Introduction to Mathematical Philosophy

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INTRODUCTION TO MATHEMATICAL PHILOSOPHY

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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 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 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, *