. This is an example merely " entities are not to be of Occam's razor, namely, multiplied without necessity." But when we refuse to assert that there
pelled to
We
are classes,
we must not be supposed
to be asserting dogmatically
We
are merely agnostic as regards them : " like Laplace, we can say, je n'ai pas besoin de cette hypotbese." set us forth the conditions that a symbol must fulfil if Let
that there are none.
it is
to serve as a class.
be found necessary and
I
think the following conditions will
sufficient
:
must determine a class, Every prepositional of for which the function is true. those arguments consisting Given any proposition (true or false), say about Socrates, we function
(i)
can imagine Socrates replaced by Plato or Aristotle or a
man
moon
gorilla
any other individual in the world. In general, some of these substitutions will give a true proposition
or the
and some a
in the
false one.
or
The
class
determined
will consist of all
those substitutions that give a true one. Of course, " still to decide what we mean by all those which, etc."
we have All that
Classes
we
are observing at present
by
a prepositional function,
determines an appropriate (2)
Two
formally
determine the same
is
1
that a class
is
85
rendered determinate
and that every propositional function
class.
equivalent
propositional
functions
must
and two which are not formally equiva lent must determine different classes. That is, a class is deter class,
mined by its membership, and no two different classes can have the same membership. (If a class is determined by a function x,
we say
We
that a
a
is
" member " of the
class
if
c/>a
is
true.)
some way of defining not only classes, but (3) classes of classes. We saw in Chapter II. that cardinal numbers are to be defined as classes of classes. The ordinary phrase " of elementary mathematics, The combinations of n things
m
must
at a time
all
classes of
of
n terms.
"
find
represents a class of classes, namely, the class of terms that can be selected out of a given class
m
Without some symbolic method of dealing with mathematical logic would break down.
classes of classes, It
(4)
must under
all
This
circumstances be meaningless (not of itself or not a member of
member
to suppose a class a
false) itself.
from the contradiction which we discussed
results
in
Chapter XIII. (5)
and
Lastly
this is the condition
must be possible
which
is
most
difficult
make
propositions about all the classes that are composed of individuals, or about all the " classes that are composed of objects of any one logical type."
of fulfilment,
it
were not the
to
uses of classes would go astray for example, mathematical induction. In defining the posterity of a given term, we need to be able to say that a member of the
If this
case,
many
posterity belongs to all hereditary classes to which the given
term belongs, and
this requires the sort of totality that is in
The reason
there is a difficulty about this condition question. is that it can be proved to be impossible to speak of all the propositional functions that can have arguments of a given type.
We
will,
to begin with, ignore this last condition
problems which
it
raises.
The
first
and the
two conditions may be
1
86
Introduction
taken together.
more and no
They
Mathematical Philosophy
to
state that there
less, for each
positional functions
e.g.
;
group
to be one class,
is
no
of formally equivalent pro-
the class of
men
is
to be the
same
as
that of featherless bipeds or rational animals or Yahoos or what ever other characteristic may be preferred for defining a human
Now, when we say that two formally equivalent pro-
being.
positional functions
may
be not identical, although they define
we may prove
same
the truth of the assertion by point class, a statement out that ing may be true of the one function and " " I false of the other ; e.g. believe that all men are mortal " may be true, while I believe that all rational animals are " mortal may be false, since I may believe falsely that the the
Phoenix
an immortal rational animal.
is
Thus we are
led to
consider statements about functions, or (more correctly) functions
offunctions.
Some
of the things that
may
be said about a function
may
be regarded as said about the class defined by the function, " all men are mortal " whereas others cannot. The statement " " " and " x is mortal ; or, x is human involves the functions if
we
choose,
mortals. its
We
we can say that it involves the classes men and can interpret the statement in either way, because " x is human " if we for
truth-value
"x
is
unchanged
mortal
substitute
"
any formally equivalent function. But, the statement " I believe that all men are just seen, " cannot be regarded as being about the class determined mortal or for as
is
we have
by either function, because its truth-value may be changed by the substitution of a formally equivalent function (which
We will call a statement involving " function of the function <#, if an " extensional
leaves the class unchanged).
a function "
it is
like
all
men
are mortal,"
i.e. if its
truth-value
is
unchanged
by the substitution of any formally equivalent function ; and when a function of a function is not extensional, we will call it " "I " believe that all men are mortal intensional," so that " " " x is human or x is mortal." is an intensional function of Thus
extensional functions of a function x
may,
for practical
Classes
187
purposes, be regarded as functions of the class determined by while intensional functions cannot be so regarded. It is to be observed that all the specific functions of functions
x,
we have
that
extensional.
occasion to introduce in mathematical logic are Thus, for example, the two fundamental functions
" and " <j>x is sometimes always true Each of these has its truth-value unchanged if any
of functions are
true."
formally
"
:
x
function
equivalent
language of classes,
is
if
a
is
is
substituted
for
In the " is
x.
the class determined by
x 9
(f>x
" " is always true equivalent to everything is a member of a," " " and x is sometimes true is equivalent to " a has members " " or (better) a has at least one member." Take, again, the with in the preceding chapter, for the existence dealt condition, " the term of <#." The condition is that there is a satisfying
term
such that $x
c
is
"x
always equivalent to
is
c"
This
It is equivalent to the assertion obviously extensional. that the class defined by the function (f>x is a unit class, i.e. a
is
having one member; in other words, a
class
member
of
Given a function extensional, certainly
class
which
is
a
I.
of a function
we can always
extensional
which
derive from
may it
same
function of the
or
may
not be
connected and
a
function,
by the
Let our original function of a function be one following plan which attributes to <j>x the property f\ then consider the asser :
tion
"
there
is
/
a function having the property and formally This is an extensional function of x ; it
equivalent to <#."
true when our original statement is true, and it is formally equivalent to the original function of x if this original function is extensional ; but when the original function is intensional,
is
the
new one
is
more often true than the old one.
For example,
"
I believe that all men are mortal," regarded consider again " is of a function as x human." The derived extensional function
"
There
a function formally equivalent to and such that I believe that whatever satisfies
is
:
is
This remains true
when we
substitute
"
x
is
*
it
x
is
is
human
'
mortal." "
a rational animal
1
88
Introduction
"
for
x
human," even and immortal.
is
rational
Mathematical Philosophy
to
if
I
believe falsely that the Phoenix
"
We
derived extensional function give the name of function constructed as above, namely, to the function
:
a function having the property
"
is
to the
" There
/ and
formally equivalent to " the function $x," where the original function was j>x has the property/." is
We may for its
regard the derived extensional function as having argument the class determined by the function x, and
as asserting/ of this class.
about a To assert that " the
of a proposition
This
class.
class
may
I.e.
be taken as the definition
we may
define
:
determined by the function
has the property/" is to assert that function derived from/.
<j>x
satisfies
x
the extensional
This gives a meaning to any statement about a class which
can be made significantly about a function ; and it will be found that technically it yields the results which are required
make a theory symbolically satisfactory. 1 What we have said just now as regards the definition of classes is sufficient to satisfy our first four conditions. The
in order to
way
in
which
it
bility of classes of classes,
or not being a
explained in
granted here.
and fourth, namely, the possi
secures the third
member
and the impossibility of a class being is somewhat technical it is
of itself,
Principia Mathematics It results that,
;
but
but for our
may
be taken for
fifth condition,
we
might regard our task as completed. But this condition at once the most important and the most difficult is not fulfilled in virtue of anything
we have
said as yet.
The
difficulty is
connected with the theory of types, and must be briefly discussed. 2 We saw in Chapter XIII. that there is a hierarchy of logical types, and that it is a fallacy to allow an object belonging to one of these to be substituted for an object belonging to another. See Principia Mathematica, vol. i. pp. 75-84 and * 20. desires a fuller discussion should consult Principia Mathematica, Introduction, chap, ii.; also * 12. 1 2
The reader who
Classes
Now
it is
not
difficult to
189
show that the various functions which
can take a given object a as argument are not all of one type. Let us call them all ^-functions. We may take first those among
them which do not involve reference to any collection of functions " If we now proceed these we will call predicative ^-functions." ;
to functions involving reference to the totality of predicative ^-functions, we shall incur a fallacy if we regard these as of the
same type as the predicative ^-functions. Take such an every " a is a How shall day statement as typical Frenchman." " " we define a Frenchman ? We may define him as typical one " possessing all qualities that are possessed by most French men." But unless we confine " all qualities " to such as do not involve a reference to any totality of qualities, we shall have to observe that most Frenchmen are not typical in the above sense, and therefore the definition shows that to be not typical is essential to a typical
Frenchman.
diction, since there
no reason why there should be any typical illustrates the need for separating off
Frenchmen;
but
is it
This
is
not a logical contra
qualities that involve reference to a totality of qualities
from
those that do not.
" Whenever, by statements about
all
"
or
" some " of the
values that a variable can significantly take, we generate a new object, this new object must not be among the values which
our previous variable could take, since, if it were, the totality which the variable could range would only be definable in terms of itself, and we should be involved in a vicious of values over
circle.
For example,
that
make
way
that
if
I
say "Napoleon had " I must define
the qualities " in such a qualities
a great general,"
it will
not include what
I
am now
all
saying,
i.e.
"
having
"
the qualities that make a great general must not be itself a in the sense This is quality supposed. fairly obvious, and is all
the principle which leads to the theory of types by which viciouscircle paradoxes are avoided. As applied to ^-functions, we " " " may suppose that qualities is to mean predicative functions." " Then when I say Napoleon had all the qualities, etc.," I mean
Introduction
190 "
Napoleon
satisfied
all
Mathematical Philosophy
to
This
the predicative functions, etc."
statement attributes a property to Napoleon, but not a pre dicative property ; thus we escape the vicious circle. But wherever " all functions which " occurs, the functions in question
must be limited and, as
type
is
to one type
if
a vicious circle
is
to be avoided
;
Napoleon and the typical Frenchman have shown, the not rendered determinate by that of the argument.
would require a much fuller discussion to set forth fully, but what has been said may suffice to make it
It
this point
clear that
the functions which can take a given argument are of an infinite series of types. We could, by various technical devices, con
would run through the first n of these is finite, but we cannot construct a variable which where n types, will run through them all, and, if we could, that mere fact would struct a variable which
at once generate a
new type
and would
whole process going again.
set the
of function with the
same arguments,
We
call predicative ^-functions the first type of ^-functions ; ^-functions involving reference to the totality of the first type
we
call
the second, type ; and so on. all these different types
can run through
some
No :
it
variable ^-function
must stop short
at
definite one.
These considerations are relevant to our definition derived extensional function.
We
formally equivalent to
It is
the type of our function. Let us is unavoidable.
fa" Any
of
the
" a function there spoke of necessary to decide upon
decision will do, but
some
decision
the supposed formally equivalent appears as a variable, and must be of call
function 0. Then ^ some determinate type.
All that we know necessarily about takes (/> arguments of a given type that it is (say) an ^-function. But this, as we have just seen, does not determine its type. If we are to be able (as our fifth requisite
the type of
is
that
it
demands) to deal with all classes whose members are of the same type as a, we must be able to define all such classes by means of functions of
some one type
type of ^-function, say the n
to say, there must be some such that any ^-function is formally
that
;
ih 9
is
Classes
191
th If this is the case, equivalent to some ^-function of the n type. then any extensional function which holds of all ^-functions
n th type
hold of any ^-function whatever. It is chiefly as a technical means of embodying an assumption leading to The assumption is called the this result that classes are useful. " be stated as follows and of the
axiom
will
of reducibility,"
" There
is
tf-function, it is
may
:
(r say) of ^-functions
a type
such that, given any
formally equivalent to some function of the type
in question."
axiom is assumed, we use functions of this type in Statements about our associated extensional function. defining all ^-classes defined all classes (i.e. by ^-functions) can be reduced If this
to statements about
all
^-functions of the type r.
So long as
only extensional functions of functions are involved, this gives us in practice results which would otherwise have required the impossible notion of
where
this is vital is
The axiom
"
all
^-functions."
One
particular region
mathematical induction.
of reducibility involves all that
in the theory of classes.
It is
is
really essential
therefore worth while to ask
whether there
is any reason to suppose it true. This axiom, like the multiplicative axiom and the axiom of infinity, is necessary for certain results, but not for the bare
existence
of
deductive reasoning.
as explained in Chapter XIV.,
The theory
and the laws
of
deduction,
for propositions
" all " and " some," are of the very texture of mathe involving matical reasoning without them, or something like them, :
we should not merely not obtain
the
same
not obtain any results at
We
cannot use them as hypo
theses,
all.
results,
and deduce hypothetical consequences,
but we should
for
they
are
rules of deduction as well as premisses. They must be absolutely or else what we deduce according to them does not even true,
On the other hand, the axiom of two previous mathematical axioms, could perfectly well be stated as an hypothesis whenever it is used, instead of being assumed to be We can deduce actually true.
follow from the premisses. reducibility, like our
Introduction
192
Mathematical Philosophy
to
consequences hypothetically ; we can also deduce the con sequences of supposing it false. It is therefore only convenient, not necessary. And in view of the complication of the theory
its
and
most general principles, impossible as yet to say whether there may not be some way of dispensing with the axiom of reducibility of types,
it
altogether.
of the uncertainty of all except its
is
However, assuming the correctness of the theory what can we say as to the truth or falsehood of
outlined above, the axiom ?
The axiom, we may
observe,
identity of indiscernibles.
is
a generalised form of Leibniz's
Leibniz assumed, as a logical principle,
that two different subjects must differ as to predicates. Now " predicates are only some among what we called predicative functions," which will include also relations to given terms,
and various properties not to be reckoned as predicates. Thus Leibniz's assumption is a much stricter and narrower one than ours. (Not, of course, according to his logic, which regarded all
as
propositions
But there
is
reducible
no good reason
to
the subject-predicate form.)
for believing his form, so far as I
can
There might quite well, as a matter of abstract logical possibility, be two things which had exactly the same predicates, see.
narrow sense in which we have been using the word " pre dicate." How does our axiom look when we pass beyond pre
in the
dicates in this narrow sense
no way
?
In the actual world there seems
of doubting its empirical truth as regards particulars,
no two particulars owing to spatio-temporal differentiation have exactly the same spatial and temporal relations to all other :
particulars.
But
this
is,
as
it
were, an accident, a fact about
we happen to find ourselves. Pure logic, and pure mathematics (which is the same thing), aims at being the world in which
true, in Leibnizian phraseology, in all possible worlds, not only
in this higgledy-piggledy job-lot of a world in which chance has imprisoned us. There is a certain lordliness which the logician
should preserve
:
he must not condescend to derive arguments
from the things he sees about him.
Classes
Viewed from
193
this strictly logical point of view, I
do not see
any reason to believe that the axiom of reducibility is logically necessary, which is what would be meant by saying that it is true in
all
The admission
possible worlds.
a system of logic ically true.
is
therefore a defect, even
It is for this reason that the
axiom into
of this
if
the axiom
is
empir
theory of classes cannot
be regarded as being as complete as the theory of descriptions. is need of further work on the theory of types, in the hope
There
which does not require such a reasonable to regard the theory
of arriving at a doctrine of classes
dubious assumption.
But
it is
outlined in the present chapter as right in its main lines, i.e. in reduction of propositions nominally about classes to pro
its
The avoidance of positions about their defining functions. classes as entities by this method must, it would seem, be sound in principle, It is
however the
detail
may
still
because this seems indubitable that
require adjustment. the
we have included
theory of classes, in spite of our desire to exclude, as far as possible,
whatever seemed open to serious doubt. The theory of classes, as above outlined, reduces
axiom and one
definition.
For the sake of
here repeat them. The axiom is There is a type r such that if $
itself to
one
we
will
definiteness,
:
is
a function which can take a
given object a as argument, then there is a Junction
r which
The If and r
the is
is formally
definition is
is
equivalent
to
$
of the type
<j>.
:
a function which can take a given object a as argument, mentioned in the above axiom, then to say that
the type
class
determined by
a function of type
property f.
T,
<j>
has the property
formally equivalent
f is
to
to <,
say that there
and having
the
CHAPTER
XVIII
MATHEMATICS AND LOGIC
MATHEMATICS and distinct studies.
logic, historically speaking,
have been entirely
Mathematics has been connected with
science,
But both have developed in modern times has become more mathematical and mathematics has logic become more logical. The consequence is that it has now become wholly impossible to draw a line between the two ; in fact, the two are one. They differ as boy and man logic is the youth of mathematics and mathematics is the manhood of logic. This
logic
with Greek.
:
:
view
is
resented
by
logicians
who, having spent their time in
the study of classical texts, are incapable of following a piece of symbolic reasoning,
and by mathematicians who have learnt
a technique without troubling to inquire into
its meaning or are Both now justification. types fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and
formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail starting with :
premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously
belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathe matics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and 194
Mathematics and Logic
195
deductions of Principia Maihematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.
In the earlier chapters of this book, starting from the natural " defined number " and shown
numbers, we have
how
cardinal
first
to generalise the
conception of number, and have then
analysed the conceptions involved in the definition, until we found In a synthetic, ourselves dealing with the fundamentals of logic.
deductive treatment
these fundamentals
come
first,
and the
natural numbers are only reached after a long journey. Such treatment, though formally more correct than that which we
more difficult for the reader, because the ultimate logical concepts and propositions with which it starts are remote and unfamiliar as compared with the natural numbers. Also they represent the present frontier of knowledge, beyond which is the still unknown ; and the dominion of knowledge over them have adopted,
is
is
not as yet very secure. It used to be said that mathematics
"
"
Quantity
is
we may
is
the science of
"
a vague word, but for the sake of
quantity."
argument
"
number." The statement replace by the word that mathematics is the science of number would be untrue
in
two
it
different ways.
On
the one hand, there are recognised
branches of mathematics which have nothing to do with number all geometry that does not use co-ordinates or measurement,
example : projective and descriptive geometry, down to the point at which co-ordinates are introduced, does not have to do with number, or even with quantity in the sense of greater
for
On the other hand, through the definition of cardinals, less. through the theory of induction and ancestral relations, through and
the general theory of series, and through the definitions of the arithmetical operations, it has become possible to generalise much that used to be proved only in connection with numbers. The result is that what was formerly the single study of Arithmetic has now become divided into numbers of separate studies, no
one of which
is
specially concerned with numbers.
The most
196
Introduction
to
Mathematical Philosophy
elementary properties of numbers are concerned with one-one Addition is concerned relations, and similarity between classes.
with the construction of mutually exclusive classes respectively similar to a set of classes which are not known to be mutually " exclusive. selec Multiplication is merged in the theory of a kind i.e. of certain of relations. Finitude tions," one-many
merged in the general study of ancestral relations, which yields the whole theory of mathematical induction. The ordinal
is
properties of the various kinds of number-series, and the elements of the theory of continuity of functions and the limits of functions,
can be generalised so as no longer to involve any essential reference to numbers. It is a principle, in all formal reasoning, to generalise to the utmost, since we thereby secure that a given process of deduction shall have more widely applicable results ; we are, therefore,
in
thus
generalising
the
reasoning
of
arithmetic,
merely following a precept which is universally admitted in mathematics. And in thus generalising we have, in effect, created a set of arithmetic
is
new deductive
at once dissolved
systems, in which traditional
and enlarged
;
but whether any
one of these new deductive systems for example, the theory of is to be said to belong to logic or to arithmetic is
selections
entirely arbitrary,
and incapable
of being decided rationally.
We is
are thus brought face to face with the question : this subject, which may be called indifferently either
What mathe
matics or logic ? Is there any way in which we can define it ? Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties : we deal formally with what can be said are prepared to say that about any thing or any property.
We
one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians,
we have never heard
of Socrates
and Plato.
there were no such individuals would
one and one are two.
It is
still
A
world in which
be a world in which
not open to us, as pure mathematicians
or logicians, to mention anything at
all,
because,
if
we do
so,
Mathematics and Logic
197
We may
we introduce something irrelevant and not formal. make this clear by applying it to the case of the " All men are Traditional logic says mortal, Socrates it is clear that is mortal." Now therefore Socrates :
syllogism. is a man,
what we
mean
to assert, to begin with, is only that the premisses imply the conclusion, not that premisses and conclusion are actually true ; even the most traditional logic points out that the actual
truth of the premisses is irrelevant to logic. Thus the first change to be made in the above traditional syllogism is to state in the
it
form
then Socrates
" :
If all
is
mortal."
convey that
to
this
men
are mortal
and Socrates
is
a
man,
We may now observe that it is intended
argument
is
valid in virtue of
in virtue of the particular terms occurring in
it.
its
form, not
If
we had
" from our is a man premisses, we should a non-formal argument, only admissible because
omitted " Socrates
have
had
Socrates
is
in fact a
ised the argument.
man
;
in that case
But when,
we could not have
general
as above, the
argument is formal, Thus we may in it. that occur nothing depends upon the terms substitute a for men, j8 for mortals, and x for Socrates, where and j3 are any classes whatever, and x is any individual. We " No matter what then arrive at the statement possible values :
a's are j8's and x is an a, then x ' " is a the prepositional function if all a's in other words, ; j8 Here at last are ]8 and x is an a, then x is a j8 is always true."
x and a and "
j3
may
have,
if all
'
we have
a proposition of logic the one which the traditional statement about Socrates and It is clear that,
we in
shall
is
is
only suggested by
men and
mortals.
what we are aiming
if formal reasoning at, always arrive ultimately at statements like the above,
which no actual things or properties are mentioned ; this happen through the mere desire not to waste our time proving It would be a particular case what can be proved generally.
will
in
argument about Socrates, and then go through precisely the same argument again about Plato. If our argument is one (say) which holds of all men, we shall prove " it x" with the hypothesis " if x is a man." With concerning ridiculous to go through a long
Introduction
198
this hypothesis, the
even when x
is
Mathematical Philosophy
to
argument will retain But now we
not a man.
its
hypothetical validity
shall find that our argu
ment would still be valid if, instead of supposing x to be a man, we were to suppose him to be a monkey or a goose or a Prime
We
Minister.
"x premiss class
is
waste our time taking as our but shall take " x is an a," where a is any " " is or where
shall therefore not
man "
a
of individuals,
(f>x
is
"
a necessary result of the fact that this study purely formal."
At is
this point
we
The problem
we
say,
problem which
find ourselves faced with a
than to solve.
easier to state
as
is,
is
" :
What
are
"
the constituents of a logical proposition ? I do not know the but I to how the answer, explain problem arises. propose Take (say) the proposition " Socrates was before Aristotle." Here it seems obvious that we have a relation between two terms,
and that the constituents
of the proposition (as well as of the are simply the two terms and the relation, corresponding fact) i.e. Socrates, Aristotle, and before. (I ignore the fact that
Socrates and Aristotle are not simple ; also the fact that what appear to be their names are really truncated descriptions. is relevant to the present issue.) We may " such x form of the y," general propositions by represent " to y." This general x has the relation which may be read form may occur in logical propositions, but no particular instance
Neither of these facts
R
R
of it
can occur.
Are we to
infer that the general
constituent of such logical propositions ? " Socrates Given a proposition, such as
we have form
is
certain constituents
not
new form
itself
to
a
new
and
it
;
if it
were,
itself is
But the
we should need
and the other constituents.
can, in fact, turn all the constituents
of a
a
before Aristotle,"
also a certain form.
constituent
embrace both
is
form
proposition
a
We into
form unchanged. This is what we " do when we use such a schema as x R y," which stands for any
variables, while keeping the
Mathematics and Logic
199
one of a certain class of propositions, namely, those asserting between two terms. We can proceed to general asser " " such i.e. there are cases as x R y is sometimes true tions,
relations
where dual relations hold.
This assertion will belong to logic
(or mathematics) in the sense in which we are using the word. But in this assertion we do not mention any particular things or particular relations ; no particular things or relations can
ever enter into a proposition of pure logic. We are left with pure forms as the only possible constituents of logical propositions. I do not wish to assert e.g. the positively that pure forms
form " x
we
Ry
"
do actually enter into propositions of the kind The question of the analysis of such pro
are considering.
positions
is
a difficult one, with conflicting considerations on the
one side and on the other.
now, but we
may
We cannot embark upon this question
accept, as a first approximation, the view
what enter into logical propositions as their constituents. And we may explain (though not formally define) what we mean by the " form " of a proposition as follows The " form " of a proposition is that, in it, that remains un changed when every constituent of the proposition is replaced that forms are
:
by another. Thus " Socrates as
"
is earlier
than Aristotle " has the same form
is greater than Wellington," though every con two propositions is different. may thus lay down, as a necessary (though not sufficient)
Napoleon
stituent of the
We
characteristic of logical or mathematical propositions, that they
are to be such as can be obtained
no variables
no such words as
from a proposition containing
some, a, the, etc.) by turning every constituent into a variable and asserting that the result is
(i.e.
all,
always true or sometimes true, or that
respect of
some
it is
of the variables that the result
is
always true in sometimes true
any variant of these forms. And same thing is to say that logic (or concerned only with forms, and is concerned
in respect of the others, or
another
way
of stating the
mathematics) is with them only in the way of stating that they are always or
2oo
Introduction
Mathematical Philosophy
to
sometimes true with all the permutations of " always " and " sometimes " that may occur.
There are
language some words whose sole function is These words, broadly speaking, are commonest in languages having fewest inflections. Take " Socrates is " " human." Here is is in every
to indicate form.
not a constituent of the proposition,
but merely indicates the subject-predicate form. Similarly " in Socrates is earlier than Aristotle," " is " and " than " " Socrates merely indicate form ; the proposition is the same as in which these words have disappeared precedes Aristotle,"
and the form
is
otherwise indicated.
Form, as a
indicated otherwise than by specific words
:
rule,
can be
the order of the
words can do most of what is wanted. But this principle must not be pressed. For example, it is difficult to see how we could (i.e.
We
conveniently
express
molecular forms
of
propositions
what we call " truth-functions ") without any word at saw in Chapter XIV. that one word or symbol is enough
this purpose,
namely, a
for
word or symbol expressing incompati
But without even one we should
bility.
all.
find ourselves in
not the point that
diffi
important for our present purpose. What is important for us is to observe that form may be the one concern of a general proposition, culties.
This, however,
is
is
even when no word or symbol in that proposition designates the form. If we wish to speak about the form itself, we must
have a word about
all
;
but
if,
as in mathematics,
we wish
to speak
propositions that have the form, a word for the form
will usually is
for it
be found not indispensable
;
probably in theory
it
never indispensable.
Assuming
as I think
we may
that the forms of propositions
can be represented by the forms of the propositions in which they are expressed without any special word for forms, we should arrive at a language in which everything formal belonged to
syntax and not to vocabulary. In such a language we could express all the propositions of mathematics even if we did not
know one
single
word
of the language.
The language
of
mathe-
Mathematics and Logic
201
matical logic, if it were perfected, would be such a language. We should have symbols for variables, such as " x " and " R " and " y," arranged in various ways ; and the way of arrange ment would indicate that something was being said to be true of all
some values
values or
of the variables.
We
should not need
know any
words, because they would only be needed for giving values to the variables, which is the business of the applied to
mathematician, not of the pure mathematician or logician. It is one of the marks of a proposition of logic that, given a suitable language, such a proposition can be asserted in such a language by a person a single
word
who knows
the syntax without knowing
of the vocabulary.
" " is But, after all, there are words that express form, such as " and than." And in every symbolism hitherto invented for mathematical logic there are symbols having constant formal meanings. We may take as an example the symbol for in compatibility which is employed in building Such words or symbols may occur in logic.
How
are
we
to define
them
up truth-functions. The question is :
?
Such words or symbols express what are called " constants."
Logical
we denned forms
A
fundamental
among
a
;
constants
may
defined
in fact, they are in essence the
logical constant will
number
be
of propositions,
be that which
any one
of
is
logical
exactly
same in
as
thing.
common
which can result
from any other by substitution of terms one for another. For " " results from example, Napoleon is greater than Wellington " " Socrates is earlier than Aristotle the substitution of
by
for
"Socrates," "Wellington" for "Aristotle," and " greater " for " earlier." Some propositions can be obtained in this way from the prototype " Socrates is earlier than Aris " totle and some cannot ; those that can are those that are of " R the form x We cannot obtain y," i.e. express dual relations.
"Napoleon"
from the above prototype by term-for-term substitution such " " " the Athenians Socrates is human or propositions as gave the hemlock to Socrates," because the first is of the subject-
2O2
Introduction
to
Mathematical Philosophy
predicate form and the second expresses a three-term relation. If we are to have any words in our pure logical language, they
"
and "
must be such
as
" constants
always either be, or be derived from, what
will
express
constants,"
logical
logical is
in
common among
a group of propositions derivable from each And other, in the above manner, by term-for-term substitution. this
which
in
is
In this sense
" is what we call form." " " the constants that occur in pure mathe
common all
matics are logical constants.
The number
for
I,
" There
derivative from propositions of the form : is true when, and (f>x only when, x
such that
is
example, is
c"
is
a term c
This
is
a
function of ^, and various different propositions result from giving different values to <. may (with a little omission of intermediate steps not relevant to our present purpose) take
We
the above function of
mined by ^
member
as
what
is
meant by " the
" or " a unit class the
is
class
"
class deter
determined by is a In this way, proposi <j>
being a class of classes). tions in which I occurs acquire a meaning which is derived from a certain constant logical form. And the same will be found of
I
(i
to be the case with
all
mathematical constants
:
all
are logical
constants, or symbolic abbreviations whose full use in a proper context is defined by means of logical constants.
But although all logical (or mathematical) propositions can be expressed wholly in terms of logical constants together with variables, it is not the case that, conversely, all propositions
We
that can be expressed in this way are logical. have found so far a necessary but not a sufficient criterion of mathematical have sufficiently defined the character of the propositions.
We
primitive ideas in terms of
which
all
the ideas of mathematics
can be defined, but not of the primitive propositions from which all the propositions of mathematics can be deduced. This is a
more full
difficult
answer
We may
matter, as to which
it is
not yet
known what
the
is.
take the axiom of infinity as an example of a pro position which, though it can be enunciated in logical terms,
Mathematics and Logic
203
cannot be asserted by logic to be true. All the propositions of logic have a characteristic which used to be expressed by saying that they were analytic, or that their contradictories were selfcontradictory. factory.
mode
This
The law it
of statement, however, is not satis
of contradiction is merely
one among logical
has no special pre-eminence
propositions ; that the contradictory of
some proposition
is
;
and the proof
self-contradictory
likely to require other principles of deduction besides the law of contradiction. Nevertheless, the characteristic of logical is
propositions that we are in search of and intended to be defined, by those in deducibility
is
the one which was
who
said that
from the law of contradiction.
moment, we may
istic, which, for the
it
felt,
consisted
This character
call tautology,
obviously does not belong to the assertion that the number of individuals in the universe is whatever number n may be. But for the , diversity of types, it would be possible to prove logically that there are classes of n terms, where n is any finite integer ; or even
N terms.
But, owing to types, such are left fallacious. are Chapter XIII., to empirical observation to determine whether there are as many " " as n individuals in the world. worlds, Among possible that there are classes of
proofs, as
we saw
We
in
be worlds having one, two, There does not even seem any logical there should be even one individual 1 why, in
in the Leibnizian sense, there will three,
.
.
.
individuals.
necessity why fact, there should be of the existence of
any world
God,
if
existence
i.e. it fails
and
at
all.
The
ontological proof
were valid, would establish the
one individual.
logical necessity of at least
recognised as invalid,
it
in fact rests
upon
But it is generally a mistaken view of
to realise that existence can only be asserted
of
something described, not of something named, so that it is " " this and " the is the so-and-so meaningless to argue from " " so-and-so exists to this exists."
If
we
reject the ontological
1 The primitive propositions in Principia Mathematica are such as to allow the inference that at least one individual exists. But I now view
this as
a defect in
logical purity.
Introduction
204
to
Mathematical Philosophy
argument, we seem driven to conclude that the existence of a world is an accident i.e. it is not logically necessary. If that be so, no principle of logic can assert " existence " except under a hypothesis, i.e. none can be of the form " the prepositional function
so-and-so
is
form, when they occur
sometimes true." in logic, will
of
Propositions
this
have to occur as hypotheses
or consequences of hypotheses, not as complete asserted pro The complete asserted propositions of logic will all positions.
be such as affirm that some prepositional function
For example,
is
always true.
always true that p implies q and q implies p implies r, or that, if all a's are jS's and x is an a then a ]8. Such propositions may occur in logic, and their truth it is
if
r then
x
is
independent of the existence of the universe. We may lay that, if there were no universe, all general propositions
is
it
down
would be true
we saw
;
for the contradictory of a general proposition
Chapter XV.) is a proposition asserting existence, and would therefore always be false if no universe existed. (as
in
Logical propositions are such as can be
study of the actual world.
known
a
'priori,
without
We
only know from a study of a man, but we know the correct
empirical facts that Socrates is ness of the syllogism in its abstract form
(i.e.
when
it is
stated
in terms of variables) without needing
any appeal to experience. a characteristic, not of logical propositions in themselves, but of the way in which we know them. It has, however, a This
is
bearing upon the question what their nature may be, since there are some kinds of propositions which it would be very difficult to suppose It
is
we could know without
clear that the definition of
experience.
"
" logic
or
" mathematics "
must be sought by trying to give a new definition of the old " " propositions. Although we can no longer analytic
notion of
be satisfied to define logical propositions as those that follow still admit that
from the law of contradiction, we can and must
they are a wholly different class of propositions from those that
we come which, a
know empirically. They all have the characteristic moment ago, we agreed to call " tautology." This,
to
Mathematics and Logic combined with the of variables
and
205
fact that they can be expressed wholly in terms
logical constants (a logical constant being
some
thing which remains constant in a proposition even when all will give the definition of logic constituents are changed)
its
For the moment, I do not know how to It would be easy to offer a definition
or pure mathematics. define
" tautology."
1
which might seem satisfactory for a while ; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of
At
which a definition
moment, we reach
this point, therefore, for the
knowledge on our backward journey into the tions of mathematics.
of
We
have now come to an end
of our
is
wanted.
the frontier
logical
founda
somewhat summary intro
duction to mathematical philosophy. It is impossible to convey adequately the ideas that are concerned in this subject so long Since ordinary as we abstain from the use of logical symbols.
language has no words that naturally express exactly what we wish to express, it is necessary, so long as we adhere to ordinary language, to strain words into unusual meanings ; and the reader to lapse into attaching the usual meanings to words, thus arriving at wrong notions as to what is
is
sure, after a time
if
not at
first,
Moreover, ordinary grammar and syntax extraordinarily misleading. This is the case, e.g., as regards numbers ; " ten men " is grammatically the same form as " white men," so that 10 might be thought to be an adjective " men." It is the wherever
intended to be said. is
qualifying functions are involved,
case, again,
and
propositional
in particular as regards existence
and
Because language is misleading, as well as because descriptions. and inexact when applied to logic (for which it was diffuse is it never intended), logical symbolism is absolutely necessary to any exact or thorough treatment of our subject. Those readers, " " of for a definition of mathematics was tautology me out to former by my Ludwig Wittgenstein, who was pupil pointed working on the problem. I do not know whether he has solved it, or even whether he is alive or dead. 1
The importance
206
Introduction
therefore,
who wish
mathematics,
to
to acquire a mastery of the principles of to be hoped, not shrink from the labour
will, it is
of mastering the symbols
than might be thought.
made
Mathematical Philosophy
a labour which
is,
in fact,
less
As the above hasty survey must have
evident, there are innumerable unsolved problems in the
subject,
and much work needs to be done.
If
led into a serious study of mathematical logic
book,
much
it will
written.
have served the
chief purpose for
any student
by
which
this it
is
little
has been
INDEX Aggregates, 12. Alephs, 83, 92, 97, 125.
Equivalence, 183. Euclid, 67. Existence, 164, 171, 177. Exponentiation, 94, 120.
Aliorelatives, 32. All, 158 &.
Analysis, 4. Ancestors, 25, 33. Argument of a function, 47, 108. Arithmetising of mathematics, 4. Associative law, 58, 94.
Axioms,
i.
Between, 38
ff.,
58.
Bolzano, 138 n.
Cantor, Georg, 77, 79, 85
86, 89, ., 95, 102, 136. 181 ff. ; Classes, 12, 137, reflexive, 80, 127, 138 ; similar, 15, 16. Clifford, W. K., 76. Collections, infinite, 13. Commutative law, 58, 94. Conjunction, 147. Consecutiveness, 37, 38, 81. Constants, 202.
Construction, method of, 73. Continuity, 86, 97 ff. ; Cantorian, 102 ff. ; Dedekindian, 101 ; in philos ophy, 105 ; of functions, 106 ff. Contradictions, 135 ff.
Convergence, 115. Converse, 16, 32, 49.
99.
Geometry, 29, 59, 67, 74, 100, 145
;
analytical, 4, 86. less, 65, 90.
Greater and Hegel, 107.
Hereditary properties, 21. Implication, 146, 153
;
formal, 163.
Incommensurables, 4, 66. Incompatibility, 147 ff., 200. Incomplete symbols, 182.
ff.,
87, 93,
185.
extensional and inten
Inductive properties, 21. Inference, 148 ff. Infinite, 28 ; of ratipnals, 65 ; Can torian, 65 ; of cardinals, 77 ff. ; and series and ordinals, 89 ff. 77, 131 ff., Infinity, axiom of, 66 n., 202. Instances, 156. Integers, positive Intervals, 115. Intuition, 145.
ff.
Dimensions, 29. Disjunction, 147. Distributive law, 58, 94. Diversity, 87.
Domain,
ff.,
Generalisation, 156.
Induction, mathematical, 20
Dedekind, 69, 99, 138 n. Deduction, 144 ff. Derivatives, 100. Descriptions, 139, 144, 167
Gap, Dedekindian, 70
Indiscernibles, 192. Individuals, 132, 141, 173.
Correlators, 54.
Counterparts, objective, 61. Counting, 14, 16.
;
Fictions, logical, 14 n., 45, 137. Field of a relation, 32, 53. Finite, 27. Flux, 105. Form, 198. Fractions, 37, 64. Frege, 7, 10, 25 n., 77, 95, 146 Functions, 46 ; descriptive, 46, 180 ; intensional and extensional, 186 ; predicative, 189 ; prepositional, 46, 144, 155 ff. .
Boots and socks, 126. Boundary, 70, 98, 99.
Definition, 3 sion al, 12.
Extension of a relation, 60.
and negative,
Irrationals, 66, 72.
16, 32, 49.
307
64.
208
Introduction
Mathematical Philosophy
to
Kant, 145.
Quantity, 97, 195.
Leibniz, 80, 107, 192. Lewis, C. I., 153, 154. Likeness, 52. Limit,29,69ff.,97ff.; of functions, 1 06 ff. Limiting points, 99. Logic, 159, 169, 194 ff. ; mathematical, v, 201, 206. Logicising of mathematics, 7.
Ratios, 64, 71, 84, 133. Reducibility, axiom of, 191. Referent, 48.
Maps, 52, 60 ff., 80. Mathematics, 194 ff.
44 transitive, 16, 32. Relatum, 48.
Maximum,
Representatives, 120. Rigour, 144.
Relation numbers, 56 ff. Relations, asymmetrical, 31, 42 ; con nected, 32 ; many-one, 15 ; onemany, 15, 45; one-one, 15, 47, 79 J reflexive, 16 ; serial, 34 ; similar, 52 ff ; squares of, 32 symmetrical, ;
Median
16,
70, 98.
class, 104.
;
Meinong, 169. Method, vi.
Royce, 80.
Minimum,
Section, Dedekindian, 69
70, 98.
Multiplication, 118 ff. Multiplicative axiom, 92, 117
ff.
173, 182. Necessity, 165. Neighbourhood, 109. Nicod, 148, 149, 151 H. Null-class, 23, 132. Number, cardinal, 10 ff., 56, 77 ff.
finite,
;
ductive, 27, 78, 131 irrational, 66, 72 multipliable, 130
; ;
;
20
closed, 103 ; compact, ; 100 condensed in itself, 102 ; Dedekindian, 71, 73, 101 ; generation of, 41 ; infinite, 89 ff. ; perfect, 102, 103 ; well-ordered, 92,
Series, 29 66, 93,
ff.,
ff.
infinite,
maximum natural, 2
;
95
;
in
77 135 22 ff.
?
ff.,
; ; ;
non-inductive, 88, 127 ; real, 66, 72, 84 ; reflexive, 80, 127 ; relation, 56, 94 ; serial, 57-
Occam,
ultimate,
;
Segments, 72, 98. Selections, 117 ff. Sequent, 98.
Names,
complex, 74
ff.
in.
Modality, 165.
184.
Occurrences, primary and secondary, 179-
Ontological proof, 203. Order, 29 ff. ; cyclic, 40. Oscillation, ultimate, in.
Parmenides, 138. Particulars, 140 ff., 173. Peano, 5 ff., 23, 24, 78, 81, 131, 163. Peirce, 32 n. Permutations, 50. Philosophy, mathematical, v, i. Plato, 138. Plurality, 10. Poincare, 27. Points, 59. Posterity, 32 ff. ; proper, 36. Postulates, 71, 73.
ff.
;
123. Sheffer, 148. Similarity, of classes, tions, 52 ff., 83.
Some, 158
15
ff.
;
of rela
ff.
Space, 61, 86, 140. Structure, 60 ff. Sub-classes, 84 ff. Subjects, 142. Subtraction, 87. Successor of a number, 23, 35. Syllogism, 197.
Tautology, 203, 205. The, 167, 172 ff.
Time, 61,
86, 140.
Truth-function, 147. Truth-value, 146. Types, logical, 53, 135
ff.,
185, i&8.
Unreality, 168.
Value of a function, 47, 108. Variables, 10, 161, 199.
Veblen, 58. Verbs, 141.
Precedent, 98. Premisses of arithmetic, 5. Primitive ideas and propositions, 5, 202. Progressions, 8, 81 ff. Propositions, 155 ; analytic, 204 ; ele
Weierstrass, 97, 107. Wells, H. G., 114. Whitehead, 64, 76, 107, 119. Wittgenstein, 205 n.
mentary, 161. Pythagoras, 4, 67.
Zermelo, 123, 129. Zero, 65.
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EDINBURGH.
UA y KO snc Russe Bertrand, 1872-1970. Introduction to mathematical phi losophy AIC-3633 (mbab) .
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