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Full text of "Introduction to mathematical philosophy"

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EDITED BY J. H. MUIRHEAD, LL.D. 



INTRODUCTION TO MATHEMATICAL 
PHILOSOPHY 




the same j4uthor. 



PRINCIPLES OF SOCIAL RECONSTRUC 
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London : George Allen &** Unwin, Ltd. 



INTRODUCTION TO 

MATHEMATICAL 

PHILOSOPHY 



BY 



BERTRAND RUSSELL 




LONDON : GEORGE ALLEN & UNWIN, LTD. 
NEW YORK: THE MACMILLAN CO, 






First published May 1919 
Second Edition April 1920 



[All rights reserved} 



PREFACE 

THIS book is intended essentially as an " Introduction," and 
does not aim at giving an exhaustive discussion of the problems 
with which 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 
at least equal philosophical interest. Much of what is set forth 
in the following chapters is 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 
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 
to 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 
are known. A book dealing with those parts may, therefore, 
claim to be an introduction to mathematical philosophy, though 
it can hardly claim, except where it steps outside its province, 
to be actually dealing with a part of philosophy. It does deal, 



vi Introduction to Mathematical Philosophy 

however, with a body of knowledge which, to those who accept 
it, 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 
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 
a book as the following. It is to be hoped that some readers 
may be sufficiently interested to advance to a study of the 
method by which mathematical logic can be made helpful in 
investigating the traditional problems of philosophy. But that 
is a topic with which the following pages have not attempted 
to deal. 

BERTRAND RUSSELL. 



EDITOR'S NOTE 

THOSE who, relying on the distinction between Mathematical 
Philosophy and the Philosophy of Mathematics, think that this 
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 
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 bearing of the definitions and 
discussions that follow on the work of " traditional philosophy." 
If philosophers cannot consent to relegate the criticism of these 
categories to any of the special sciences, it is essential, at any 
rate, that they should know the precise meaning that the science 
of mathematics, in which these concepts play so large a part, 
assigns to them. If, on the other hand, there be mathematicians 
to whom these definitions and discussions seem to be an elabora 
tion and complication of the simple, it may be well to remind 
them from the side of philosophy that here, as elsewhere, apparent 
simplicity may conceal a complexity which it is the business of 
somebody, whether philosopher or mathematician, or, like the 
author of this volume, both in one, to unravel. 




vii 



CONTENTS 



CHAP. PAGE 

PREFACE ........ V 

EDITOR'S NOTE ....... vii 

1. THE SERIES OF NATURAL NUMBERS .... I 

2. DEFINITION OF NUMBER . . . . . ,11 

3. FINITUDE AND MATHEMATICAL INDUCTION 2O 

4. THE DEFINITION OF ORDER ..... 29 

5. KINDS OF RELATIONS ...... 42 

6. SIMILARITY OF RELATIONS . . . . S 2 

7. RATIONAL, REAL, AND COMPLEX NUMBERS ... 63 

8. INFINITE CARDINAL NUMBERS ..... 77 

9. INFINITE SERIES AND ORDINALS .... 89 

10. LIMITS AND CONTINUITY ...... 97 

11. LIMITS AND CONTINUITY OF FUNCTIONS . . . 107 

12. SELECTIONS AND THE MULTIPLICATIVE AXIOM . IJ 7 

13. THE AXIOM OF INFINITY AND LOGICAL TYPES . . 131 

14. INCOMPATIBILITY AND THE THEORY OF DEDUCTION . 144 

15. PROPOSITIONAL FUNCTIONS ..... 155 

16. DESCRIPTIONS ........ 167 

17. CLASSES .... . l8l 

18. MATHEMATICS AND LOGIC ...... 194 

INDEX 207 



Viii 



Introduction to 
Mathematical Philosophy 

CHAPTER I 

THE SERIES OF NATURAL NUMBERS 

MATHEMATICS is a study which, when we 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 : from integers to fractions, 
real numbers, complex numbers ; from addition and multi 
plication to differentiation and integration, and on to higher 
mathematics. The other direction, which is less familiar, 
proceeds, by analysing, to greater and greater abstractness 
and logical simplicity ; instead of asking what can be defined 
and deduced from what 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. 
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 is one, not in 
the subject matter, but in the state of mind 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 to Mathematical Philosophy 

ordinary sense. The distinction between mathematics and 
mathematical philosophy is one which depends upon the interest 
inspiring the research, and upon the stage which the research 
has reached ; not upon the propositions with which the research 
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 " simple " in a 
logical sense). And as we need two sorts of instruments, the 
telescope and the microscope, for the enlargement of our visual 
powers, so we need two sorts of instruments for the enlargement 
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 fresh lines of advance after our backward journey. 
It is the purpose of this 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 scarcely possible. A full treatment will be found 
in Principia Mathematica ; * the treatment in the present volume 
is 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 
whole numbers, 

i, 2, 3, 4, ... etc. 

1 Cambridge University Press, vol. i., 1910 ; vol. ii., 1911 ; vol. iii., 1913. 
By Whitehead and Russell. 



The Series of Natural Numbers 3 

Probably only a person with some mathematical knowledge 
would think of beginning with o instead of with i, but we will 
presume this degree of knowledge ; we will take as our starting- 
point the series : 

o, i, 2, 3, . . . n, n+ 1, . . . 

and it is this series that we shall mean when we speak of the 
" series of natural numbers." 

It is only at a high stage of civilisation that we could take 
this series as 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 
involved is far from easy. And the discovery that I is a number 
must have been difficult. As for o, it is a very recent addition ; 
the Greeks and Romans had no such digit. If we had been 
embarking upon mathematical philosophy in earlier days, we 
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 is easiest 
and most familiar in mathematics. 

But though familiar, they are not understood. Very few 
people are prepared with a definition of what is meant by 
" number," or " o," or " I." It is not very difficult to see that, 
starting from o, any other of the natural numbers can be reached 
by repeated additions of I, but we shall have to define what 
we mean by " adding I," and what we mean by " repeated." 
These questions are by no means easy. It was believed until 
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 clear that human knowledge must always be content 
to accept some terms as intelligible without definition, in order 



4 Introduction to Mathematical Philosophy 

to have a starting-point for its definitions. It is not clear that 
there must be terms which are incapable of definition : it is 
possible that, however far back we go in defining, we always 
might go further still. On the other hand, it is also possible 
that, when analysis has been pushed far enough, we can reach 
terms that really are simple, and therefore logically incapable 
of the sort of definition that consists in analysing. This is a 
question which it is not necessary for us to decide ; for our 
purposes it is sufficient to observe that, since human powers 
are finite, the definitions known to us must always begin some 
where, with terms undefined for the moment, though perhaps 
not permanently. 

All traditional pure mathematics, including analytical geom 
etry, may be regarded as consisting wholly of propositions 
about the natural numbers. That is to say, the terms which 
occur can be defined by means of the natural numbers, and 
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 is a fairly recent discovery, though it had 
long been suspected. Pythagoras, who believed that not only 
mathematics, but everything else could be deduced from 
numbers, was the discoverer of the most serious obstacle in 
the way of what is called the " arithmetising " of mathematics. 
It was Pythagoras who discovered the existence of incom- 
mensurables, and, in particular, the incommensurability of the 
side of a square and the diagonal. If the length of the side is 
I inch, the number of inches in the diagonal is the square root 
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 
completely by the help of the reduction of arithmetic to logic, 
which will be explained in following chapters. For the present, 
we shall take for granted the arithmetisation of mathematics, 
though this was a feat of the very greatest importance. 



The Series of Natural Numbers 5 

Having reduced all traditional pure mathematics to the 
theory of the natural numbers, the next step in logical analysis 
was to reduce this theory itself to the smallest set of premisses 
and undefined terms from which it could be derived. This work 
was accomplished by Peano. He showed that the entire theory 
of the natural numbers could be derived from three primitive 
ideas and five primitive propositions in addition to those of 
pure logic. These three ideas and five propositions thus became, 
as it were, hostages for the whole of traditional pure mathe 
matics. If they could be defined and proved in terms of others, 
so could all pure mathematics. Their logical " weight," if one 
may use such an expression, is equal to that of the whole series 
of sciences that have been deduced from the theory of the natural 
numbers ; the truth of this whole series is assured if the truth 
of the five primitive propositions is guaranteed, provided, of 
course, that there is nothing erroneous in the purely logical 
apparatus which is also involved. The work of analysing mathe 
matics is extraordinarily facilitated by this work of Peano's. 

The three primitive ideas in Peano's arithmetic are : 

o, number, successor. 

By " successor " he means the next number in the natural 
order. That is to say, the successor of o is I, the successor of 
I is 2, and so on. By " number " he means, in this connection, 
the class of the natural numbers. 1 He is not assuming that 
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 is a number. 

(2) The successor of any number is a number. 

(3) No two numbers have the same successor. 

1 We shall use " number " in this sense in the present chapter. After 
wards the word will be used in a more general sense. 



6 Introduction to Mathematical Philosophy 

(4) o is not the successor of any number. 

(5) Any property which belongs to o, and also to the successor 

of every number which has the property, belongs to all 
numbers. 

The last of these is the principle of mathematical induction. 
We shall have much to say concerning mathematical induction 
in the sequel ; for the present, we are concerned with it only 
as it 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 
propositions. To begin with, we define I as " the successor of o," 
2 as " the successor of I," and so on. We can obviously go 
on as long as we like with these definitions, since, in virtue of 
(2), every number that we reach will have a successor, and, in 
virtue of (3), this cannot be any of the numbers already defined, 
because, if it were, two different numbers would have the same 
successor ; and in virtue of (4) none of the numbers we reach 
in the series of successors can be o. Thus the series of successors 
gives us an endless series of continually new numbers. In virtue 
of (5) all numbers come in this series, which begins with o and 
travels 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 
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. In virtue of (5) this gives a definition of 
the sum of m and n, whatever number n may be. Similarly 
we can define the product of any two numbers. The reader can 
easily convince himself that any ordinary elementary proposition 
of arithmetic can be proved by means of our five premisses, 
and if he has any difficulty he can find the proof in Peano. 

It is time now to turn to the considerations which make it 
necessary to advance beyond the standpoint of Peano, who 



The Series of Natural Numbers 7 

represents the last perfection of the " arithmetisation " of 
mathematics, to that of Frege, who first succeeded in " 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 
number and of particular numbers, but we shall give some of the 
reasons why Peano's treatment is less final than it appears to be. 
In the first place, Peano's three primitive ideas namely, " o," 
" number," and " successor " are capable of an infinite number 
of different interpretations, all of which will satisfy the five 
primitive propositions. We will give some examples. 

(1) Let " o " be taken to mean loo, and let " number " be 
taken to mean the numbers from 100 onward in the series of 
natural numbers. Then all our primitive propositions are 
satisfied, even the fourth, for, though 100 is the successor of 
99, 99 is not a " number " in the sense which we are now giving 
to the word " number." It is obvious that any number may be 
substituted for 100 in this example. 

(2) Let " o " have its usual meaning, 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 
for the number four, and so on ; the series of " numbers " now 
will be 

o, two, four, six, eight . . . 

All Peano's five premisses are satisfied still. 

(3) Let " o " mean the number one, let " number " mean 
the set 

!> i> 1> i T V 

and let "successor" mean "half." Then all Peano's five 
axioms will be true of this set. 

It is clear that such examples might be multiplied indefinitely. 
In fact, given any series 



8 Introduction to Mathematical Philosophy 

which is endless, contains 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 
axioms. This is easily seen, though the formal proof is some 
what long. Let " o " mean # , let " number " mean the whole 
set of terms, and let the " successor " of # n mean x n+l . Then 

(1) " o is a number," i.e. x is a member of the set. 

(2) " The successor of any number is a number," i.e. taking 
any term x n in the set, x n+l is also in the set. 

(3) " No two numbers have the same successor," i.e. if x m 
and x n are two different members of the set, x m+l and x n+l are 
different ; this results from the fact that (by hypothesis) there 
are no repetitions in the set. 

(4) " o is not the successor of any number," i.e. no term in 
the set comes before x . 

(5) This becomes : Any property which belongs to x 09 and 
belongs to x n+l provided it belongs to x n , belongs to all the x's. 

This follows from the corresponding property for numbers. 
A series of the form 



in which there is a first term, a successor to each term (so that 
there is no last term), no repetitions, and every term can be 
reached from the start in a finite number of steps, is called a 
progression. Progressions are of great importance in the princi 
ples of mathematics. As we have just seen, every progression 
verifies Peano's five axioms. It can be proved, conversely, 
that every series which verifies Peano's five axioms is a pro 
gression. Hence these five axioms may be used to define the 
class of progressions : " progressions " are " those series which 
verify these five axioms." Any progression may be taken as 
the basis of pure mathematics : we may give the name " o " 
to its first term, the name " number " to the whole set of its 
terms, and the name " successor " to the next in the progression. 
The progression need not be composed of numbers : it may be 



The Series of Natural Numbers 9 

composed of points in space, or moments of time, or any other 
terms of which there is an infinite supply. Each different 
progression will give rise to a different interpretation of all the 
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 different interpretations of his primitive ideas. 
It is assumed that we know what is meant by " o," and that 
we shall not suppose that this symbol means 100 or Cleopatra's 
Needle or any of the other things that it might mean. 

This point, that " o " and " number " and "successor " 
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 
ten fingers and two eyes and one nose. A system in which " I " 
meant 100, and " 2 " meant 101, and so on, might be all right 
for pure mathematics, but would not suit daily life. We want 
" o " and " number " and " successor " to have meanings which 
will give us the right allowance of fingers and eyes and noses. 
We have already some knowledge (though not sufficiently 
articulate or analytic) of what we mean by " I " and " 2 " and 
so on, and our use of numbers in arithmetic must conform to 
this knowledge. We cannot secure that this shall be the case 
by Peano's method ; all that we can do, if we adopt his method, 
is to say " we know what we mean by * o ' and ' number ' and 
' successor,' though we cannot explain what we mean in terms 
of other simpler concepts." It is quite legitimate to say this 
when we must, and at some point we all must ; but it is the 
object of mathematical philosophy to put off saying it as long 
as possible. By the logical theory of arithmetic we are able to 
put it off for a very long time. 

It might be suggested that, instead of setting up " o " and 
" number " and " successor " as terms of which we know the 
meaning although we cannot define them, we might let them 



io Introduction to Mathematical Philosophy 

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 
five axioms, but which are otherwise undetermined. If we adopt 
this plan, our theorems will not be proved concerning an ascer 
tained set of terms called " the natural numbers," but concerning 
all sets of terms having certain properties. Such a procedure 
is not fallacious ; indeed for certain purposes it represents a 
valuable generalisation. But from two points of view it fails 
to give an adequate basis for arithmetic. In the first place, it 
does not enable us to know whether there are any sets of terms 
verifying Peano's axioms ; it does not even give the faintest 
suggestion of any way of discovering whether there are such sets. 
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 II 

DEFINITION OF NUMBER 

THE question " What is a number ? " 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 
der Arithmetik* Although this book is quite short, not difficult, 
and of the very highest importance, it attracted almost no 
attention, and the definition of number which it contains re 
mained practically unknown until it was rediscovered by the 
present author in 1901. 

In seeking a definition of number, the first thing to be clear 
about is what we may call the grammar of our inquiry. Many 
philosophers, when attempting to define number, are really 
setting to work to define plurality, which is quite a different 
thing. Number is what is characteristic of numbers, as man 
is what is characteristic of men. A plurality is not an instance 
of number, but of some particular number. A trio of men, 
for example, is an instance of the number 3, and the number 
3 is an instance of number ; but the trio is not an instance of 
number. This point may seem elementary and scarcely worth 
mentioning ; yet it has proved too subtle for the philosophers, 
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 trio consisting of Brown, Jones, and Robinson. The number 
3 is 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 
that number. 

Instead of speaking of a " collection," we shall as a rule speak 
of a " class," or sometimes a " set." Other words used in 
mathematics for the same thing are " aggregate " and " mani 
fold." We shall have much to say later on about classes. For 
the present, we will say as little as possible. But there are 
some remarks that must be made immediately. 

A class or collection may be defined in two ways that at first 
sight seem quite distinct. We may enumerate its members, as 
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 which enumerates is called a definition by " exten 
sion," and the one which mentions a defining property is called 
a definition by " intension." Of these two kinds of 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 
the extensional one. Each of these points needs a word of 
explanation. 

(i) Brown, Jones, and Robinson all of them possess a certain 
property which is possessed by nothing else in the whole universe, 
namely, the property of being either Brown or Jones or Robinson. 
This property can be used to give a definition by intension of 
the class consisting of Brown and Jones and Robinson. Con 
sider such a formula as " x is Brown or x is Jones or x is Robinson." 
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 Number 13 

men, and peculiar to them. A similar treatment can obviously 
be applied to any other class given in extension. 

(2) It is obvious that in practice we can often know a great 
deal about a class without being able to enumerate its members. 
No one man could actually enumerate all men, or even all the 
inhabitants of London, yet a great deal is known about each of 
these classes. This is 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 
theoretically possible for beings who only live for a finite time. 
We cannot enumerate all the natural numbers : they are o, I, 2, 
3, and so on. At some point we must content ourselves with 
" and so on." We cannot enumerate all fractions or all irrational 
numbers, or all of any other infinite collection. Thus our know 
ledge in regard to all such collections can only be derived from a 
definition by 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 
infinite collection : it is to be presumed, for example, that there 
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 
finite, which, though possible, seems unlikely. In the third 
place, we wish to define " number " in such a 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 all 
its members and peculiar to them. 

For many purposes, a class and a defining characteristic of 
it are practically interchangeable. The vital difference between 
the two consists in the fact that there is 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 



14 Introduction to Mathematical Philosophy 

may be defined as featherless bipeds, or as rational animals, 
or (more correctly) by the traits by which Swift delineates the 
Yahoos. It is this fact that a defining characteristic is never 
unique which makes classes useful ; otherwise we could be 
content with the properties common and peculiar to their 
members. 1 Any one of these properties can be used in place 
of the class whenever uniqueness is not important. 

Returning now to the definition of number, it is clear that 
number is a way of bringing together certain collections, namely, 
those that have a given number of terms. We can suppose 
all couples in one bundle, all trios in another, and so on. In 
this way we obtain various bundles of collections, each bundle 
consisting of all the collections that have a certain number of 
terms. Each bundle is a class whose members are collections, 
i.e. classes ; thus each is a class of classes. The bundle con 
sisting of all couples, for example, is a class of classes : each 
couple is a class with two members, and the whole bundle of 
couples is a class with an infinite number of members, each of 
which is a class of two members. 

How shall we decide whether two collections are to belong 
to the same bundle ? The answer that suggests itself is : " Find 
out how many members each has, and put them in the same 
bundle if they have the same number of members." But this 
presupposes that we have defined numbers, and that we know 
how 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, 
though familiar, is logically a very complex operation ; more 
over it is only available, as a 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, 

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. 



Definition of Number 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 
collections have the same number of terms than it is to define 
what that number is. An illustration will make this clear. 
If 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 
not need a census to assure us of this, nor do we need to know 
what is the actual number of husbands and of wives. We know 
the 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." 

A relation is said to be " one-one " when, if x has the relation 
in question to y, 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 
fulfilled, it is called " many-one." It should be observed that 
the number I is not used in these definitions. 

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 many-one, but that of eldest son to father 
is one-one. If n is any number, the relation of n to -|-i is 
one-one ; so is the relation of n to 2n or to 3. When we are 
considering only positive numbers, the relation of n to 2 is 
one-one ; but when negative numbers are admitted, it becomes 
two-one, since n and n have the same square. These instances 
should suffice to make clear the notions of one-one, one-many, 
and many-one relations, which play a great part in the princi 
ples of 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 



1 6 Introduction to Mathematical Philosophy 

relation which correlates the terms of the one class each with 
one term of the other class, in the same manner in which the 
relation of marriage correlates husbands with wives. A few 
preliminary definitions will help us to state this definition more 
precisely. The class of those terms that have a given relation 
to something or other is called the domain 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 : thus the class of wives is the converse domain 
of the relation of husband to wife. We may now state our 
definition of similarity as follows : 

One class is said to be " similar " to another when there is a 
one-one relation of which the one class is the domain, while the 
other is the converse domain. 

It is easy to prove (i) that every class is similar to itself, (2) 
that if a class a is similar to a class j3, then j3 is similar to a, (3) 
that if a is similar to j3 and j8 to y, then a is similar to y. A 
relation is said to be reflexive when it possesses the first of these 
properties, symmetrical when it possesses the second, and transi 
tive when it possesses the third. It is obvious that a relation 
which is symmetrical and transitive must be reflexive throughout 
its domain. Relations which possess these properties are an 
important kind, and it is worth while to note that similarity is 
one of this kind of relations. 

It is obvious to common sense that two finite classes have 
the same number of terms if they are similar, but not otherwise. 
The 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 
set to be counted as there are numbers up to the last number 
used in the counting. And we also know that, so long as 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 
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 (say) 10 objects is to 
show that the set of these objects is similar to the set of numbers 
I to 10. The notion of similarity is logically presupposed in 
the operation of counting, and is logically simpler though less 
familiar. In counting, it is necessary to take the objects counted 
in a certain order, as first, second, third, etc., but order is not 
of the essence of number : it is an irrelevant addition, an un 
necessary complication from the logical point of view. The 
notion of similarity does not demand an order : for example, 
we saw that the number of husbands is the same as the number 
of wives, without having to establish an order of precedence 
among them. The notion of similarity also does not require 
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 correlate 2 with J, 3 with J, and 
so on, thus proving that the two classes are similar. 

We may thus use the notion of " similarity " to decide when 
two collections are to belong to the same bundle, in the sense 
in which we were asking this question earlier in this chapter. 
We want to make one bundle containing the class that has no 
members : this will be for the number o. Then we want a bundle 
of all the classes that have one member : this will be for the 
number I. Then, for the number 2, we want a bundle consisting 

2 



1 8 Introduction to Mathematical Philosophy 

of all couples ; then one of all trios ; and so on. Given any collec 
tion, we can define the bundle it is to belong to as being the class 
of all those collections that are " similar " to it. It is very easy 
to see that if (for example) a collection has three members, the 
class of all those collections that are similar to it will be the 
class of trios. And whatever number of terms a collection may 
have, those collections that are " similar " to it will have the same 
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. 

So far 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, 
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 
couples : it is indubitable and not difficult to define, whereas 
the number 2, in any other sense, is a metaphysical entity about 
which we can never feel sure that it exists or that we have tracked 
it down. 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 up the following definition : 

The number of a class is the class of all those classes that are 
similar 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 
to prove that numbers so defined have all the properties that we 
expect numbers to have. 

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 



Definition of Number 1 9 

other, and none outside the set are similar to any inside the set. 
In other words, a number (in general) is any collection which is 
the number of one of its members ; or, more simply still : 

A number is anything which is the number of some class. 

Such a definition has a verbal appearance of being circular, 
but in fact it is not. We define " the number of a given class " 
without using the notion of number in general ; therefore we may 
define number in general in terms of " the number of a given 
class " without committing any logical error. 

Definitions of this sort are in fact very common. The class 
of fathers, for example, would have to be defined by first defining 
what it is to be the father of somebody ; then the class of fathers 
will be all those who are somebody's father. Similarly if we want 
to define square numbers (say), we must first define what we 
mean by saying that one number is the square 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 will serve 
for finite collections. It remains to be seen how it will serve 
for infinite collections. But first we must decide what we mean 
by " finite " and " infinite," which cannot be done within the 
limits of the present chapter. 



CHAPTER III 

FINITUDE AND MATHEMATICAL INDUCTION 

THE series of natural numbers, as we saw in Chapter I., can all 
be defined if we know what we mean by the three terms " o," 
" number," and " successor." But we may go a step farther : 
we can define all the natural numbers if we know what we mean 
by " o " and " successor." It will help us to understand the 
difference between finite and infinite to see how this can be done, 
and why the method by which it is done cannot be extended 
beyond the finite. We will not yet consider how " o " and " suc 
cessor " are to be defined : we will for the moment assume that 
we know what these terms mean, and show how thence all other 
natural numbers can be obtained. 

It is easy to see that we can reach any assigned number, say 
30,000. We first define " I " as " the successor of o," then we 
define " 2 " as " the successor of I," and so on. In the case of 
an assigned number, such as 30,000, the proof that we can reach 
it by proceeding step by step in this fashion may be made, if we 
have the patience, by actual experiment : we can go on until 
we actually arrive at 30,000. But although the method of 
experiment is available for each particular natural number, it 
is not available for proving the general proposition that all such 
numbers can be reached in this way, i.e. by proceeding from o 
step by step from each number to its successor. Is there any 
other way by which this can be proved ? 

Let us consider the question the other way round. What are 
the numbers that can be reached, given the terms " o " and 



Finitude and Mathematical Induction 21 

" successor " ? Is there any way by which we can define the 
whole class of such numbers ? We reach I, as the successor of o ; 
2, as the successor of I ; 3, as the successor of 2 ; and so on. It 
is this " and so on " that we wish to replace by something less 
vague and indefinite. We might be tempted to say that " and 
so on " means that the process of proceeding to the successor 
may be repeated any finite number of times ; but the problem 
upon which we are engaged is the problem of defining " finite 
number," and therefore we must not use this notion in our defini 
tion. Our definition must not assume that we know what a 
finite 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 
numbers. It stated that any property which belongs to o, and 
to the successor of any number which has the property, belongs 
to all the natural numbers. 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 
that can be reached from o by successive steps from next to 
next, but as the point is important we will set forth the matter 
in some detail. 

We shall do well to begin with some definitions, which will be 
useful in other connections also. 

A property is said to be " hereditary " in the natural-number 
series if, whenever it belongs to a number , it also belongs to 
n-j-i, the successor of n. Similarly a class is said to be " heredi 
tary " if, whenever n is a member of the class, so is n+i. It is 
easy to see, though we 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 all that are not less than 100, or all that are 
less than 1000, or it may be that it belongs to all that are not 
less than o, i.e. to all without exception. 

A property is said to be " inductive " when it is a hereditary 



22 Introduction to Mathematical Philosophy 

property which belongs to o. Similarly a class is " inductive " 
when it is a hereditary class of which o is a member. 

Given a hereditary class of which o is a member, it follows 
that I is a member of it, because a hereditary class contains the 
successors of its 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 step- 
by-step procedure that any assigned natural number, say 30,000, 
is 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 the converse of " 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 
sists of itself and all greater natural numbers ; but this also we 
do not yet officially know. 

By the above definitions, the posterity of o will consist of those 
terms which belong to every inductive class. 

It is now not difficult to make it obvious that the posterity of 
o is the same set as those 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 
our terms) ; in the second place, if n belongs to both sets, so does 
n+i. It is to be observed that we are dealing here with the 
kind of matter that does not admit of precise proof, namely, the 
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 
as if it conveyed a definite meaning ; on the other hand, " the 
posterity of o " is precise and explicit just where the other idea 
is hazy. It may be taken as giving what we meant to mean 
when we spoke of the terms that can be reached from o by 
successive steps. 

We now lay down the following definition : 

The " natural numbers " are the -posterity of o with respect to the 



Finitude and Mathematical Induction 23 

relation " immediate predecessor " (which is the converse of 
" successor " ). 

We have thus arrived at a definition of one of Peano's three 
primitive ideas in terms of the other two. As a result of this 
definition, two of his primitive propositions namely, the one 
asserting that o is a number and the one asserting mathematical 
induction become unnecessary, since they result from the defini 
tion. The one asserting that the successor of a natural number 
is a natural number is only needed in the weakened form " every 
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 
which has no members, i.e. in the class which is called the " null- 
class." By the general definition of number, the number of terms 
in the null-class is the set of all classes similar to the null-class, 
i.e. (as is easily proved) the set consisting of the null-class all 
alone, i.e. the class whose only member is the null-class. (This 
is not identical with the null-class : it has one member, namely ? 
the null-class, whereas the null-class itself has no members. A 
class which has one member is never identical with that one 
member, as we shall explain when we come to the theory of 
classes.) Thus we have the following purely logical definition : 

o is the class whose only member is the null-class. 

It remains to define " successor." Given any number n, let 
a be a class which has n members, and let x be a term which 
is not a member of a. Then the class consisting of a with x 
added on will have n-\-i members. Thus we have the following 
definition : 

The successor of the number of terms in the class a is the number 
of terms in the class consisting of a together with x, where x is any 
term not belonging to the class. 

Certain niceties are required to make this definition perfect, 
but they need not concern us. 1 It will be remembered that we 
1 See Principia Mathematical, vol. ii. * no, 



24 Introduction to Mathematical Philosophy 

have already given (in Chapter II.) a logical definition of the 
number of terms in a class, namely, we defined it as the set of all 
classes that are similar to the given class. 

We have thus reduced Peano's three primitive ideas to ideas 
of logic : we have given definitions of them which make them 
definite, no longer capable of an infinity of different meanings, 
as they were when they were only determinate to the extent of 
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 
of " natural number." How stands it with the remaining three ? 
It is very easy to prove that o is 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 difficulty 
does not arise unless the total number of individuals in the 
universe is finite ; for given two numbers m and n, neither of 
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 
in the universe were (say) 10 ; then there would be no class of 
II individuals, and the number 1 1 would be the null-class. So 
would the number 12. 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 
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, 



Finitude and Mathematical Induction 25 

three primitive ideas, but in seeing how to prove his five primitive 
propositions, by means of primitive ideas and propositions belong 
ing to logic. It follows that all pure mathematics, in so far 
as it is deducible from the theory of the natural numbers, is only 
a prolongation of logic. The extension of this result to those 
modern branches of 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 m will have this relation 
to w+i. A property is "hereditary with respect to N," or 
simply " N-hereditary," if, whenever the property belongs to a 
number m, it also belongs to m-fi, i.e. to the number to which 
m has the relation N. And a number n will be said to belong to 
the " posterity " of 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 
if R is any relation whatever, we can lay down the following 
definitions : 2 

A property is called " R-hereditary " when, if it belongs to 
a term x, and x has the relation R to y, then it belongs to y. 

A class is R-hereditary when its defining property is R- 
hereditary. 

A term x is 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.) 

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 twenty years after its publication. 



26 Introduction to Mathematical Philosophy 

The " R-posterity " of x is all the terms of which x is an R- 
ancestor. 

We have framed the above definitions so that if a term is the 
ancestor of anything it is its own ancestor and belongs to its own 
posterity. This is merely for convenience. 

It will be observed that if we take for R the relation " parent," 
" ancestor " and " posterity " will have the usual meanings, 
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. A person confronted for the first time with the 
problem of defining " ancestor " in terms of " parent " would 
naturally say that A is an ancestor of Z if, between A and Z, 
there are a certain number of people, B, C, . . ., of whom 
B is a child of A, each is a parent of the next, until the last, who 
is a parent of Z. But this definition is not adequate unless we 
add that the number of intermediate terms is to be finite. Take, 
for example, such a series as the following : 

I, f, J, 8 9 . g> > 2? M 

Here we have first a series of negative fractions with no end, 
and then a series of positive fractions with no beginning. Shall 
we say 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. 
But, as we saw, " finite " is to be defined by means of mathe 
matical induction, and it is simpler to define the ancestral relation 
generally at once than to define it first only for the case of the 
relation of n to n-f-i, and then extend it to other cases. Here, 
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 of mathematical induction in demonstrations was, 
in the past, something of a mystery. There seemed no reason 
able doubt that it was a valid method of proof, but no one quite 
knew why it was valid. Some believed it to be really a case 
of induction, in the sense in which that word is used in logic. 
Poincare * considered it to be a principle of the utmost import 
ance, by means of which an infinite number of syllogisms could be 
condensed into one argument. We now know that all such views 
are mistaken, and that mathematical induction is a definition, 
not a principle. There are some numbers to which it can be 
applied, and there are others (as we shall see in Chapter VIII.) 
to which it cannot be applied. We define the " natural numbers " 
as those to which proofs by mathematical induction can be 
applied, i.e. as those that possess all inductive properties. It 
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 
defined as animals having four legs, it will follow that animals 
that have four legs are quadrupeds ; and the case of numbers 
that obey mathematical induction is exactly similar. 

We shall use the phrase " inductive numbers " to 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 affords, more than anything else, 
the essential characteristic by which the finite is distinguished 
from the infinite. The principle of mathematical induction 
might be stated popularly in some such form as " what can be 
inferred from next to next can be inferred from first to last." 
This is true when the number of intermediate steps between 
first and last is finite, not otherwise. Anyone who has ever 
1 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 is very long, it is a very long time before the last 
truck moves. If the train were infinitely long, there would be 
an infinite succession of jerks, and the time would never come 
when the whole train 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 
of infinites), every truck 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 
numbers, where arguments from mathematical induction will 
be no longer valid, the properties of such numbers will help to 
make clear, by contrast, the almost unconscious use that is made 
of mathematical induction where finite numbers are concerned. 



CHAPTER IV 

THE DEFINITION OF ORDER 

WE have now carried our analysis of the series of natural numbers 
to the point where we have obtained logical definitions of the 
members of this series, of the whole class of its members, and 
of the relation of a number to its immediate successor. We 
must now consider the serial character of the natural numbers 
in the order o, I, 2, 3, . . . We ordinarily think of the num 
bers as in this order, and it is an essential part of the work 
of analysing our data to seek a definition of " order " or " series " 
in logical terms. 

The notion of order is one which has enormous importance 
in mathematics. 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 
order of points on a line is essential to geometry ; so is the 
slightly more complicated order of lines through a point in a 
plane, or of planes through a line. Dimensions, in geometry, 
are a development of order. The conception of a limit, which 
underlies all higher mathematics, is 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 
in which this notion 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 all the orders of which it is capable. Some 
times one order is so much more familiar and natural to our 



30 Introduction 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 
or the " inductive " numbers, as we shall also call them occur 
to us most readily in order of magnitude ; but they are capable 
of an infinite number of other arrangements. We might, for 
example, consider first all the odd numbers and then all the 
even numbers ; or first 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 
" arrange " the numbers in these various orders, that is an 
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 
is equally true of points on a line or of the moments of time : 
one order is more familiar, but others are equally valid. We 
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 all those that have algebraic non- 
rational co-ordinates, and so on, through any set of complica 
tions we please. The resulting order will be one which the 
points of the line certainly have, whether we choose to notice 
it or not ; the only thing that is arbitrary about the various 
orders of a set of terms is our attention, for the terms themselves 
have always all the orders of which they are capable. 

One important result of this consideration is that 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 

the members of the class, in respect of which some appear as 
earlier and some as later. The fact that a class may have many 
orders is due to the fact that there can be many relations holding 
among the members of one single class. 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 
to order may be discovered by considering that in respect of 
such a relation we must be able to say, of any two terms in 
the class which is to be ordered, that one " precedes " and the 
other " follows." Now, in order that we may be able to use 
these words in the way in which we should naturally understand 
them, we require that the ordering relation should have three 
properties : 

(1) If x precedes y, y must not also precede x. This is an 
obvious characteristic of the kind of relations that lead to series. 
If x is less than y, y is not also less than x. If x is earlier in 
time than y, y is not also earlier than x. If x is to the left of 
y, 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 
x is a brother or sister of y, y is a brother or sister of x. If x is 
of the same height as y, y is of the same height as x. If x is of a 
different height from y, y is of a different height from x. In 
all these cases, when the relation holds between x and y, it also 
holds between y and x. But with serial relations such a thing 
cannot happen. A relation having this first property is called 
asymmetrical. 

(2) If x precedes y and y precedes z, x must precede z. This 
may be illustrated by the same instances as before : less, earlier, 
left of. But as instances of relations which do not have this 
property only two of our previous three instances will serve. 
If x is brother or sister of y, and y of z, x may not be brother 
or sister of z, since x and z may be the same person. The same 
applies to difference of height, but not to sameness of height, 
which has our second property but not our first. The relation 
" father," on the other hand, has our first property but not 



32 Introduction to Mathematical Philosophy 

our second. A relation having our second property is called 
transitive. 

(3) Given any two terms of the class which is to be ordered, 
there 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 
numbers this is not true. Of any two moments in time, one 
must be earlier than the other ; but of events, which may be 
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 is called connected. 

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 relation having these 
three properties can be found generating it. 

Before illustrating this thesis, we will introduce a few 
definitions. 

(1) A relation is said to be an aliorelative, 1 or to be contained 
in or imply diversity, if no term has this relation to itself. 
Thus, for example, " greater," " different in size," " brother," 
" husband," " father " are aliorelatives ; but " equal," " born 
of the same parents," " dear friend " are not. 

(2) The square of a relation is that relation which holds between 
two terms x and z when there is 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," 
" greater by 2 " is the square of " greater by I," and so on. 

(3) The domain of a relation consists of all those terms that 
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, but are 
recalled here for the sake of the following definition : 

(4) The field of a relation consists of its domain and converse 
domain together. 

1 This term is due to C. S. Peirce. 



The Definition of Order 33 

(5) One relation is said to contain or be implied by another if 
it holds whenever the other holds. 

It will be seen that an asymmetrical relation is the same thing 
as a relation 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 
example, " spouse " is an aliorelative, but is symmetrical, 
since if x is the spouse of y, y is the spouse of x. But among 
transitive relations, all aliorelatives are asymmetrical as well 
as vice versa. 

From the definitions it will be seen that a transitive relation 
is one which is implied by its square, or, as we also say, " con 
tains " its square. Thus " ancestor " is transitive, because 
an ancestor's ancestor is an ancestor ; but " father " is not 
transitive, because a father's father is not a father. A transitive 
aliorelative is one which contains its square and is contained 
in diversity ; or, what comes to the same thing, one whose 
square implies both it and diversity because, when a relation 
is transitive, asymmetry is equivalent to being an aliorelative. 

A relation is connected when, given any two different terms 
of its field, the relation holds between the first and the second 
or between the second and the first (not excluding the possibility 
that both may happen, though both cannot happen if the relation 
is asymmetrical). 

It will be seen that the relation " ancestor," for example, 
is 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 relation " less than or equal to," among numbers, is 
transitive 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 
or less than y, and y is greater or less than z, it may happen 
that x and z are the same number. 

Thus the three properties of being (i) an aliorelative, (2) 

3 



34 Introduction to Mathematical Philosophy 

transitive, and (3) connected, are mutually independent, since 
a relation may have any two without having the third. 

We now lay down the following definition : 

A relation is serial when it is an aliorelative, transitive, and 
connected ; or, what is equivalent, when it is asymmetrical, 
transitive, and connected. 

A series is the same thing as a serial relation. 

It might have been thought that a series should be the field 
of a serial relation, not the serial relation itself. But this would 
be an error. For example, 

I, 2, 3 ; i, 3, 2 ; 2, 3, I ; 2, i, 3 ; 3, I, 2 ; 3, 2, I 

are six different series which all have the same field. If the 
field were the series, there could only be one series with a given 
field. What distinguishes the above six series is simply the 
different ordering relations in the six cases. Given the ordering 
relation, the field and the order are both determinate. Thus 
the ordering relation may be taken to be the series, but the field 
cannot be so taken. 

Given any serial relation, say P, we shall say that, in respect 
of this relation, x " precedes " y if x has the relation P to y, 
which we shall write " xPy " for short. The three characteristics 
which P must have in order to be serial are : 

(1) We must never have xPx, i.e. no term must precede 

itself. 

(2) P 2 must imply P, i.e. if x precedes y and y precedes z, x must 

precede z. 

(3) If 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 
we expect of series will also be found, and vice versa. We are 
therefore justified in taking the above as a definition of order 



The Definition of Order 35 

or series. And it will be observed that the definition 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 series may serve as an illustration. The 
relation we assumed in considering the natural numbers was 
the relation of immediate succession, i.e. the relation between 
consecutive integers. This relation is asymmetrical, but not 
transitive or connected. We can, however, derive from it, 
by the method of mathematical induction, the " ancestral " 
relation which we considered in the preceding chapter. This 
relation will be the same as " less than or equal to " among 
inductive integers. For purposes of generating the series of 
natural numbers, we want the relation " less than," excluding 
" equal to." This is the relation oimton when m is an ancestor 
of n but not identical with n, or (what comes to the same thing) 
when the successor of m is an ancestor of n in the sense in which 
a number is its own ancestor. That is to say, we shall lay down 
the following definition : 

An inductive number m is said to be less than another 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 
" less than," so defined, is asymmetrical, transitive, and con 
nected, and has the inductive numbers for its field. Thus by 
means of this relation the inductive numbers acquire an order 
in the sense in which we defined the term " order," and this order 
is the so-called " natural " order, or order of magnitude. 

The generation of series by means of relations more or less 
resembling that of n to n-j-i is very common. The series of the 
Kings of England, for example, is generated by relations of each 
to his successor. This is probably the easiest way, where it is 
applicable, of conceiving the generation of a series. In this 
method we pass on from each term to the next, as long as there 



36 Introduction 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 " earlier " and 
" later " in a series so generated. On the analogy of " proper 
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 
posterity. Reverting to the fundamental definitions, we find that 
the " proper posterity " may be defined as follows : 

The " proper posterity " of x with respect to R consists of 
all terms that possess every R-hereditary property possessed by 
every term to which x has the relation R. 

It is to be observed that this definition has to be so framed 
as to 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 
child) where there may be many terms to which x has the relation 
R. We define further : 

A term x is a " proper ancestor " of y with respect to R if y 
belongs to the proper posterity of x with respect to R. 

We shall speak for short of " R-posterity " and " R-ancestors " 
when these terms seem more convenient. 

Reverting now to the generation of series by the relation R 
between consecutive terms, we see that, if this method is to be 
possible, the relation " proper R-ancestor " must be an aliorela- 
tive, transitive, and connected. Under what circumstances will 
this occur ? It will always be transitive : no matter what sort 
of relation R may be, " R-ancestor " and " proper R-ancestor " 
are always both transitive. But it is only under certain circum 
stances that it will be an aliorelative or connected. Consider, 
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 
a case, though the relation " proper R-ancestor " is connected, 
and though R itself is an aliorelative, we do not get a series 
because " proper R-ancestor " is not an aliorelative. It is for 
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 
ordinary sense of the word " ancestor." If x is a proper ancestor 
of y, x and y cannot be the same person ; but it is not true that 
of 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 con- 
secutiveness 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 ; 
therefore all that remains to ensure its being serial is that it shall 
be contained in diversity. This is a generalisation of the instance 
of the dinner-table. Another generalisation consists in taking 
R to be a one-one relation, and including the ancestry of x as 
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 be contained in diversity. 

The generation of order by means of relations of consecutive- 
ness, though important in its own sphere, is less general than the 
method which uses a transitive relation to define the order. It 
often happens in a series that there are an infinite number of inter 
mediate terms between any two that may be selected, however 
near together these may be. Take, for instance, fractions in order 
of magnitude. Between any two fractions there are others for 
example, the arithmetic mean of the two. Consequently there is 
no such thing as a pair of consecutive fractions. If we depended 



38 Introduction to Mathematical Philosophy 

upon consecutiveness for defining order, we should not be able 
to define the order of magnitude among fractions. But in fact 
the relations of greater and less among fractions do not demand 
generation from relations of consecutiveness, and the relations 
of greater and less among fractions have the three characteristics 
which we need for defining serial relations. In all such cases 
the order must be defined by means 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 number of a collection, is appro 
priate to the finite ; it may even be extended to certain infinite 
series, namely, those in which, though the total number of terms is 
infinite, the number of terms between any two is always finite ; 
but it must not be regarded as general. Not only so, but care 
must be taken to eradicate from the imagination all habits of 
thought resulting from supposing it general. If this is not done, 
series in which there are no consecutive terms will remain difficult 
and puzzling. And such series are of vital importance for the 
understanding of continuity, space, time, and motion. 

There are many ways in which series may be generated, but 
all depend upon the finding or construction of an asymmetrical 
transitive connected relation. Some of these ways have con 
siderable importance. We may take as illustrative the genera 
tion of series by means of a three-term relation which we may 
call " between." This method is very useful in geometry, and 
may serve as an introduction to relations having more than two 
terms ; it is best introduced in connection with elementary 
geometry. 

Given any three points on a straight line in ordinary space, 
there must be one of them which is between the other two. This 
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. 
In fact, the notion " between " is characteristic of open series 
or series in the strict sense as opposed to what may be called 



The Definition of Order 39 

" cyclic " series, where, as with people at the dinner-table, a 
sufficient journey brings us back to our starting-point. This 
notion of " between " may be chosen as the fundamental notion 
of ordinary geometry ; but for the present we will only consider 
its application to a single straight line and to the ordering of the 
points on a straight line. 1 Taking any two points #, b, the line 
(ab) consists of three parts (besides a and b themselves) : 

(1) Points between a and b. 

(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 (ab) can be defined in terms of the relation 
" between." 

In order that this relation " between " may arrange the points 
of the line in an order from left to right, we need certain assump 
tions, namely, the following : 

(1) If anything is between a and b, a and b are not identical. 

(2) Anything between a and b is also between b and a. 

(3) Anything between a and b is not identical with a (nor, 
consequently, with b, in virtue of (2)). 

(4) If x is between a and b, anything between a and x is also 
between a and b. 

(5) If x is between a and b, and b is between x and y, then b 
is 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. 

(7) If b is between a and x and also between a and y, then either 
a: and y are identical, or x is between and y, or y is between 
b and #. 

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 may be seen from the 
following definitions. For the sake of definiteness, let us assume 

1 Cf . Rivista di Matematica, iv. pp. 55 ft. ; Principles of Mathematics, p. 
394 ( 375). 



4-O Introduction to Mathematical Philosophy 

that a is to the left of b. Then the points of the line (ab) are (i) 
those between which and b, a lies these we will call to the left 
of a ; (2) a itself ; (3) those between a and b ; (4) b itself ; (5) 
those between which and a lies b these we will call to the right 
of b. We may now define generally that of two points x, y, on 
the line (ab), we shall say that x is " to the left of " y in any 
of the following cases : 

(1) When x and y are both to the left of a, and y is between 

x and a ; 

(2) When x is to the left of a, and y is a or b or between a and 

b or to the right of b ; 

(3) When x is a, and y is between a and b or is b or is to the 

right of b ; 

(4) When x and y are both between a and , and y is between 

# and b ; 

(5) When x is between <z and b, and y is 3 or to the right of b ; 

(6) When x is and y is to the right of b ; 

(7) When x and y are both to the right of b and x is between 

b and y. 

It will be found that, from the seven properties which we have 
assigned to the relation " between," it can be deduced that the 
relation " to the left of," as above defined, is a serial relation as 
we defined that term. It is important to notice that nothing 
in the definitions or the argument depends upon our meaning 
by " between " the actual relation of that name which occurs in 
empirical space : any three-term relation having the above seven 
purely formal properties will serve the purpose of the argument 
equally well. 

Cyclic order, such as that of the points on a circle, cannot be 
generated by means of three-term relations of " between." We 
need a relation of four terms, which may be called " separation 
of couples." The point may be illustrated by considering a 
journey round the world. One may go from England to New 
Zealand by way of Suez or by way of San Francisco ; we cannot 



The Definition of Order 41 

say definitely that either of these two places is " between " 
England and New Zealand. But if a man chooses that route 
to go round the world, whichever way round he goes, his times in 
England and New Zealand are separated from each other by his 
times in Suez and San Francisco, and conversely. Generalising, 
if we take any four points on a circle, we can separate them into 
two couples, say a and b and x and y, such that, in order to get 
from a to b one must pass through either x or y, and in order to 
get from x to y one must pass through either a or b. Under these 
circumstances we say that the couple (a, b) are " separated " by 
the couple (x, y). Out of this relation a cyclic order can be gen 
erated, in a way resembling that in which we generated an open 
order from " between," but somewhat more complicated. 1 

The purpose of the latter half of this chapter has been to suggest 
the subject which one may call " generation of serial relations." 
When such relations have been defined, the generation of them 
from other relations possessing only some of the properties 
required for series becomes very important, especially in the 
philosophy of geometry and physics. But we cannot, within 
the limits of the present volume, do more than make the reader 
aware that such a subject exists. 

1 Cf. Principles of Mathematics, p. 205 ( 194), and references there given. 



CHAPTER V 

KINDS OF RELATIONS 

A GREAT part of the philosophy of mathematics is concerned with 
relations, and many different kinds of relations have different 
kinds of uses. It often happens that a property which belongs 
to all relations is only important as regards relations of certain 
sorts ; in these cases the reader will not see the bearing of the 
proposition asserting such a property unless he has in mind the 
sorts of relations for which it is useful. For reasons of this 
description, as well as from the intrinsic interest of the subject, 
it is well to have in our minds a rough list of the more 
mathematically serviceable varieties of relations. 

We dealt in the preceding chapter with a supremely important 
class, namely, serial relations. Each of the three properties which 
we combined in defining series namely, asymmetry, transitiveness, 
and connexity has its own importance. We will begin by saying 
something on each of these three. 

Asymmetry, i.e. the property of being incompatible with the 
converse, is a characteristic of the very greatest interest and 
importance. In order to develop its functions, we will consider 
various examples. The relation husband is asymmetrical, and 
so is the relation wife ; i.e. if a is husband of b, b cannot be husband 
of a, and similarly in the case of wife. On the other hand, the 
relation " spouse " is symmetrical : if a is spouse of b, then b is 
spouse of a. Suppose now we are given the relation spouse, and 
we wish to derive the relation husband. Husband is the same as 
male spouse or spouse of a female ; thus the relation husband can 

4* 



Kinds of Relations 43 

be derived from spouse either by limiting the domain to males 
or by limiting the converse to females. We see from this instance 
that, when a symmetrical relation is given, it is sometimes possible, 
without the help of any further relation, to separate it into two 
asymmetrical relations. But the cases where this is possible are 
rare and exceptional : they are cases where there are two mutually 
exclusive classes, say a and j3, such that whenever the relation 
holds between two terms, one of the terms is a member of a and 
the other is a member of )3 as, in the case of spouse, one term 
of the relation belongs to the class of males and one to the class 
of females. In such a case, the relation with its domain confined 
to a will be asymmetrical, and so will the relation with its domain 
confined to j3. But such cases are not of the sort that occur 
when we are dealing with series of more than two terms ; for in 
a series, all terms, except the first and last (if these exist), belong 
both to the domain and to the converse domain of the generating 
relation, so that a relation like husband, where the domain and 
converse domain do not overlap, is excluded. 

The question how to construct relations having some useful 
property by means of operations upon relations which only have 
rudiments of the property is one of considerable importance. 
Transitiveness and connexity are easily constructed in many cases 
where the originally given relation does not possess them : for 
example, if R is any relation whatever, the ancestral relation 
derived from R by generalised induction is transitive ; and if R 
is a many-one relation, the ancestral relation will be connected 
if confined to the posterity of a given term. But asymmetry is 
a much more difficult property to secure by construction. The 
method by which we derived husband from spouse is, as we have 
seen, not available in the most important cases, such as greater, 
before, to the right of, where domain and converse domain overlap. 
In all these cases, we can of course obtain a symmetrical relation 
by adding together the given relation and its converse, but we 
cannot pass back from this symmetrical relation to the original 
asymmetrical relation except by the help of some asymmetrical 



44 Introauction to Mathematical Philosophy 

relation. Take, for example, the relation greater : the relation 
greater or less i.e. unequal is symmetrical, but there is nothing 
in this relation to show that it is the sum of two asymmetrical 
relations. Take such a relation as " differing in shape." This 
is not the sum of an asymmetrical relation and its converse, since 
shapes do not form a single series ; but there is nothing to show 
that it differs from " differing in magnitude " if we did not already 
know that magnitudes have relations of greater and less. This 
illustrates the fundamental character of asymmetry as a property 
of relations. 

From the point of view of the classification of relations, being 
asymmetrical is a much more important characteristic than 
implying diversity. Asymmetrical relations imply diversity, 
but the converse is not the case. " Unequal," for example, 
implies diversity, but is symmetrical. Broadly speaking, we 
may say that, if we wished as far as possible to dispense with 
relational propositions and replace them by such as ascribed 
predicates to subjects, we could succeed in this so long as we 
confined ourselves to symmetrical relations : those that do not 
imply diversity, if they are transitive, may be regarded as assert 
ing a common predicate, while those that do imply diversity 
may be regarded as asserting incompatible predicates. For 
example, consider the relation of similarity between classes, 
by means of which we defined numbers. This relation is sym 
metrical and transitive and does not imply diversity. It would 
be possible, though less simple than the procedure we adopted, 
to regard the number of a collection as a predicate of the collec 
tion : then two similar classes will be two that have the same 
numerical predicate, while two that are not similar will be two 
that have different numerical predicates. Such a method of 
replacing relations by predicates is formally possible (though 
often very inconvenient) so long as the relations concerned are 
symmetrical ; but it is formally impossible when the relations 
are asymmetrical, because both sameness and difference of predi 
cates are symmetrical. Asymmetrical relations are, we may 



Kinds of Relations 45 

say, the most characteristically relational of relations, and the 
most important to the philosopher who wishes to study the 
ultimate logical nature of relations. 

Another class of relations that is of the greatest use is the 
class of one-many relations, i.e. relations which at most one 
term can have to a given term. Such are father, mother, 
husband (except in Tibet), square of, sine of, and so on. But 
parent, square root, and so on, are not one-many. It is possible, 
formally, to replace all relations by one-many relations by means 
of a device. Take (say) the relation less among the inductive 
numbers. Given any number n greater than I, there will not 
be only one number having the relation less to n, but we can 
form the whole class of numbers that are less than n. This 
is one class, and its relation to n is not shared by any other class. 
We may call the class of numbers that are less than n the " proper 
ancestry " of n, in the sense in which we spoke of ancestry and 
posterity in connection with mathematical induction. Then 
" proper ancestry " is a one-many relation (one-many will always 
be used so as to include one-one), since each number determines 
a single class of numbers as constituting its proper ancestry. 
Thus the relation less than can be replaced by being a member of 
the proper ancestry of. In this way a one-many relation in which 
the one is a class, together with membership of this class, can 
always formally replace a relation which is not one-many. Peano, 
who for some reason always instinctively conceives of a relation 
as one-many, deals in this way with those that are naturally 
not so. Reduction to one-many relations by this method, 
however, though possible as a matter of form, does not represent 
a technical simplification, and there is every reason to think 
that it does not represent a philosophical analysis, if only because 
classes must be regarded as " logical fictions." We shall there 
fore continue to regard one-many relations as a special kind of 
relations. 

One-many relations are involved in all phrases of the form 
" the so-and-so of such-and-such." " The King of England," 



46 Introduction to Mathematical Philosophy 

" the wife of Socrates," " the father of John Stuart Mill," and 
so on, all describe some person by means of a one-many relation 
to a given term. A person cannot have more than one father, 
therefore " the father of John Stuart Mill " described some one 
person, even if we did not know whom. There is much to 
say on the subject of descriptions, but for the present it is 
relations that we are concerned with, and descriptions are only 
relevant as exemplifying the uses of one-many relations. It 
should be observed that all mathematical functions result from 
one-many relations : the logarithm of x, the cosine of x, etc., 
are, like the father of x, terms described by means of a one-many 
relation (logarithm, cosine, etc.) to a given term (x). The 
notion of function need not be confined to numbers, or to the 
uses to which mathematicians have accustomed us ; it can be 
extended to all cases of one-many relations, and " the father of x " 
is just as legitimately a function of which x is the argument as 
is " the logarithm of x." Functions in this sense are descriptive 
functions. As we shall see later, there are functions of a still 
more general and more fundamental sort, namely, prepositional 
functions ; but for the present we shall confine our attention 
to descriptive functions, i.e. " the term having the relation R 
to x," or, for short, " the R of x" where R is any one-many 
relation. 

It will be observed that if " the R of x " is to describe a definite 
term, x must be a term to which something has the relation R, 
and there must not be more than one term having the relation 
R to x, since " the," correctly used, must imply uniqueness. 
Thus we may speak of " the father of x " if x is any human being 
except Adam and Eve ; but we cannot speak of " the father 
of x " if x is a table or a chair or anything else that does not 
have a father. We shall say that the R of x " exists " when 
there is just one term, and no more, having the relation R to x. 
Thus if R is a one-many relation, the R of x exists whenever 
x belongs to the converse domain of R, and not otherwise. 
Regarding " the R of x " as a function in the mathematical 



Kinds of Relations 47 

sense, we say that x is the " argument " of the function, and if 
y is the term which has the relation R to x, i.e. if y is the R of x, 
then y is the " value " of the function for the argument x. If 
R is a one-many relation, the range of possible arguments to 
the function is the converse domain of R, and the range of values 
is the domain. Thus the range of possible arguments to the 
function " the father of x " is all who have fathers, i.e. the con 
verse domain of the relation father, while the range of possible 
values for the function is all fathers, i.e. the domain of the relation. 

Many of the most important notions in the logic of relations 
are descriptive functions, for example : converse, domain, con 
verse domain, field. Other examples will occur as we proceed. 

Among one-many relations, one-one relations are a specially 
important class. We have already had occasion to speak of 
one-one relations in connection with the definition of number, 
but it is necessary to be familiar with them, and not merely 
to know their formal definition. Their formal definition may 
be derived from that of one-many relations : they may be 
defined as one-many relations which are also the converses of 
one-many relations, i.e. as relations which are both one-many 
and many-one. One-many relations may be defined as relations 
such that, if x has the relation in question to y, there is no other 
term x' which also has the relation to y. Or, again, they may 
be defined as follows : Given two terms x and x', the terms to 
which x has the given relation and those to which x' has it have 
no member in common. Or, again, they may be defined as 
relations such that the relative product of one of them and 
its converse implies identity, where the " relative product " 
of two relations R and S is that relation which holds between 
x and 2 when there is an intermediate term y, such that x has 
the relation R to y and y has the relation S to 2. Thus, for 
example, if R is the relation of father to son, the relative product 
of R and its converse will be the relation which holds between 
x and a man 2 when there is a person y, such that x is the father 
of y and y is the son of 2. It is obvious that x and z must be 



48 Introduction to Mathematical Philosophy 

the same person. If, on the other hand, we take the relation 
of parent and child, which is not one-many, we can no longer 
argue that, if x is a parent of y and y is a child of z, x and z must 
be the same person, because one may be the father of y and the 
other the mother. This illustrates that it is characteristic of 
one-many relations when the relative product of a relation and 
its converse implies identity. In the case of one-one relations 
this happens, and also the relative product of the converse and 
the relation implies identity. Given a relation R, it is convenient, 
if x has the relation R to y, to think of y as being reached from 
x by an " R-step " or an " R-vector." In the same case x will 
be reached from y by a " backward R-step." Thus we may 
state the characteristic of one-many relations with which we 
have been dealing by saying that an R-step followed by a back 
ward R-step must bring us back to our starting-point. With 
other relations, this is by no means the case ; for example, if 
R is the relation of child to parent, the relative product of R and 
its converse is the relation " self or brother or sister,'* and if R 
is the relation of grandchild to grandparent, the relative product 
of R and its converse is " self or brother or sister or first cousin." 
It will be observed that the relative product of two relations 
is not in general commutative, i.e. the relative product of R 
and S is not in general the same relation as the relative product 
of S and R. E.g. the relative product of parent and brother is 
uncle, but the relative product of brother and parent is parent. 

One-one relations give a correlation of two classes, term for 
term, so that each term in either class has its correlate in the 
other. Such correlations are simplest to grasp when the two 
classes have no members in common, like the class of husbands 
and the class of wives ; for in that case we know at once whether 
a term is to be considered as one from which the correlating 
relation R goes, or as one to which it goes. It is convenient 
to use the word referent for the term from which the relation 
goes, and the term relatum for the term to which it goes. Thus 
if x and y are husband and wife, then, with respect to the relation 



Kinas of Relations 49 

" husband," x is referent and y relatum, but with respect to the 
relation " wife," y is referent and x relatum. We say that a 
relation and its converse have opposite " senses " ; thus the 
" sense " of a relation that goes from x to y is the opposite of 
that of the corresponding relation from y to x. The fact that a 
relation has a " sense " is fundamental, and is part of the reason 
why order can be generated by suitable relations. It will be 
observed that the class of all possible referents to a given relation 
is its domain, and the class of all possible relata is its converse 
domain. 

But it very often happens that the domain and converse 
domain of a one-one relation overlap. Take, for example, 
the first ten integers (excluding o), and add I to each ; thus 
instead of the first ten integers we now have the integers 

2, 3, 4> 5 6 > 7> 8 > 9 I0 > 

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 
end. There are still 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 
ten integers, 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 five of our previous set of integers, namely, 
2, 4, 6, 8, 10. The correlating relation in this case is the relation 
of a number to its double, which is again a one-one relation. 
Or we might have replaced each number by its square, thus 
obtaining the set 

i, 4, 9, 16, 25, 36, 49, 64, 81, 100. 

On this occasion only three of our original set are left, namely, 
I, 4, 9. Such processes of correlation may be varied endlessly. 

The most interesting case of the above kind is the case where 
our one-one relation has a converse domain which is part, but 

4 



50 Introduction to Mathematical Philosophy 

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, 
putting the correlate directly under the number whose correlate 
it is. Thus when the correlator is the relation of n to n-{-i y we 
have the two rows : 

1, 2, 3, 4, 5, ... n ... 

2, 3>4> 5> 6 > "+ 1 - - 

When the correlator is the relation of a number to its double, 
we have the two rows : 

1, 2, 3, 4, 5, ... n . . . 

2, 4, 6, 8, 10, ... 2w ... 

When the correlator is the relation of a number to its square, 
the rows are : 

i, 2, 3, 4, 5, ... 

i, 4, 9, 1 6, 25, ..... n 2 ... 

In all these cases, all inductive numbers occur in the top row, 
and only some in the bottom row. 

Cases of this sort, where the converse domain is a " proper 
part " of the domain (i.e. a part not the whole), will occupy us 
again when we come to deal with infinity. For the present, we 
wish only to note that they exist and demand consideration. 

Another class of correlations which are often important is 
the class called " permutations," where the domain and converse 
domain are identical. Consider, for example, the six possible 
arrangements 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, 
(a, b, c) and (c, b, a). Here a is correlated with c, b with itself, 
and c with a. It is obvious that the combination of two permu 
tations is again a permutation, i.e. the permutations of a given 
class form what is called a " 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 
in the philosophy of mathematics, as we have partly seen already, 
but shall see much more fully as we proceed. One of its uses 
will occupy us in our next chapter. 



CHAPTER VI 

SIMILARITY OF RELATIONS 

WE saw in Chapter II. that two classes have the same number 
of terms when they are " similar," i.e. when there is a one-one 
relation whose domain is the one class and whose converse 
domain is the other. In such a case we say that there is a 
" one-one correlation " between the two classes. 

In the present chapter we have to define a relation between 
relations, which will play the same part for them that similarity 
of classes plays for classes. We will call this relation " similarity 
of relations," or " likeness " when it seems desirable to use a 
different word from that which we use for classes. How is 
likeness to be defined ? 

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 
of these two terms. The easiest example of the sort of thing 
we desire is a map. When one place is north of another, the 
place on the map corresponding to the one is above the place 
on the map corresponding to the other ; when one place is west 
of another, the place on the map corresponding to the one is 
to the left of the place on the map corresponding to the other ; 
and so on. The structure of the map corresponds with that of 

52 



Similarity of Relations 53 

the country of which it is a map. The space-relations in the 
map have " likeness " to the space-relations in the country 
mapped. It is this kind of connection between relations that 
we wish to define. 

We may, in the first place, profitably introduce a certain 
restriction. We will confine ourselves, in defining likeness, to 
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 
relation has to the relation. This relation has all classes for its 
domain, since every class is the domain of some relation ; and 
it has all relations for its converse domain, since every relation 
has a domain. But classes and relations cannot be added to 
gether to form a new single class, because they are of different 
logical " types." We do not need to enter upon the difficult 
doctrine of types, but it is well to know when we are abstaining 
from entering upon it. We may say, without entering upon 
the grounds for the assertion, that a relation only has a " field " 
when it is what we call " homogeneous," i.e. when its domain 
and converse domain are of the same logical type ; and as a 
rough-and-ready indication of what we mean by a " type," 
we may say that individuals, classes of individuals, relations 
between individuals, relations between classes, relations of 
classes to individuals, and so on, are different types. Now the 
notion of likeness is not very useful as applied to relations that 
are not homogeneous ; we shall, therefore, in defining likeness, 
simplify our problem by speaking of the " field " of one of the 
relations concerned. This somewhat limits the generality of 
our definition, but the limitation is not of any practical impor 
tance. And having been stated, it need no longer be remembered. 
We may define two relations P and Q as " similar," or as 
having " likeness," when there is a one-one relation S whose 
domain is the field of P and whose converse domain is the field 
of Q. and which is such that, if one term has the relation P 



54 Introduction to Mathematical Philosophy 

to another, the correlate of the one has the relation Q to the 
correlate of the other, and vice versa. A figure will make this 

clearer. Let x and v be two 
x, P y 

. > . terms having the relation P. 

Then there are to be two terms 
z, w, such that x has the rela 
tion S to z, y has the relation 
S to zv, and z has the relation 

> Q to 20. If this happens with 

z Q w . 

every pair of terms such as x 

and y, and if the converse happens with every pair of terms such 
as z and w, it is clear that for every instance in which the relation 
P holds there is a corresponding instance in which the relation 
Q holds, and vice versa ; and this is what we desire to secure by 
our definition. We 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 z, the Q-step from z to w, and the backward S-step from 
w to y. Thus we may set up the following definitions : 

A relation S is said to be a " correlator " or an " ordinal 
correlator " of two relations P and Q if S is one-one, has the 
field of Q for its converse domain, and is such that P is the 
relative product of S and Q and the converse of S. 

Two relations P and Q are said to be " similar," or to have 
" likeness," when there is at least one correlator of P and Q. 

These definitions will be found to yield what we above decided 
to be necessary. 

It will be found that, when two relations are similar, they 
share all properties which do not depend upon the actual terms 
in their fields. For instance, if one implies diversity, so does 
the other ; if one is transitive, so is the other ; if one is con 
nected, so is the other. Hence if one is serial, so is the other. 
Again, if one is one-many or one-one, the other is one-many 



Similarity of Relations 55 

or one-one ; and so on, through all the general properties of 
relations. Even statements involving the actual terms of the 
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. We are led 
by such considerations to a problem which has, in mathematical 
philosophy, an importance by no means adequately recognised 
hitherto. Our problem may be stated as follows : 

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 
ings of the 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 
knowledge of nature. We know that certain scientific pro 
positions which, in the most advanced sciences, are expressed 
in mathematical symbols are more or less true of the world, 
but we are very much at sea as to the interpretation to be put 
upon the terms which occur in these propositions. We know 
much more (to use, for a moment, an old-fashioned pair of 
terms) about the form of nature than about the matter. 
Accordingly, what we really know when we enunciate a law 
of nature is only that there is probably some interpretation of 
our terms which will make the law approximately true. Thus 
great importance attaches to the question : What are the 
possible meanings of a law expressed in terms of which we do 
not know the substantive meaning, but only the grammar and 
syntax ? And this question is the one suggested above. 

For the present we will ignore the general question, which 
will occupy us again at a later stage; the subject of likeness 
itself must first be further investigated. 

Owing to the fact that, when two relations are similar, their 
properties are the same except when they depend upon the 
fields being composed of just the terms of which they are com 
posed, it is desirable to have a nomenclature which collects 



56 Introduction to Mathematical Philosophy 

together all the relations that are similar to a given relation. 
Just as we called the set of those classes that are similar to a 
given class the " number " of that class, so we may call the set 
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 
a " relation-number." Thus we have the 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 ; or, what comes to 
the same thing, a relation number is a class of relations consisting 
of all those relations that are similar to one member of the class. 

When it is necessary to speak of the numbers of classes in 
a way which makes it impossible to confuse them with relation- 
numbers, we shall call them " cardinal numbers." Thus cardinal 
numbers are the numbers appropriate to classes. These include 
the ordinary integers of daily life, and also certain infinite 
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 of a cardinal number, 
it will be remembered, is as follows : 

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. 
Two 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 
number of terms, and only then i.e. a series of (say) 15 terms 
will have the same relation-number as any other series of fifteen 
terms, but will not have the same relation-number as a series 
of 14 or 1 6 terms, nor, of course, the same relation-number 
as a relation which is not serial. Thus, in the quite special case 
of finite series, there is parallelism between cardinal and relation- 
numbers. The relation-numbers applicable to series may be 



Similarity of Relations 57 

called " serial numbers " (what are commonly called " ordinal 
numbers " are a sub-class of these) ; thus a finite serial number 
is determinate when we know the cardinal number of terms 
in the field of a series having the serial number in question. 
If n is a finite cardinal number, the relation-number of a series 
which has n terms is called the " ordinal " number n. (There 
are also infinite ordinal numbers, but of them we shall speak 
in a later chapter.) When the cardinal number of terms in 
the field of a series is infinite, the relation-number of the series 
is not determined merely by the cardinal number, indeed an 
infinite number of relation-numbers exist for one infinite cardinal 
number, as we shall see when we come to consider infinite series. 
When a series is infinite, what we may call its " length," i.e. 
its relation-number, may vary without change in the cardinal 
number ; but when a series is finite, this cannot happen. 

We can define addition and multiplication for relation- 
numbers as well as for cardinal numbers, and a whole arithmetic 
of relation-numbers can be developed. The manner in which 
this is to be done is 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 
of the 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 
that there is an order involved as between the two series : one 
of them must be placed before the other. Thus if P and Q 
are the generating relations of the two series, in the series which 
is their sum with P put before Q, every member of the field of 
P will 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 not serial, being not connected, since it does not hold between 
a member of the field of P and a member of the field of Q. Thus 
the sum of P and Q, as above defined, is what we need in order 



58 Introduction to Mathematical Philosophy 

to define the sum of two relation-numbers. Similar modifica 
tions are needed for products and powers. The resulting arith 
metic does not obey the commutative law : the sum or product 
of two relation-numbers generally depends upon the order in 
which they are taken. But it obeys the associative law, one 
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, is a thoroughly respectable branch of mathematics. 

It 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 
brought fully into relations of likeness with a system applying 
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. 
We may illustrate this by the relations of the sort that may be 
called " between," which we considered in Chapter IV. We 
there saw that, provided a three-term relation has certain formal 
logical properties, it will give rise to series, and may be called 
a " between-relation." Given any two points, we can use the 
between-relation to define the straight line determined by those 
two points ; it consists of a and b together with all points x, 
such that the between-relation holds between the three points 
a, b, x in some order or other. It has been shown by 0. Veblen 
that we may regard our whole space as the field of a three-term 
between-relation, and define our geometry by the properties we 
assign to our between-relation. 1 Now likeness is just as easily 

1 This does not apply to elliptic space, but only to spaces in which 
the straight line is an open series. Modern Mathematics, edited by 
J. W. A. Young, pp. 3-51 (monograph by O. Veblen on " The Foundations of 
Geometry"). 



Similarity of Relations 59 

definable between three-term relations as between two-term 
relations. If B and B' are two between-relations, so that 
" xB(y, z) " means " x is between y and z with respect to B," 
we shall call S a correlator of B and B 7 if it has the field of B' 
for its converse domain, and is such that the relation B holds 
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 
convince himself that, if B is like B' in this sense, there can be 
no difference between the geometry generated by B and that 
generated by B'. 

It follows from this that the mathematician need not concern 
himself with the particular being or intrinsic nature of his points, 
lines, and planes, even when he is speculating as an applied 
mathematician. We may 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 
is a matter of indifference, so long as it satisfies the axioms. If 
we can, out of empirical material, construct a logical structure, 
no matter how complicated, which will satisfy our geometrical 
axioms, that structure may legitimately be called a " point." 
We must not say that there is nothing else that could legitimately 
be called a " point " ; we must only say : " This object we have 
constructed is sufficient for the geometer ; it may be one of 
many objects, any of which would be sufficient, but that is no 
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 
principle that what matters in mathematics, and to a very great 
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 to Mathematical Philosophy 

" 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 
class may be defined by various different but co-extensive concepts 
e.g. " man " and " featherless biped," so two relations which 
are conceptually different may hold in the same set of instances. 
An " instance " in which a relation holds is to be conceived as a 
couple of terms, with an order, so that one of the terms comes 
first 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 
call the " extension " of this relation as the class of all ordered 
couples (Xy y) which are such that x is the father of y. From 
the mathematical point of view, the only thing of importance 
about the relation " father " is that it defines this set of ordered 
couples. Speaking generally, we say : 

The " extension " of a relation is the class of those ordered 
couples (x, y) which are such that x has the relation in question 
to y. 

We can now go a step further in the process of abstraction, 
and consider what we mean by " structure." Given any relation, 
we can, if it is a sufficiently simple one, construct a map of it. 
For the sake of definiteness, let us take a relation of which the 
extension is the following couples : ab y aCy ad y be, ce, dcy de y where 
<z, by Cy dy e ale five terms, no matter what. We may make a 
" map " of this relation by taking five points 

a . > . on a plane and connecting them by arrows, 

as in the accompanying figure. What is 
revealed by the map is what we call the 
" structure " of the relation. 

It is clear that the " structure " of the 
relation does not depend upon the particular 
terms that make up the field of the relation. 
The field may be changed without changing the structure, and 
the structure may be changed without changing the field for 




Similarity of Relations 61 

example, if we were to add the couple ae in the above illustration 
we should alter the structure but not the field. Two relations 
have the same " structure," we shall say, when the same map 
will do for both or, what comes to the same thing, when either 
can be a map for the other (since every relation can be its own 
map). And that, as a moment's reflection shows, is the very 
same thing as what we have called " likeness." That is to say, 
two relations have the same structure when they have likeness, 
i./. when they have the same relation-number. Thus what we 
defined as the " relation-number " is the very same thing as is 
obscurely intended by the word " structure " a word which, 
important as it is, is never (so far as we know) defined in precise 
terms by those who use it. 

There has been a great deal of speculation in traditional 
philosophy which might have been avoided if the importance of 
structure, and the difficulty of getting behind it, had been realised. 
For example, it is often said that space and time are subjective, 
but they have objective counterparts ; or that phenomena are 
subjective, but are caused by things in themselves, which must 
have differences inter se corresponding with the differences in 
the phenomena to which they give rise. Where such hypotheses 
are made, it is generally supposed that we can know very little 
about the objective counterparts. In actual fact, however, if 
the hypotheses as stated were correct, the objective counterparts 
would form a world having the same structure as the phenomenal 
world, and allowing us to infer from phenomena the truth of all 
propositions that can be stated in abstract terms and are known 
to be true of phenomena. If the phenomenal world has three 
dimensions, so must the world behind phenomena ; if the pheno 
menal world is Euclidean, so must the other be ; and so on. 
In short, every proposition having a communicable significance 
must be true of both worlds or of neither : the only difference 
must lie in just that essence of individuality which always eludes 
words and bafHes description, but which, for that very reason, 
is irrelevant to science. Now the only purpose that philosophers 



62 Introduction to Mathematical Philosophy 

have in view in condemning phenomena is in order to persuade 
themselves and others that the real world is very different from 
the world of appearance. We can all sympathise with their wish 
to prove such a very desirable proposition, but we cannot con 
gratulate them on their success. It is true that many of them 
do not assert objective counterparts to phenomena, and these 
escape from the above argument. Those who do assert counter 
parts are, as a rule, very reticent on the subject, probably because 
they feel instinctively that, if pursued, it will bring about too 
much of a rapprochement between the real and the phenomenal 
world. If they were to pursue the topic, they could hardly avoid 
the conclusions which we have been suggesting. In such ways, 
as well as in many others, the notion of structure or relation- 
number is important. 



CHAPTER VII 

RATIONAL, REAL, AND COMPLEX NUMBERS 

WE have now seen how to define cardinal numbers, and also 
relation-numbers, of which what are commonly called ordinal 
numbers are a particular species. It will be found that each 
of these kinds of number may be infinite just as well as finite. 
But neither is capable, as it stands, of the more familiar exten 
sions of the idea of number, namely, the extensions to negative, 
fractional, irrational, and complex numbers. In the present 
chapter we shall briefly supply logical definitions of these various 
extensions. 

One of the mistakes that have delayed the discovery of correct 
definitions in this region is the common idea that each extension 
of number included the previous sorts as special cases. It was 
thought that, in dealing with positive and negative integers, the 
positive integers might be identified with the original signless 
integers. Again it was thought that a fraction whose denominator 
is I may be identified with the natural number which is its 
numerator. And the irrational numbers, such as the square 
root of 2, were supposed to find their place among rational frac 
tions, as being greater than some of them and less than the others, 
so that rational and irrational numbers could be taken together 
as one class, called " real numbers." And when the idea of 
number was further extended so as to include " complex " 
numbers, i.e. numbers involving the square root of I, it was 
thought that real numbers could be regarded as those among 
complex numbers in which the imaginary part (i.e. the part 

63 



64 Introduction to Mathematical Philosophy 

which was a multiple of the square root of i) was zero. All 
these suppositions were erroneous, and must be discarded, as we 
shall find, if correct definitions are to be given. 

Let us begin with positive and negative integers. It is obvious 
on a moment's consideration that +1 and I must both be 
relations, and in fact must be each other's converses. The 
obvious and sufficient definition is that -f-i is the relation of 
tt-f I to n, and I is the relation of n to n-f-l. Generally, if m 
is any inductive number, -\-m will be the relation of n-\-m to n 
(for any n), and m will be the relation of n to n-\-m. Accord 
ing to this definition, -\-m is a relation which is one-one so 
long as n is a cardinal number (finite or infinite) and m is an 
inductive cardinal number. But -\-m is under no circumstances 
capable of being identified with m y which is not a relation, but 
a class of classes. Indeed, -f m is every bit as distinct from m 
as m is. 

Fractions are more interesting than positive or negative integers. 
We need fractions for many purposes, but perhaps most obviously 
for purposes of measurement. My friend and collaborator Dr 
A. N. Whitehead has developed a theory of fractions specially 
adapted for their application to measurement, which is set forth 
in Principia Mathematical But if all that is needed is to define 
objects having the required purely mathematical properties, this 
purpose can be achieved by a simpler method, which we shall 
here adopt. We shall define the fraction m/n as being that 
relation which holds between two inductive numbers x t y when 
xn=ym. This definition enables us to prove that m/n is a one- 
one relation, provided neither m or n is zero. And of course n/m 
is the converse relation to m/n. 

From the above definition it is clear that the fraction m/i is 
that relation between two integers x and y which consists in the 
fact that x=my. This relation, like the relation -f-w, is by no 
means capable of being identified with the inductive cardinal 
number m % because a relation and a class of classes are objects 
1 Vol. iii. * 300 ff., especially 303. 



Rational) Real, and Complex Numbers 65 

of utterly different kinds. 1 It will be seen that o/ is always the 
same relation, whatever inductive number n may be; it is, in short, 
the relation of o to any other inductive cardinal. We may call 
this the zero of rational numbers ; it is not, of course, identical 
with the cardinal number o. Conversely, the relation ra/o is 
always the same, whatever inductive number m may be. There 
is not any inductive cardinal to correspond to m/o. We may call 
it " the infinity of rationals." It is an instance of the sort of 
infinite that is traditional in mathematics, and that is represented 
by " oo ." This is a totally different sort from the true Cantorian 
infinite, which we shall consider in our next chapter. The in 
finity of rationals does not demand, for its definition or use, any 
infinite classes or infinite integers. It is not, in actual fact, a 
very important notion, and we could dispense with it altogether 
if there were any object in doing so. The Cantorian infinite, on 
the other hand, is of the greatest and most fundamental impor 
tance ; the understanding of it opens the way to whole new realms 
of mathematics and philosophy. 

It will be observed that zero and infinity, alone among ratios, 
are not one-one. Zero is one-many, and infinity is many-one. 

There is not any difficulty in defining greater and less among 
ratios (or fractions). Given two ratios mjn and p/q, we shall say 
that m/n is less than p/q if mq is less than pn. There is no 
difficulty in proving that the relation " less than," so defined, is 
serial, so that the ratios form a series in order of magnitude. In 
this series, zero is the smallest term and infinity is the largest. 
If we omit zero and infinity from our series, there is no longer 
any smallest or largest ratio ; it is obvious that if m/n is any ratio 
other than zero and infinity, m/2n is smaller and 2m/n is larger, 
though neither is zero or infinity, so that m/n is neither the smallest 

1 Of course in practice we shall continue to speak of a fraction as (say) 
greater or less than i, meaning greater or less than the ratio i/i. So 
long as it is understood that the ratio i/i and the cardinal number i are 
different, it is not necessary to be always pedantic in emphasising the 
difference. 

5 



66 Introduction to Mathematical Philosophy 

nor the largest ratio, and therefore (when zero and infinity are 
omitted) there is no smallest or largest, since m/n was chosen 
arbitrarily. In like manner we can prove that however nearly 
equal two fractions may be, there are always other fractions 
between them. For, let m/n and p/q be two fractions, of which 
p/q is the greater. Then it is easy to see (or to prove) that 
(m+p)/(n-}-q) will be greater than m/n and less than p/q. Thus 
the series of ratios is one in which no two terms are consecutive, 
but there are always other terms between any two. Since there 
are other terms between these others, and so on ad infinitum, it 
is obvious that there are an infinite number of ratios between 
any two, however nearly equal these two may be. 1 A series 
having the property that there are always other terms between 
any two, so that no two are consecutive, is called " compact." 
Thus the ratios in order of magnitude form a " compact " series. 
Such series have many important properties, and it is important 
to observe that ratios afford an instance of a compact series 
generated purely logically, without any appeal to space or time 
or any other empirical datum. 

Positive and negative ratios can be defined in a way analogous 
to that in which we defined positive and negative integers. 
Having first defined the sum of two ratios m/n and p/q as 
(mq+pn)/nq, we define -{-p/q as the relation of m/n-\-p/q to m/n, 
where m/n is any ratio ; and p/q is of course the converse of 
-\-p/q- This is not the only possible way of defining positive and 
negative ratios, but it is a way which, for our purpose, has the 
merit of being an obvious adaptation of the way we adopted in 
the case of integers. 

We come now to a more interesting extension of the idea of 
number, i.e. the extension to what are called " real " numbers, 
which are the kind that embrace irrationals. In Chapter I. we 
had occasion to mention " incommensurables " and their dis- 

1 Strictly speaking, this statement, as well as those following to the end 
of the paragraph, involves what is called the " axiom of infinity," which 
will be discussed in a later chapter. 



Rational, Reat y and Complex Numbers 67 

covery by Pythagoras. It was through them, i.e. through 
geometry, that irrational numbers were first thought of. A 
square of which the side is one inch long will have a diagonal of 
which the length is the square root of 2 inches. But, as the 
ancients discovered, there is no fraction of which the square is 2. 
This proposition is proved in the tenth book of Euclid, which is 
one of those books that schoolboys supposed to be fortunately lost 
in the days when Euclid was still used as a text-book. The proof 
is extraordinarily simple. If possible, let mjn be the square root 
of 2, so that ra 2 /ft 2 =2, i.e. m 2 2n 2 . Thus m 2 is an even number, 
and therefore m must be an even number, because the square of 
an odd number is odd. Now if m is even, m* must divide by 4, 
for if m=2p, then m 2 =^.p 2 . Thus we shall have 4 2 =2 2 , where 
p is half of m. Hence 2p 2 =n 2 9 and therefore n/p will also be the 
square root of 2. But then we can repeat the argument : if 
n=2q, pjq will also be the square root of 2, and so on, through 
an unending series of numbers that are each half of its predecessor. 
But this is impossible ; if we divide a number by 2, and then 
halve the half, and so on, we must reach an odd number after a 
finite number of steps. Or we may put the argument even more 
simply by assuming that the m/n we start with is in its lowest 
terms ; in that case, m and n cannot both be even ; yet we have 
seen that, if m 2 /n 2 2, they must be. Thus there cannot be any 
fraction m/n whose square is 2. 

Thus no fraction will express exactly the length of the diagonal 
of a square whose side is one inch long. This seems like a 
challenge thrown out by nature to arithmetic. However the 
arithmetician may boast (as Pythagoras did) about the power 
of numbers, nature seems able to baffle him by exhibiting lengths 
which no numbers can estimate in terms of the unit. But the 
problem did not remain in this geometrical form. As soon as 
algebra was invented, the same problem arose as regards the 
solution of equations, though here it took on a wider form, 
since it also involved complex numbers. 

It is clear that fractions can be found which approach nearer 



68 Introduction to Mathematical Philosophy 

and nearer to having their square equal to 2. We can form an 
ascending series of fractions all of which have their squares 
less than 2, but differing from 2 in their later members by 
less than any assigned amount. That is to say, suppose I assign 
some small amount in advance, say one-billionth, it will be 
found that all the terms of our series after a certain one, say the 
tenth, have squares that differ from 2 by less than this amount. 
And if I had assigned a still smaller amount, it might have been 
necessary to go further along the series, but we should have 
reached sooner or later a term in the series, say the twentieth, 
after which all terms would have had squares differing from 2 
by less than this still smaller amount. If we set to work to 
extract the square root of 2 by the usual arithmetical rule, we 
shall obtain an unending decimal which, taken to so-and-so 
many places, exactly fulfils the above conditions. We can 
equally well form a descending series of fractions whose squares 
are all greater than 2, but greater by continually smaller amounts 
as we come to later terms of the series, and differing, sooner or 
later, by less than any assigned amount. In this way we seem 
to be drawing a cordon round the square root of 2, and it may 
seem difficult to believe that it can permanently escape us. 
Nevertheless, it is not by this method that we shall actually 
reach the square root of 2. 

If we divide all ratios into two classes, according as their 
squares are less than 2 or not, we find that, among those whose 
squares are not less than 2, all have their squares greater than 2. 
There is no maximum to the ratios whose square is less than 2, 
and no minimum to those whose square is greater than 2. There 
is no lower limit short of zero to the difference between the 
numbers whose square is a little less than 2 and the numbers 
whose square is a little greater than 2. We can, in short, divide 
all ratios into two classes such that all the terms in one class 
are less than all in the other, there is no maximum to the one 
class, and there is no minimum to the other. Between these 
two classes, where V2 ought to be, there is nothing. Thus our 



Rational, Real, and Complex Numbers 69 

cordon, though we have drawn it as tight as possible, has been 
drawn in the wrong place, and has not caught v 2. 

The above method of dividing all the terms of a series into 
two classes, of which the one wholly precedes the other, was 
brought into prominence by Dedekind, 1 and is therefore called 
a " Dedekind cut." With respect to what happens at the point 
of section, there are four possibilities : (i) there may be a 
maximum to the lower section and a minimum to the upper 
section, (2) there may be a maximum to the one and no minimum 
to the other, (3) there may be no maximum to the one, but a 
minimum to the other, (4) there may be neither a maximum to 
the one nor a minimum to the other. Of these four cases, the 
first is illustrated by any series in which there are consecutive 
terms : in the series of integers, for instance, a lower section 
must end with some number n and the upper section must 
then begin with n+i. The second case will be illustrated 
in the series of ratios if we take as our lower section all ratios 
up to and including I, and in our upper section all ratios greater 
than I. The third case is illustrated if we take for our lower 
section all ratios less than I, and for our upper section all ratios 
from I upward (including I itself). The fourth case, as we have 
seen, is illustrated if we put in our lower section all ratios whose 
square is less than 2, and in our upper section all ratios whose 
square is greater than 2. 

We may neglect the first of our four cases, since it only arises 
in series where there are consecutive terms. In the second of 
our four cases, we say that the maximum of the lower section 
is the lower limit of the upper section, or of any set of terms 
chosen out of the upper section in such a way that no term of 
the upper section is before all of them. In the third of our 
four cases, we say that the minimum of the upper section is the 
upper limit of the lower section, or of any set of terms chosen 
out of the lower section in such a way that no term of the lower 
section is after all of them. In the fourth case, we say that 
1 Stetigkeit und irrationale Zahlen, 2nd edition, Brunswick, 1892. 



70 Introduction to Mathematical Philosophy 

there is a " gap " : neither the upper section nor the lower has 
a limit or a last term. In this case, we may also say that we 
have an " irrational section," since sections of the series of ratios 
have " gaps " when they correspond to irrationals. 

What delayed the true theory of irrationals was a mistaken 
belief that there must be " limits " of series of ratios. The 
notion of " limit " is of the utmost importance, and before 
proceeding further it will be well to define it. 

A term x is said to be an " upper limit " of a class a with 
respect to a relation P if (i) a has no maximum in P, (2) every 
member of a which belongs to the field of P precedes x, (3) every 
member of the field of P which precedes x precedes some member 
of a. (By " precedes " we mean " has the relation P to.") 

This presupposes the following definition of a " maximum " : 

A term x is said to be a " maximum " of a class a with respect 
to a relation P if x is a member of a and of the field of P and does 
not have the relation P to any other member of a. 

These definitions do not demand that the terms to which 
they are applied should be quantitative. For example, given 
a series of moments of time arranged by earlier and later, their 
" maximum " (if any) will be the last of the moments ; but if 
they are arranged by later and earlier, their " maximum " (if 
any) will be the first of the moments. 

The " minimum " of a class with respect to P is its maximum 
with respect to the converse of P ; and the " lower limit " with 
respect to P is the upper limit with respect to the converse of P. 

The notions of limit and maximum do not essentially demand 
that the relation in respect to which they are defined should 
be serial, but they have few important applications except to 
cases when the relation is serial or quasi-serial. A notion which 
is often important is the notion " upper limit or maximum," 
to which we may give the name " upper boundary." Thus the 
" upper boundary " of a set of terms chosen out of a series is 
their last member if they have one, but, if not, it is the first 
term after all of them, if there is such a term. If there is neither 



Rational) Real, and Complex Numbers 71 

a maximum nor a limit, there is no upper boundary. The 
" lower boundary " is the lower limit or minimum. 

Reverting to the four kinds of Dedekind section, we see that 
in the case of the first three kinds each section has a boundary 
(upper or lower as the case may be), while in the fourth kind 
neither has a boundary. It is also clear that, whenever the 
lower section has an upper boundary, the upper section has 
a lower boundary. In the second and third cases, the two 
boundaries are identical ; in the first, they are consecutive 
terms of the series. 

A series is called " Dedekindian " when every section has a 
boundary, upper or lower as the case may be. 

We have seen that the series of ratios in order of magnitude 
is not Dedekindian. 

From the habit of being influenced by spatial imagination, 
people have supposed that series must have limits in cases where 
it seems odd if they do not. Thus, perceiving that there was 
no rational limit to the ratios whose square is less than 2, they 
allowed themselves to " postulate " an irrational limit, which 
was to fill the Dedekind gap. Dedekind, in the above-mentioned 
work, set up the axiom that the gap must always be filled, i.e. 
that every section must have a boundary. It is for this reason 
that series where his axiom is verified are called " Dedekindian." 
But there are an infinite number of series for which it is not 

4 

verified. 

The method of " postulating " what we want has many advan 
tages ; they are the same as the advantages of theft over honest 
toil. Let us leave them to others and proceed with our honest toil. 

It is clear that an irrational Dedekind cut in some way " repre 
sents " an irrational. In order to make use of this, which to 
begin with is no more than a vague feeling, we must find some 
way of eliciting from it a precise definition ; and in order to do 
this, we must disabuse our minds of the notion that an irrational 
must be the limit of a set of ratios. Just as ratios whose de 
nominator is i are not identical with integers, so those rational 



72 Introduction to Mathematical Philosophy 

numbers which can be greater or less than irrationals, or can 
have irrationals as their limits, must not be identified with ratios. 
We have to define a new kind of numbers called " real numbers," 
of which some will be rational and some irrational. Those that 
are rational " correspond " to ratios, in the same kind of way 
in which the ratio n/i corresponds to the integer n ; but they are 
not the same as ratios. In order to decide what they are to be, 
let us observe that an irrational is represented by an irrational 
cut, and a cut is represented by its lower section. Let us confine 
ourselves to cuts in which the lower section has no maximum ; 
in this case we will call the lower section a " segment." Then 
those segments that correspond to ratios are those that consist 
of all ratios less than the ratio they correspond to, which is 
their boundary ; while those that represent irrationals are those 
that have no boundary. Segments, both those that have 
boundaries and those that do not, are such that, of any two 
pertaining to one series, one must be part of the other ; hence 
they can all be arranged in a series by the relation of whole and 
part. A series in which there are Dedekind gaps, i.e. in which 
there are segments that have no boundary, will give rise to more 
segments than it has terms, since each term will define a segment 
having that term for boundary, and then the segments without 
boundaries will be extra. 

We are now in a position to define a real number and an 
irrational number. 

A " real number " is a segment of the series of ratios in order 
of magnitude. 

An " irrational number " is a segment of the series of ratios 
which has no boundary. 

A " rational real number " is a segment of the series of ratios 
which has a boundary. 

Thus a rational real number consists of all ratios less than a 
certain ratio, and it is the rational real number corresponding 
to that ratio. The real number I, for instance, is the class of 
proper fractions. 



Rational, Real, and Complex Numb en 73 

In the cases in which we naturally supposed that an irrational 
must be the limit of a set of ratios, the truth is that it is the limit 
of the corresponding set of rational real numbers in the series 
of segments ordered by whole and part. For example, ^/^ is 
the upper limit of all those segments of the series of ratios that 
correspond to ratios whose square is less than 2. More simply 
still, \/2 is the segment consisting of all those ratios whose square 
is less than 2. 

It is easy to prove that the series of segments of any series 
is Dedekindian. For, given any set of segments, their boundary 
will be their logical sum, i.e. the class of all those terms that 
belong to at least one segment of the set. 1 

The above definition of real numbers is an example of " con 
struction " as against " postulation," of which we had another 
example in the definition of cardinal numbers. The great 
advantage of this method is that it requires no new assumptions, 
but enables us to proceed deductively from the original apparatus 
of logic. 

There is no difficulty in defining addition and multiplication 
for real numbers as above defined. Given two real numbers 
\L and v, each being a class of ratios, take any member of JJL and 
any member of v and add them together according to the rule 
for the addition of ratios. Form the class of all such sums 
obtainable by varying the selected members of p and v. This 
gives a new class of ratios, and it is easy to prove that this new 
class is a segment of the series of ratios. We define it as the 
sum. of p and v. We may state the definition more shortly as 
follows : 

The arithmetical sum of two real numbers is the class of the 
arithmetical sums of a member of the one and a member of the 
other chosen in all possible ways. 

1 For a fuller treatment of the subject of segments and Dedekindian 
relations, see Principia Mathematical, vol. ii. * 210-214. For a fuller 
treatment of real numbers, see ibid., vol. iii. * 310 ff., and Principles of 
Mathematics, chaps, xxxiii. and xxxiv. 



74 Introduction to Mathematical Philosophy 

We can define the arithmetical product of two real numbers 
in exactly the same way, by multiplying a member of the one by 
a member of the other in all possible ways. The class of ratios 
thus generated is defined as the product of the two real numbers. 
(In all such definitions, the series of ratios is to be defined as 
excluding o and infinity.) 

There is no difficulty in extending our definitions to positive 
and negative real numbers and their addition and multiplication. 

It remains to give the definition of complex numbers. 

Complex numbers, though capable of a geometrical interpreta 
tion, are not demanded by geometry in the same imperative way 
in which irrationals are demanded. A " complex " number means 
a number involving the square root of a negative number, whether 
integral, fractional, or real. Since the square of a negative 
number is positive, a number whose square is to be negative has 
to be a new sort of number. Using the letter i for the square 
root of I, any number involving the square root of a negative 
number can be expressed in the form x-\-yi, where x and y are 
real. The part yi is called the " imaginary " part of this number, 
x being the " real " part. (The reason for the phrase " real 
numbers " is that they are contrasted with such as are " ima 
ginary.") Complex numbers have been for a long time habitually 
used by mathematicians, in spite of the absence of any precise 
definition. It has been simply assumed that they would obey 
the usual arithmetical rules, and on this assumption their employ 
ment has been found profitable. They are required less for 
geometry than for algebra and analysis. We desire, for example, 
to be able to say that every quadratic equation has two roots, 
and every cubic equation has three, and so on. But if we are 
confined to real numbers, such an equation as # 2 -|-i=o has no 
roots, and such an equation as x^io has only one. Every 
generalisation of number has first presented itself as needed for 
some simple problem : negative numbers were needed in order 
that subtraction might be always possible, since otherwise a b 
would be meaningless if a were less than b ; fractions were needed 



Rational) Real, and Complex Numbers 75 

in order that division might be always possible ; and complex 
numbers are needed in order that extraction of roots and solu 
tion of equations may be always possible. But extensions of 
number are not created by the mere need for them : they are 
created by the definition, and it is to the definition of complex 
numbers that we must now turn our attention. 

A complex number may be regarded and defined as simply an 
ordered couple of real numbers. Here, as elsewhere, many 
definitions are possible. All that is necessary is that the defini 
tions adopted shall lead to certain properties. In the case of 
complex numbers, if they are defined as ordered couples of real 
numbers, we secure at once some of the properties required, 
namely, that two real numbers are required to determine a com 
plex number, and that among these we can distinguish a first 
and a second, and that two complex numbers are only identical 
when the first real number involved in the one is equal to the 
first involved in the other, and the second to the second. What 
is needed further can be secured by defining the rules of addition 
and multiplication. We are to have 



Thus we shall define that, given two ordered couples of real 
numbers, (#, y) and (#', y'), their sum is to be the couple (x+x r , 
y+y')> and their product is to be the couple (xx f yy', xy'-\-x'y). 
By these definitions we shall secure that our ordered couples 
shall have the properties we desire. For example, take the 
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 is o are those which, according to the usual 
nomenclature, have their imaginary part zero ; in the notation 
x-\- yi, they are x+oi, which it is natural to write simply x. Just 
as it is natural (but erroneous) to identify ratios whose de 
nominator is unity with integers, so it is natural (but erroneous) 



j6 Introduction to Mathematical Philosophy 

to identify complex numbers whose imaginary part is zero with 
real 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 
the couple (o, l) is represented by *, and the couple (1, o) is 
represented by I. Now our rules of multiplication make the 
square of (o, l) equal to (1, o), i.e. the square of i is i. This 
is what we desired to secure. Thus our definitions serve all 
necessary purposes. 

It is easy to give a geometrical interpretation of complex 
numbers in the geometry of the plane. This subject was agree 
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 of order n is obtained by an 
obvious extension of the definition we have given. We define a 
complex number of order n as a one-many relation whose domain 
consists of certain real numbers and whose converse domain 
consists of the integers from I to n. 1 This is what would ordi 
narily be indicated by the notation (x l9 x 2 , # 3 , . . . x n ), where the 
suffixes denote correlation with the integers used as suffixes, and 
the correlation is one-many, not necessarily one-one, because x r 
and x a may be equal when r and s are not equal. The above 
definition, with a suitable rule of multiplication, will serve all 
purposes for which complex numbers of higher orders are needed. 

We have now completed our review of those extensions of 
number which do not involve infinity. The application of number 
to infinite collections must be our next topic. 

1 Cf . Principles of Mathematics, 360, p. 379. 



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 
natural numbers. To these we gave the name " inductive 
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 
inductive number of terms, nor have we inquired whether such 
collections can be said to have a number at all. This is an 
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. The assumption that there are is what 
we call the " axiom of infinity." 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 
time, there is certainly no logical reason against infinite collections, 
and we are therefore 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 form of our assumption with 

77 



7 8 Introduction to Mathematical Philosophy 

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 is not equal to n-\-i. This is involved in Peano's 
assumption that no two inductive numbers have the same suc 
cessor ; for, if n=n-}-i, then n I and n have the same successor, 
namely n. Thus we are assuming nothing that was not involved 
in Peano's primitive propositions. 

Let us now consider the collection of the inductive numbers 
themselves. This 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, *<? those which possess every property 
possessed by o and by the successors of possessors, meaning by 
the "successor" of n the number n-\-\. Thus the class of 
" inductive numbers " is perfectly definite. By our general 
definition of cardinal numbers, the number of terms in the class 
of inductive numbers is to be defined as " all those classes that 
are similar to the class of inductive numbers " i.e. this set of 
classes is the number of the inductive numbers according to our 
definitions. 

Now it is easy to see that this number is not one of the inductive 
numbers. If n is any inductive number, the number of numbers 
from o to n (both included) is n-\-i ; therefore the total number 
of inductive numbers is greater than n, no matter which of the 
inductive numbers n may be. If we arrange the inductive 
numbers in a series in order of magnitude, this series has no last 
term ; but if n is an inductive number, every series whose field 
has n terms has a last term, as it is easy to prove. Such differences 
might be multiplied a<L lib. Thus the number of inductive 
numbers is a new number, different from all of them, not possess 
ing all inductive properties. It may happen that o has a certain 



Infinite Cardinal Numbers 79 

property, and that if n has it so has w+i, and yet that this new 
number does not have it. The difficulties that so long delayed 
the theory of infinite numbers were largely due to the fact that 
some, at least, of the inductive properties were wrongly judged 
to be such as must belong to all numbers ; indeed it was thought 
that they could not be denied without contradiction. The first 
step in understanding infinite numbers consists in realising the 
mistakenness of this view. 

The most noteworthy and astonishing difference between an 
inductive number and this new number is that this new number 
is unchanged by adding I or subtracting I or doubling or halving 
or any of a number of other operations which we think of as 
necessarily making a number larger or smaller. The fact of being 
not altered by the addition of I is used by Cantor for the defini 
tion of what he calls " transfinite " cardinal numbers ; but for 
various reasons, some of which will appear as we proceed, it is 
better to define an infinite cardinal number as one which does 
not possess all inductive properties, i.e. simply as one which is 
not an inductive number. Nevertheless, the property of being 
unchanged by the addition of I is a very important one, and we 
must dwell on it for a time. 

To say that a class has a number which is not altered by the 
addition of I is the same thing as to say that, if we take a term x 
which does not belong to the class, we can find a one-one relation 
whose domain is the class and whose converse domain is obtained 
by adding x to the class. For in that case, the class is similar 
to the sum of itself and the term x, i.e. to a class having one extra 
term ; so that it has the same number as a class with one extra 
term, so that if n is this number, n=n-\-\. In this case, we shall 
also have nn I, i.e. there will be one-one relations whose 
domains consist of the whole class and whose converse domains 
consist of just one term short of the whole class. It can be shown 
that the cases in which this happens are the same as the apparently 
more general cases in which some part (short of the whole) can be 
put into one-one relation with the whole. When this can be done, 



8o Introduction to Mathematical Philosophy 

the correlator by which it is done may be said to " reflect " the 
whole class into a part of itself ; for this reason, such classes will 
be called " reflexive." Thus : 

A " reflexive " class is one which is similar to a proper part 
of itself. (A " proper part " is a part short of the whole.) 

A " reflexive " cardinal number is the cardinal number of a 
reflexive class. 

We have now to consider this property of reflexiveness. 

One of the most striking instances of a " reflexion " is Royce's 
illustration of the map : he imagines it decided to make a map 
of England upon a part of the surface of England. A map, if 
it is accurate, has a perfect one-one correspondence with its 
original ; thus our map, which is part, is in one-one relation with 
the whole, and must contain the same number of points as the 
whole, which must therefore be a reflexive number. Royce is 
interested in the fact that the map, if it is correct, must contain 
a map of the map, which must in turn contain a map of the map 
of the map, and so on ad infinitum. This point is interesting, 
but need not occupy us at this moment. In fact, we shall do 
well to pass from picturesque illustrations to such as are more 
completely definite, and for this purpose we cannot do better 
than consider the number-series itself. 

The relation of n to w-f-i, confined to inductive numbers, is 
one-one, has the whole of the inductive numbers for its domain, 
and all except o for its converse domain. Thus the whole class 
of inductive numbers is similar to what the same class becomes 
when we omit o. Consequently it is a " reflexive " class according 
to the definition, and the number of its terms is a " reflexive " 
number. Again, the relation of n to 2n, confined to inductive 
numbers, is one-one, has the whole of the inductive numbers for 
its domain, and the even inductive numbers alone for its converse 
domain. Hence the total number of inductive numbers is the 
same as the number of even inductive numbers. This property 
was used by Leibniz (and many others) as a proof that infinite 
numbers are impossible ; it was thought self-contradictory that 



Infinite Cardinal Numbers 81 

" the part should be equal to the whole." But this is one of those 
phrases that depend for their plausibility upon an unperceived 
vagueness : the word " equal " has many meanings, but if it is 
taken to mean what we have called " similar," there is no contra 
diction, since an infinite collection can perfectly well have parts 
similar to itself. Those who regard this as impossible have, 
unconsciously as a rule, attributed to numbers in general pro 
perties which can only be proved by mathematical induction, 
and which only their familiarity makes us regard, mistakenly, 
as true beyond the region of the finite. 

Whenever we can " reflect " a class into a part of itself, the 
same relation will necessarily reflect that part into a smaller 
part, and so on ad infinitum. For example, we can reflect, 
as we have just seen, all the inductive numbers into the even 
numbers ; we can, by the same relation (that of n to 2n) reflect 
the even numbers into the multiples of 4, these into the multiples 
of 8, and so on. This is an abstract analogue to Royce's problem 
of the map. The even numbers are a " map " of all the inductive 
numbers ; the multiples of 4 are a map of the map ; the multiples 
of 8 are a map of the map of the map ; and so on. If we had 
applied the same process to the relation of to w+l," our " map " 
would have consisted of all the inductive numbers except o ; 
the map of the map would have consisted of all from 2 onward, 
the map of the map of the map of all from 3 onward ; and so on. 
The chief use of such illustrations is in order to become familiar 
with the idea of reflexive classes, so that apparently paradoxical 
arithmetical propositions can be readily translated into the 
language of reflexions and classes, in which the air of paradox 
is much less. 

It will be useful to give a definition of the number which is 
that of the inductive cardinals. For this purpose we will 
first define the kind of series exemplified by the inductive cardinals 
in order of magnitude. The kind of series which is called a 
" progression " has already been considered in Chapter I. It is a 
series which can be generated by a relation of consecutiveness : 

6 



82 Introduction to Mathematical Philosophy 

every member of the series is to have a successor, but there is 
to be just one which has no predecessor, and every member of 
the series is to be in the posterity of this term with respect to 
the relation " immediate predecessor." These characteristics 
may be summed up in the following definition : * 

A " progession " is a one-one relation such that there is just 
one term belonging to the domain but not to the converse domain, 
and the domain is identical with the posterity of this one term. 

It is easy to see that a progression, so defined, satisfies Peano's 
five axioms. The term belonging to the domain but not to the 
converse domain will be what he calls " o " ; the term to which 
a term has the one-one relation will be the " successor " of the 
term ; and the domain of the one-one relation will be what 
he calls " number." Taking his five axioms in turn, we have 
the following translations : 

(1) " o is a number " becomes : " The member of the domain 
which is not a member of the converse domain is a member of 
the domain." This is equivalent to the existence of such a 
member, which is given in our definition. We will call this 
member " the first term." 

(2) " The successor of any number is a number " becomes : 
" The term to which a given member of the domain has the rela 
tion in question is again a member of the domain." This is 
proved as follows : By the definition, every member of the 
domain is a member of the posterity of the first term ; hence 
the successor of a member of the domain must be a member of 
the posterity of the first term (because the posterity of a term 
always contains its own successors, by the general definition of 
posterity), and therefore a member of the domain, because by 
the definition the posterity of the first term is the same as the 
domain. 

(3) " No two numbers have the same successor." This is 
only to say that the relation is one-many, which it is by definition 
(being one-one). 

1 Cf. Pnncipia Mathematica, vol. ii. # 123. 



Infinite Cardinal Numbers 83 

(4) " o is not the successor of any number " becomes : " The 
first term is not a member of the converse domain," which is 
again an immediate result of the definition. 

(5) This is mathematical induction, and becomes : " Every 
member of the domain belongs to the posterity of the first term," 
which was part of our definition. 

Thus progressions as we have defined them have the five 
formal properties from which Peano deduces arithmetic. It is 
easy to show that two progessions are " similar " in the sense 
defined for similarity of relations in Chapter VI. We can, of 
course, derive a relation which is serial from the one-one relation 
by which we define a progression : the method used is that 
explained in Chapter IV., and the relation is that of a term to 
a member of its proper posterity with respect to the original 
one-one relation. 

Two transitive asymmetrical relations which generate pro 
gressions are similar, for the same reasons for which the cor 
responding one-one relations are similar. The class of all such 
transitive generators of progressions is a " serial number " in 
the sense of Chapter VI.; it is in fact the smallest of infinite 
serial numbers, the number to which Cantor has given the name 
o>, by which he has made it famous. 

But we are concerned, for the moment, with cardinal numbers. 
Since two progressions are similar relations, it follows that their 
domains (or their fields, which are the same as their domains) 
are similar classes. The domains of progressions form a cardinal 
number, since every class which is similar to the domain of a 
progression is easily shown to be itself the domain of a progression. 
This cardinal number is the smallest of the infinite cardinal 
numbers ; it is the one to which Cantor has appropriated the 
Hebrew Aleph with the suffix o, to distinguish it from larger 
infinite cardinals, which have other suffixes. Thus the name of 
the smallest of infinite cardinals is N . 

To say that a class has N terms is the same thing as to say 
that it is a member of N , and this is the same thing as to say 



84 Introduction to Mathematical Philosophy 

that the members of the class can be arranged in a progression. 
It is obvious that any progression remains a progression if we 
omit a finite number of terms from it, or every other term, or 
all except every tenth term or every hundredth term. These 
methods of thinning out a progression do not make it cease to 
be a progression, and therefore do not diminish the number of 
its terms, which remains N . In fact, any selection from a pro 
gression is a progression if it has no last term, however sparsely 
it may be distributed. Take (say) inductive numbers of the form 
n w , or n nW . Such numbers grow very rare in the higher parts 
of the number series, and yet there are just as many of them as 
there are inductive numbers altogether, namely, N . 

Conversely, we can add terms to the inductive numbers without 
increasing their number. Take, for example, ratios. One 
might be inclined to think that there must be many more ratios 
than integers, since ratios whose denominator is I correspond 
to the integers, and seem to be only an infinitesimal proportion 
of ratios. But in actual fact the number of ratios (or fractions) 
is exactly the same as the number of inductive numbers, namely, 
N . This is easily seen by arranging ratios in a series on the 
following plan : If the sum of numerator and denominator in 
one is less than in the other, put the one before the other ; if 
the sum is equal in the two, put first the one with the smaller 
numerator. This gives us the series 

i, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, . . . 

This series is a progression, and all ratios occur in it sooner or 
later. Hence we can arrange all ratios in a progression, and 
their number is therefore N . 

It is not the case, however, that all infinite collections have 
N terms. The number of real numbers, for example, is greater 
than N ; it is, in fact, 2^, and it is not hard to prove that 2 n 
is greater than n even when n is infinite. The easiest way of 
proving this is to prove, first, that if a class has n members, it 
contains 2 n sub-classes in other words, that there are 2 n ways 



Infinite Carainal Numbers 85 

of selecting some of its members (including the extreme cases 
where we select all or none) ; and secondly, that the number of 
sub-classes contained in a class is always greater than the number 
of members of the class. Of these two propositions, the first 
is familiar in the case of finite numbers, and is not hard to extend 
to infinite numbers. The proof of the second is so simple and 
so instructive that we shall give it : 

In the first place, it is clear that the number of sub-classes 
of a given class (say a) is at least as great as the number of 
members, since each member constitutes a sub-class, and we thus 
have a correlation of all the members with some of the sub 
classes. Hence it follows that, if the number of sub-classes is 
not equal to the number of members, it must be greater. Now 
it is easy to prove that the number is not equal, by showing that, 
given any one-one relation whose domain is the members and 
whose converse domain is contained among the set of sub 
classes, there must be at least one sub-class not belonging to 
the converse domain. The proof is as follows : x When a one- 
one correlation R is established between all the members of a 
and some of the sub-classes, it may happen that a given member 
x is correlated with a sub-class of which it is a member ; or, 
again, it may happen that x is correlated with a sub-class of 
which it is not a member. Let us form the whole class, )3 say, 
of those members x which are correlated with sub-classes of which 
they are not members. This is a sub-class of a, and it is not 
correlated with any member of a. For, taking first the members 
of ]3, each of them is (by the definition of )8) correlated with 
some sub-class of which it is not a member, and is therefore not 
correlated with j3. Taking next the terms which are not members 
of jS, each of them (by the definition of j3) is correlated with 
some sub-class of which it is a member, and therefore again 
is not correlated with j8. Thus no member of a is correlated 
with )3. Since R was any one-one correlation of all members 

1 This proof is taken from Cantor, with some simplifications : see 
Jahresbericht der deutschen Mathematiker-Vereinigung, i. (1892), p. 77. 



86 Introduction to Mathematical Philosophy 

with some sub-classes, it follows that there is no correlation 
of all members with all sub-classes. It does not matter to the 
proof if j3 has no members : all that happens in that case is that 
the sub-class which is shown to be omitted is the null-class. 
Hence in any case the number of sub-classes is not equal to the 
number of members, and therefore, by what was said earlier, 
it is greater. Combining this with the proposition that, if n is 
the number of members, 2 n is the number of sub-classes, we have 
the theorem that 2 n is always greater than n, even when n is 
infinite. 

It follows from this proposition that there is no maximum 
to the infinite cardinal numbers. However great an infinite 
number n may be, 2 n will be still greater. The arithmetic of 
infinite numbers is somewhat surprising until one becomes 
accustomed to it. We have, for example, 



N -fw=N , where n is any inductive number, 
o 2 =o- 

(This follows from the case of the ratios, for, since a ratio is 
determined by a pair of inductive numbers, it is easy to see that 
the number of ratios is the square of the number of inductive 
numbers, i.e. it is N 2 ; but we saw that it is also 



N =N 0> where n is any inductive number. 
(This follows from N 2=N o by induction ; for if N O "=N O , 
then N +i=N 2 = N .) 

But 2^0 >N . 

In fact, as we shall see later, 2^ is a very important number, 
namely, the number of terms in a series which has " continuity " 
in the sense in which this word is used by Cantor. Assuming 
space and time to be continuous in this sense (as we commonly 
do in analytical geometry and kinematics), this will be the 
number of points in space or of instants in time ; it will also be 
the number of points in any finite portion of space, whether 



Infinite Cardinal Numbers 87 

line, area, or volume. After N , 2^ is the most important and 
interesting of infinite cardinal numbers. 

Although addition and multiplication are always possible 
with infinite cardinals, subtraction and division no longer give 
definite results, and cannot therefore be employed as they are 
employed in elementary arithmetic. Take subtraction to begin 
with : so long as the number subtracted is finite, all goes well ; 
if the other number is reflexive, it remains unchanged. Thus 
N n=& , if n is finite; so far, subtraction gives a perfectly 
definite result. But it is otherwise when we subtract N from 
itself; we may then get any result, from o up to N . This is 
easily seen by examples. From the inductive , numbers, take 
away the following collections of N terms : 

(1) All the inductive numbers remainder, zero. 

(2) All the inductive numbers from n onwards remainder, 
the numbers from o to n I, numbering n terms in all. 

(3) All the odd numbers remainder, all the even numbers, 
numbering N terms. 

All these are different ways of subtracting N from N , and 
all give different results. 

As regards division, very similar results follow from the fact 
that N is unchanged when multiplied by 2 or 3 or any finite 
number n or by N . It follows that N divided by N may have 
any value from I up to N . 

From the ambiguity of subtraction and division it results 
that negative numbers and ratios cannot be extended to infinite 
numbers. Addition, multiplication, and exponentiation proceed 
quite satisfactorily, but the inverse operations subtraction, 
division, and extraction of roots are ambiguous, and the notions 
that depend upon them fail when infinite numbers are concerned. 

The characteristic by which we defined finitude was mathe 
matical induction, i.e. we defined a number as finite when it 
obeys mathematical induction starting from o, and a class as 
finite when its number is finite. This definition yields the sort 
of result that a definition ought to yield, namely, that the finite 






88 Introduction to Mathematical Philosophy 

numbers are those that occur in the ordinary number-series 
o, i, 2, 3, ... But in the present chapter, the infinite num 
bers we have discussed have not merely been non-inductive : 
they have also been reflexive. Cantor used reflexiveness as the 
definition of the infinite, and believes that it is equivalent to 
non-inductiveness ; that is to say, he believes that every class 
and every cardinal is either inductive or reflexive. This may be 
true, and may very possibly be capable of proof ; but the proofs 
hitherto offered by Cantor and others (including the present 
author in former days) are fallacious, for reasons which will be 
explained when we come to consider the " multiplicative axiom." 
At present, it is not known whether there are classes and cardinals 
which are neither reflexive nor inductive. If n were such a 
cardinal, we should not have nn-\-i y but n would not be one 
of the " natural numbers," and would be lacking in some of the 
inductive properties. All known infinite classes and cardinals 
are reflexive ; but for the present it is well to preserve an open 
mind as to whether there are instances, hitherto unknown, of 
classes and cardinals which are neither reflexive nor inductive. 
Meanwhile, we adopt the following definitions : 

A. finite class or cardinal is one which is inductive. 

An infinite class or cardinal is one which is not inductive. 
All refiexive classes and cardinals are infinite ; but it is not known 
at present whether all infinite classes and cardinals are reflexive. 
We shall return to this subject in Chapter XII. 



CHAPTER IX 

INFINITE SERIES AND ORDINALS 

AN " infinite series " may be defined as a series of which the field 
is an infinite class. We have already had occasion to consider 
one kind of infinite series, namely, progressions. In this chapter 
we shall consider the subject more generally. 

The most noteworthy characteristic of an infinite series is 
that its serial number can be altered by merely re-arranging 
its terms. In this respect there is a certain oppositeness between 
cardinal and serial numbers. It is possible to keep the cardinal 
number of a reflexive class unchanged in spite of adding terms 
to it ; on the other hand, it is possible to change the serial 
number of a series without adding or taking away any terms, 
by mere re-arrangement. At the same time, in the case of any 
infinite series it is also possible, as with cardinals, to add terms 
without altering the serial number : everything depends upon 
the way in which they are added. 

In order to make matters clear, it will be best to begin with 
examples. Let us first consider various different kinds of series 
which can be made out of the inductive numbers arranged on 
various plans. We start with the series 



3, 



which, as we have already seen, represents the smallest of in 
finite serial numbers, the sort that Cantor calls co. Let us 
proceed to thin out this series by repeatedly performing the 

89 



90 Introduction to Mathematical Philosophy 

operation of removing to the end the first even number that 
occurs. We thus obtain in succession the various series : 

i, 3, 4> 5, w > 2 > 

i, 3, 5> 6 > - n + l > 2 >4> 

i, 3> 5, 7> w + 2 > ' 2 > 4> 6 > 

and so on. If we imagine this process carried on as long as 
possible, we finally reach the series 

i, 3,5, 7, . . . 2n+i, . 2,4,6,8, .. 2n, 
in which we have first all the odd numbers and then all the even 
numbers. 

The serial numbers of these various series are w+i, co+2, 
w ~l~3> 2aj - Each of these numbers is " greater" than any 
of its predecessors, in the following sense : 

One serial number is said to be " greater " than another if 
any series having the first number contains a part having the 
second number, but no series having the second number contains 
a part having the first number. 
If we compare the two series 

i, 2, 3, 4, . . n, . . 

i, 3,4,5, . . , +i, . . 2, 

we see that the first is similar to the part of the second which 
omits the last term, namely, the number 2, but the second is 
not similar to any part of the first. (This is obvious, but is 
easily demonstrated.) Thus the second series has a greater 
serial number than the first, according to the definition i.e. 
CD+I is greater than o>. But if we add a term at the beginning 
of a progression instead of the end, we still have a progression. 
Thus I -\-a>o}. Thus i-fo> is not equal to co+i. This is 
characteristic of relation-arithmetic generally : if p, and v are 
two relation-numbers, the general rule is that p+v is not equal 
to v-\-p,. The case of finite ordinals, in which there is equality, 
is quite exceptional. 

The series we finally reached just now consisted of first all the 
odd numbers and then all the even numbers, and its serial 



Infinite Series and Ordinals 91 

number is 2o>. This number is greater than o> or a)-{-n 9 where 
n is finite. It is to be observed that, in accordance with the 
general definition of order, each of these arrangements of integers 
is to be regarded as resulting from some definite relation. E.g. 
the qne which merely removes 2 to the end will be defined by 
the following relation : " x and y are finite integers, and either 
y is 2 and x is not 2, or neither is 2 and x is less than y." The 
one which puts first all the odd numbers and then all the even 
ones will be defined by : " x and y are finite integers, and either 
x is odd and y is even or x is less than y and both are odd or both 
are even." We shall not trouble, as a rule, to give these formulae 
in future ; but the fact that they could be given is essential. 

The number which we have called 2o>, namely, the number of 
a series consisting of two progressions, is sometimes called a> .2. 
Multiplication, like addition, depends upon the order of the 
factors : a progression of couples gives a series such as 



which is itself a progression ; but a couple of progressions gives 
a series which is twice as long as a progression. It is therefore 
necessary to distinguish between 2cu and to . 2. Usage is variable ; 
we shall use 2o> for a couple of progressions and a> . 2 for a pro 
gression of couples, and this decision of course governs our 
general interpretation of " a . )3 " when a and j3 are relation- 
numbers : " a . j3 " will have to stand for a suitably constructed 
sum of a relations each having jS terms. 

We can proceed indefinitely with the process of thinning 
out the inductive numbers. For example, we can place first 
the odd numbers, then their doubles, then the doubles of these, 
and so on. We thus obtain the series 

3 5t 7 , ; 2 > 6 I0 > H> ; 4> I2 > 20 > 28, . . ; 

8, 24, 40, 56, . . ., 

of which the number is o> 2 , since it is a progression of progressions . 
Any one of the progressions in this new series can of course be 



92 Introduction to Mathematical Philosophy 

thinned out as we thinned out our original progression. We can 
proceed to o> 3 , o> 4 , . . co w , and so on ; however far we have gone, 
we can always go further. 

The series of all the ordinals that can be obtained in this way, 
i.e. all that can be obtained by thinning out a progression, is 
itself longer than any series that can be obtained by re-arranging 
the terms of a progression. (This is not difficult to prove.) 
The cardinal number of the class of such ordinals can be shown 
to be greater than N ; it is the number which Cantor calls 
Nj. The ordinal number of the series of all ordinals that can 
be made out of an N , taken in order of magnitude, is called o) v 
Thus a series whose ordinal number is coj has a field whose 
cardinal number is Nj. 

We can proceed from co x and N x to co 2 and N 2 by a process 
exactly analogous to that by which we advanced from w and N 
to o>! and M x . And there is nothing to prevent us from advancing 
indefinitely in this way to new cardinals and new ordinals. It 
is not known whether 2^ is equal to any of the cardinals in the 
series of Alephs. It is not even known whether it is comparable 
with them in magnitude ; for aught we know, it may be neither 
equal to nor greater nor less than any one of the Alephs. This 
question is connected with the multiplicative axiom, of which 
we shall treat later. 

All the series we have been considering so far in this chapter 
have been what is called "well-ordered." A well-ordered 
series is one which has a beginning, and has consecutive terms, 
and has a term next after any selection of its terms, provided 
there are any terms after the selection. This excludes, on the 
one hand, compact series, in which there are terms between 
any two, and on the other hand series which have no beginning, 
or in which there are subordinate parts having no beginning. 
The series of negative integers in order of magnitude, having 
no beginning, but ending with I, is not well-ordered; but 
taken in the reverse order, beginning with I, it is well-ordered, 
being in fact a progression. The definition is : 



Infinite Series and Ordinals 93 

A " well-ordered " series is one in which every sub-class 
(except, of course, the null-class) has a first term. 

An " ordinal " number means the relation-number of a well- 
ordered series. It is thus a species of serial number. 

Among well-ordered series, a generalised form of mathematical 
induction applies. A property may be said to be " transfinitely 
hereditary " if, when it belongs to a certain selection of the 
terms in a series, it belongs to their immediate successor pro 
vided they have one. In a well-ordered series, a transfinitely 
hereditary property belonging to the first term of the series 
belongs to the whole series. This makes it possible to prove 
many propositions concerning well-ordered series which are not 
true of all series. 

It is easy to arrange the inductive numbers in series which 
are not well-ordered, and even to arrange them in compact 
series. For example, we can adopt the following plan : consider 
the decimals from *i (inclusive) to I (exclusive), arranged in order 
of magnitude. These form a compact series ; between any 
two there are always an infinite number of others. Now omit 
the dot at the beginning of each, and we have a compact series 
consisting of all finite integers except such as divide by 10. If 
we wish to include those that divide by 10, there is no difficulty ; 
instead of starting with *i, we will include all decimals less than 
I, but when we remove the dot, we will transfer to the right any 
o's that occur at the beginning of our decimal. Omitting these, 
and returning to the ones that have no o's at the beginning, 
we can state the rule for the arrangement of our integers as 
follows : Of two integers that do not begin with the same digit, 
the one that begins with the smaller digit comes first. Of two 
that do begin with the same digit, but differ at the second digit, 
the one with the smaller second digit comes first, but first of all 
the one with no second digit ; and so on. Generally, if two 
integers agree as regards the first n digits, but not as regards 
the (n-f-i)**, that one comes first which has either no (n+i) th 
digit or a smaller one than the other. This rule of arrangement, 



94 Introduction to Mathematical Philosophy 

as the reader can easily convince himself, gives rise to a compact 
series containing all the integers not divisible by 10 ; and, 
as we saw, there is no difficulty about including those 
that are divisible by 10. It follows from this example that 
it is possible to construct compact series having N terms. 
In fact, we have already seen that there are N ratios, and 
ratios in order of magnitude form a compact series ; thus 
we have here another example. We shall resume this topic 
in the next chapter. 

Of the usual formal laws of addition, multiplication, and ex 
ponentiation, all are obeyed by transfinite cardinals, but only 
some are obeyed by transfinite ordinals, and those that are obeyed 
by them are obeyed by all relation-numbers. By the " usual 
formal laws " we mean the following : 

I. The commutative law : 

a+jS=j8+a and aX0=j8xa. 
II. The associative law : 

(a+jS)+y=a+(j3-hy) and (aXjS)Xy=aX (xy). 
III. The distributive law : 



When the commutative law does not hold, the above form 
of the distributive law must be distinguished from 



As we shall see immediately, one form may be true and the 
other false. 

IV. The laws of exponentiation : 



All these laws hold for cardinals, whether finite or infinite, 
and {QI finite ordinals. But when we come to infinite ordinals, 
or indeed to relation-numbers in general, some hold and some 
do not. The commutative law does not hold ; the associative 
law does hold ; the distributive law (adopting the convention 



Infinite Series and Ordinals 95 

we have adopted above as regards the order of the factors in a 
product) holds in the form 



but not in the form 

the exponential laws 

a? . 
still hold, but not the law 



which is obviously connected with the commutative law for 
multiplication. 

The definitions of multiplication and exponentiation that 
are assumed in the above propositions are somewhat complicated. 
The reader who wishes to know what they are and how the 
above laws are proved must consult the second volume of 
Principia Mathematics * 172-176. 

Ordinal transfinite arithmetic was developed by Cantor at 
an earlier stage than cardinal transfinite arithmetic, because it 
has various technical mathematical uses which led him to it. 
But from the point of view of the philosophy of mathematics 
it is less important and less fundamental than the theory of 
transfinite cardinals. Cardinals are essentially simpler than 
ordinals, and it is a curious historical accident that they first 
appeared as an abstraction from the latter, and only gradually 
came to be studied on their own account. This does not apply 
to Frege's work, in which cardinals, finite and transfinite, were 
treated in complete independence of ordinals ; but it was 
Cantor's work that made the world aware of the subject, while 
Frege's remained almost unknown, probably in the main on 
account of the difficulty of his symbolism. And mathematicians, 
like other people, have more difficulty in understanding and 
using notions which are comparatively " simple " in the logical 
sense than in manipulating more complex notions which are 



96 Introduction to Mathematical Philosophy 

more akin to their ordinary practice. For these reasons, it was 
only gradually that the true importance of cardinals in mathe 
matical philosophy was recognised. The importance of ordinals, 
though by no means small, is distinctly less than that of cardinals, 
and is very largely merged in that of the more general conception 
of relation-numbers. 



CHAPTER X 

LIMITS AND CONTINUITY 

THE conception of a " limit " is one of which the importance in 
mathematics has been found continually greater than had been 
thought. The whole of the differential and integral calculus, 
indeed practically everything in higher mathematics, depends 
upon limits. Formerly, it was supposed that infinitesimals were 
involved in the foundations of these subjects, but Weierstrass 
showed that this is an error : wherever infinitesimals were thought 
to occur, what really occurs is a set of finite quantities having 
zero for their lower limit. It used to be thought that " limit " 
was an essentially quantitative notion, namely, the notion of a 
quantity to which others approached nearer and nearer, so that 
among those others there would be some differing by less than any 
assigned quantity. But in fact the notion of " limit " is a purely 
ordinal notion, not involving quantity at all (except by accident 
when the series concerned happens to be quantitative). A given 
point on a line may be the limit of a set of points on the line, 
without its being necessary to bring in co-ordinates or measure 
ment or anything quantitative. The cardinal number N is the 
limit (in the order of magnitude) of the cardinal numbers I, 2, 
3, ...,..., although the numerical difference between N O 
and a finite cardinal is constant and infinite : from a quantitative 
point of view, finite numbers get no nearer to N as they grow 
larger. What makes N O the limit of the finite numbers is the 
fact that, in the series, it comes immediately after them, which 
is an ordinal fact, not a quantitative fact. 

97 7 



98 Introduction to Mathematical Philosophy 

There are various forms of the notion of " limit," of in 
creasing complexity. The simplest and most fundamental form, 
from which the rest are derived, has been already defined, but 
we will here repeat the definitions which lead to it, in a general 
form in which they do not demand that the relation concerned 
shall be serial. The definitions are as follows : 

The " minima " of a class a with respect to a relation P are 
those members of a and the field of P (if any) to which no member 
of a has the relation P. 

The " maxima " with respect to P are the minima with respect 
to the converse of P. 

The " sequents " of a class a with respect to a relation P are 
the minima of the " successors " of a, and the " successors " of 
a are those members of the field of P to which every member of 
the common part of a and the field of P has the relation P. 

The " precedents " with respect to P are the sequents with 
respect to the converse of P. 

The " upper limits " of a with respect to P are the sequents 
provided a has no maximum ; but if a has a maximum, it has no 
upper limits. 

The " lower limits " with respect to P are the upper limits with 
respect to the converse of P. 

Whenever P has connexity, a class can have at most one 
maximum, one minimum, one sequent, etc. Thus, in the cases 
we are concerned with in practice, we can speak of " the limit " 
(if any). 

When P is a serial relation, we can greatly simplify the above 
definition of a limit. We can, in that case, define first the 
" boundary " of a class a, i.e. its limits or maximum, and then 
proceed to distinguish the case where the boundary is the limit 
from the case where it is a maximum. For this purpose it is 
best to use the notion of " segment." 

We will speak of the " segment of P defined by a class a " as 
all those terms that have the relation P to some one or more of 
the members of a. This will be a segment in the sense defined 



Limits and Continuity 99 

in Chapter VII. ; indeed^ every segment in the sense there denned 
is the segment defined by some class a. If P is serial, the 
segment defined by a consists of all the terms that precede 
some term or other of a. If a has a maximum, the segment will 
be all the predecessors of the maximum. But if a has no 
maximum, every member of a precedes some other member of 
a, and the whole of a is therefore included in the segment defined 
by a. Take, for example, the class consisting of the fractions 

i i, I, if, 

i.e. of all fractions of the form I for different finite values 

2" 

of n. This series of fractions has no maximum, and it is clear 
that the segment which it defines (in the whole series of fractions 
in order of magnitude) is the class of all proper fractions. Or, 
again, consider the prime numbers, considered as a selection from 
the cardinals (finite and infinite) in order of magnitude. In this 
case the segment defined consists of all finite integers. 

Assuming that P is serial, the " boundary " of a class a will be 
the term x (if it exists) whose predecessors are the segment 
defined by a. 

A " maximum " of a is a boundary which is a member of a. 

An " upper limit" of a is a boundary which is not a member of cu 

If a class has no boundary, it has neither maximum nor limit. 
This is the case of an " irrational " Dedekind cut, or of what is 
called a " gap." 

Thus the " upper limit " of a set of terms a with respect to a 
series P is that term x (if it exists) which comes after all the a's, 
but is such that every earlier term comes before some of the a's. 

We may define all the " upper limiting-points " of a set of 
terms j3 as all those that are the upper limits of sets of terms 
chosen out of j8. We shall, of course, have to distinguish upper 
limiting-points from lower limiting-points. If we consider, for 
example, the series of ordinal numbers : 

I, 2, 3, ... CO, CO-f I, . . . 2CO, 2CO-H, ... 3^0, ... CD 2 , ... CO 3 , ..., 



ioo Introduction to Mathematical Philosophy 

the upper limiting-points of the field of this series are those that 
have no immediate predecessors, i.e. 

I, CO, 2CO, 3&>> &J 2 > to> 2 -\-O), . , . 2CO 2 , * . . CO 3 . . 

The upper limiting-points of the field of this new series will be 
I, co 2 , 2co 2 , ... co 3 , co 3 +co 2 . . . 

On the other hand, the series of ordinals and indeed every well- 
ordered series has no lower limiting-points, because there are 
no terms except the last that have no immediate successors. But 
if we consider such a series as the series of ratios, every member 
of this series is both an upper and a lower limiting-point for 
suitably chosen sets. If we consider the series of real numbers, 
and select out of it the rational real numbers, this set (the 
rationals) will have all the real numbers as upper and lower 
limiting-points. The limiting-points of a set are called its " first 
derivative," and the limiting-points of the first derivative are 
called the second derivative, and so on. 

With regard to limits, we may distinguish various grades of 
what may be called " continuity " in a series. The word " con 
tinuity " had been used for a long time, but had remained without 
any precise definition until the time of Dedekind and Cantor. 
Each of these two men gave a precise significance to the term, 
but Cantor's definition is narrower than Dedekind's : a series 
which has Cantorian continuity must have Dedekindian con 
tinuity, but the converse does not hold. 

The first definition that would naturally occur to a man seeking 
a precise meaning for the continuity of series would be to define 
it as consisting in what we have called " compactness," i.e. in the 
fact that between any two terms of the series there are others. 
But this would be an inadequate definition, because of the 
existence of " gaps " in series such as the series of ratios. We 
saw in Chapter VII. that there are innumerable ways in which 
the series of ratios can be divided into two parts, of which one 
wholly precedes the other, and of which the first has no last term, 



Limits and Continuity 101 

while the second has no first term. Such a state of affairs seems 
contrary to the vague feeling we have as to what should character 
ise " continuity," and, what is more, it shows that the series of 
ratios is not the sort of series that is needed for many mathematical 
purposes. Take geometry, for example : we wish to be able to 
say that when two straight lines cross each other they have a 
point in common, but if the series of points on a line were similar 
to the series of ratios, the two lines might cross in a " gap " and 
have no point in common. This is a crude example, but many 
others might be given to show that compactness is inadequate as 
a mathematical definition of continuity. 

It was the needs of geometry, as much as anything, that led 
to the definition of " Dedekindian " continuity. It will be re 
membered that we defined a series as Dedekindian when every 
sub-class of the field has a boundary. (It is sufficient to assume 
that there is always an upper boundary, or that there is always 
a lower boundary. If one of these is assumed, the other can be 
deduced.) That is to say, a series is Dedekindian when there 
are no gaps. The absence of gaps may arise either through 
terms having successors, or through the existence of limits in the 
absence of maxima. Thus a finite series or a well-ordered series 
is Dedekindian, and so is the series of real numbers. The former 
sort of Dedekindian series is excluded by assuming that our 
series is compact ; in that case our series must have a property 
which may, for many purposes, be fittingly called continuity. 
Thus we are led to the definition : 

A series has " Dedekindian continuity " when it is Dedekindian 
and compact. 

But this definition is still too wide for many purposes. Suppose, 
for example, that we desire to be able to assign such properties 
to geometrical space as shall make it certain that every point 
can be specified by means of co-ordinates which are real numbers : 
this is not insured by Dedekindian continuity alone. We want 
to be sure that every point which cannot be specified by rational 
co-ordinates can be specified as the limit of a progression of points 






IO2 Introduction to Mathematical Philosophy 

whose co-ordinates are rational, and this is a further property 
which our definition does not enable us to deduce. 

We are thus led to a closer investigation of series with respect 
to limits. This investigation was made by Cantor and formed 
the basis of his definition of continuity, although, in its simplest 
form, this definition somewhat conceals the considerations which 
have given rise to it. We shall, therefore, first travel through 
some of Cantor's conceptions in this subject before giving his 
definition of continuity. 

Cantor defines a series as " perfect " when all its points are 
limiting-points and all its limiting-points belong to it. But this 
definition does not express quite accurately what he means. 
There is no correction required so far as concerns the property 
that all its points are to be limiting-points ; this is a property 
belonging to compact series, and to no others if all points are to 
be upper limiting- or all lower limiting-points. But if it is only 
assumed that they are limiting-points one way, without specify 
ing which, there will be other series that will have the property 
in question for example, the series of decimals in which a decimal 
ending in a recurring 9 is distinguished from the corresponding 
terminating decimal and placed immediately before it. Such a 
series is very nearly compact, but has exceptional terms which 
are consecutive, and of which the first has no immediate prede 
cessor, while the second has no immediate successor. Apart from 
such series, the series in which every point is a limiting-point 
are compact series ; and this holds without qualification if it is 
specified that every point is to be an upper limiting-point (or 
that every point is to be a lower limiting-point). 

Although Cantor does not explicitly consider the matter, we 
must distinguish different kinds of limiting-points according to 
the nature of the smallest sub-series by which they can be defined. 
Cantor assumes that they are to be defined by progressions, or 
by regressions (which are the converses of progressions). When 
every member of our series is the limit of a progression or regres 
sion, Cantor calls our series " condensed in itself " (insichdicht). 



Limits and Continuity 103 

We come now to the second property by which perfection was 
to be defined, namely, the property which Cantor calls that of 
being " closed " (abgescblosseri). This, as we saw, was first defined 
as consisting in the fact that all the limiting-points of a series 
belong to it. But this only has any effective significance if our 
series is given as contained in some other larger series (as is the 
case, e.g., with a selection of real numbers), and limiting-points 
are taken in relation to the larger series. Otherwise, if a series 
is considered simply on its own account, it cannot fail to contain 
its limiting-points. What Cantor means is not exactly what 
he says ; indeed, on other occasions he says something rather 
different, which is what he means. What he really means is that 
every subordinate series which is of the sort that might be ex 
pected to have a limit does have a limit within the given series ; 
i.e. every subordinate series which has no maximum has a limit, 
i.e. every subordinate series has a boundary. But Cantor does 
not state this for every subordinate series, but only for progres 
sions and regressions. (It is not clear how far he recognises that 
this is a limitation.) Thus, finally, we find that the definition we 
want is the following : 

A series is said to be " closed " (abgescblossen) when every pro 
gression or regression contained in the series has a limit in the 
series. 

We then have the further definition : 

A series is " perfect " when it is condensed in itself and closed, 
i.e. when every term is the limit of a progression or regression, 
and every progression or regression contained in the series has a 
limit in the series. 

In seeking a definition of continuity, what Cantor has in mind 
is the search for a definition which shall apply to the series of 
real numbers and to any series similar to that, but to no others. 
For this purpose we have to add a further property. Among 
the real numbers some are rational, some are irrational ; although 
the number of irrationals is greater than the number of rationals, 
yet there are rationals between any two real numbers, however 



IO4 Introduction to Mathematical Philosophy 

little the two may differ. The number of rationals, as we saw, 
is >S . This gives a further property which suffices to characterise 
continuity completely, namely, the property of containing a class 
of N members in such a way that some of this class occur 
between any two terms of our series, however near together. 
This property, added to perfection, suffices to define a class of 
series which are all similar and are in fact a serial number. This 
class Cantor defines as that of continuous series. 

We may slightly simplify his definition. To begin with, 
we say : 

A " median class " of a series is a sub-class of the field such 
that members of it are to be found between any two terms of 
the series. 

Thus the rationals are a median class in the series of real 
numbers. It is obvious that there cannot be median classes 
except in compact series. 

We then find that Cantor's definition is equivalent to the 
following : 

A series is " continuous " when (i) it is Dedekindian, (2) it 
contains a median class having N terms. 

To avoid confusion, we shall speak of this kind as " Cantorian 
continuity." It will be seen that it implies Dedekindian con 
tinuity, but the converse is not the case. All series having 
Cantorian continuity are similar, but not all series having 
Dedekindian continuity. 

The notions of limit and continuity which we have been defining 
must not be confounded with the notions of the limit of a function 
for approaches to a given argument, or the continuity of a function 
in the neighbourhood of a given argument. These are different 
notions, very important, but derivative from the above and more 
complicated. The continuity of motion (if motion is continuous) 
is an instance of the continuity of a function ; on the other hand, 
the continuity of space and time (if they are continuous) is an 
instance of the continuity of series, or (to speak more cautiously) 
of a kind of continuity which can, by sufficient mathematical 






Limits and Continuity 105 

manipulation, be reduced to the continuity of series. In view 
of the fundamental importance of motion in applied mathe 
matics, as well as for other reasons, it will be well to deal 
briefly with the notions of limits and continuity as applied 
to functions ; but this subject will be best reserved for a 
separate chapter. 

The definitions of continuity which we have been considering, 
namely, those of Dedekind and Cantor, do not correspond very 
closely to the vague idea which is associated with the word in 
the mind of the man in the street or the philosopher. They 
conceive continuity rather as absence of separateness, the sort 
of general obliteration of distinctions which characterises a thick 
fog. A fog gives an impression of vastness without definite 
multiplicity or division. It is this sort of thing that a meta 
physician means by " continuity," declaring it, very truly, 
to be characteristic of his mental life and of that of children 
and animals. 

The general idea vaguely indicated by the word " continuity " 
when so employed, or by the word " flux," is one which is certainly 
quite different from that which we have been defining. Take, 
for example, the series of real numbers. Each is what it is, 
quite definitely and uncompromisingly ; it does not pass over 
by imperceptible degrees into another ; it is a hard, separate 
unit, and its distance from every other unit is finite, though 
it can be made less than any given finite amount assigned in 
advance. The question of the relation between the kind of 
continuity existing among the real numbers and the kind ex 
hibited, e.g. by what we see at a given time, is a difficult and 
intricate one. It is not to be maintained that the two kinds 
are simply identical, but it may, I think, be very well main 
tained that the mathematical conception which we have been 
considering in this chapter gives the abstract logical scheme to 
which it must be possible to bring empirical material by suitable 
manipulation, if that material is to be called " continuous " 
in any precisely definable sense. It would be quite impossible 



106 Introduction to Mathematical Philosophy 

to justify this thesis within the limits of the present volume. 
The reader who is interested may read an attempt to justify 
it as regards time in particular by the present author in the 
Monist for 1914-5, as well as in parts of Our Knowledge of the 
External World. With these indications, we must leave this 
problem, interesting as it is, in order to return to topics more 
closely connected with mathematics. 



CHAPTER XI 

LIMITS AND CONTINUITY OF FUNCTIONS 

IN this chapter we shall be concerned with the definition of the 
limit of a function (if any) as the argument approaches a given 
value, and also with the definition of what is meant by a " con 
tinuous function." Both of these ideas are somewhat technical, 
and would hardly demand treatment in a mere introduction 
to mathematical philosophy but for the fact that, especially 
through the so-called infinitesimal calculus, wrong views upon 
our present topics have become so firmly embedded in the minds 
of professional philosophers that a prolonged and considerable 
effort is required for their uprooting. It has been thought 
ever since the time of Leibniz that the differential and integral 
calculus required infinitesimal quantities. Mathematicians 
(especially Weierstrass) proved that this is an error ; but errors 
incorporated, e.g. in what Hegel has to say about mathematics, 
die hard, and philosophers have tended to ignore the work of 
such men as Weierstrass. 

Limits and continuity of functions, in works on ordinary 
mathematics, are defined in terms involving number. This is 
not essential, as Dr Whitehead has shown. 1 We will, however, 
begin with the definitions in the text-books, and proceed after 
wards to show how these definitions can be generalised so as to 
apply to series in general, and not only to such as are numerical 
or numerically measurable. 

Let us consider any ordinary mathematical function fx 9 where 

1 See Principia Mathematica, vol. ii. * 230-234. 
107 



io8 Introduction to Mathematical Philosophy 

x and/* are both real numbers, and fx is one-valued i.e. when 
x is given, there is only one value that/* can have. We call x 
the " argument," and/* the " value for the argument *." When 
a function is what we call " continuous," the rough idea for which 
we are seeking a precise definition is that small differences in * 
shall correspond to small differences in/*, and if we make the 
differences in * small enough, we can make the differences in 
/* fall below any assigned amount. We do not want, if a function 
is to be continuous, that there shall be sudden jumps, so that, 
for some value of *, any change, however small, will make a 
change in/* which exceeds some assigned finite amount. The 
ordinary simple functions of mathematics have this property : 
it belongs, for example, to * 2 , * 3 , . . . log *, sin *, and so on. 
But it is not at all difficult to define discontinuous functions. 
Take, as a non-mathematical example, " the place of birth of 
the youngest person living at time t" This is a function of t ; 
its value is constant from the time of one person's birth to the 
time of the next birth, and then the value changes suddenly 
from one birthplace to the other. An analogous mathematical 
example would be " the integer next below *," where x is a real 
number. This function remains constant from one integer to 
the next, and then gives a sudden jump. The actual fact is 
that, though continuous functions are more familiar, they are 
the exceptions : there are infinitely more discontinuous functions 
than continuous ones. 

Many functions are discontinuous for one or several values of 
the variable, but continuous for all other values. Take as an 
example sin I/*. The function sin 6 passes through all values 
from I to I every time that 6 passes from 77/2 to 77/2, or from 
77/2 to 377/2, or generally from (2w 1)77/2 to (2n-\- 1)77/2, where 
n is any integer. Now if we consider I/* when * is very small, 
we see that as * diminishes I/* grows faster and faster, so that 
it passes more and more quickly through the cycle of values from 
one multiple of 77/2 to another as * becomes smaller and smaller. 
Consequently sin i/x passes more and more quickly from I 



Limits and Continuity of Functions 109 

to i and back again, as x grows smaller. In fact, if we take 
any interval containing o, say the interval from e to -fe where 
e is some very small number, sin i/x will go through an infioite 
number of oscillations in this interval, and we cannot diminish 
the oscillations by making the interval smaller. Thus round 
about the argument o the function is discontinuous. It is easy 
to manufacture functions which are discontinuous in several 
places, or in N places, or everywhere. Examples will be found 
in any book on the theory of functions of a real variable. 

Proceeding now to seek a precise definition of what is meant 
by saying that a function is continuous for a given argument, 
when argument and value are both real numbers, let us first 
define a " neighbourhood " of a number x as all the numbers 
from x c to #-|-e, where e is some number which, in important 
cases, will be very small. It is clear that continuity at a given 
point has to do with what happens in any neighbourhood of that 
point, however small. 

What we desire is this : If a is the argument for which we wish 
our function to be continuous, let us first define a neighbourhood 
(a say) containing the value /# which the function has for the 
argument a ; we desire that, if we take a sufficiently small 
neighbourhood containing a, all values for arguments throughout 
this neighbourhood shall be contained in the neighbourhood a, 
no matter how small we may have made a. That is to say, if 
we decree that our function is not to differ from/rf by more than 
some very tiny amount, we can always find a stretch of real 
numbers, having a in the middle of it, such that throughout 
this stretch fx will not differ f rom fa by more than the pre 
scribed tiny amount. And this is to remain true whatever 
tiny amount we may select. Hence we are led to the following 
definition : 

The function f(x) is said to be " continuous " for the argu 
ment a if, for every positive number CT, different from o, but as 
small as we please, there exists a positive number e, different 
from o, such that, for all values of 8 which are numerically 



no Introduction to Mathematical Philosophy 

less 1 than e, the difference /(#+)/(#) is numerically less 
than a. 

In this definition, a first defines a neighbourhood of /(#), 
namely, the neighbourhood from/(tf) a to/(tf)-j-cr. The defini 
tion then proceeds to say that we can (by means of e) define a 
neighbourhood, namely, that from # e to a-\-e, such that, for 
all arguments within this neighbourhood, the value of the function 
lies within the neighbourhood horn f (a) a tof(a)+cr. If this 
can be done, however cr may be chosen, the function is " con 
tinuous " for the argument a. 

So far we have not defined the " limit " of a function for a 
given argument. If we had done so, we could have defined the 
continuity of a function differently : a function is continuous 
at a point where its value is the same as the limit of its value for 
approaches either from above or from below. But it is only 
the exceptionally " tame " function that has a definite limit as 
the argument approaches a given point. The general rule is 
that a function oscillates, and that, given any neighbourhood 
of a given argument, however small, a whole stretch of values 
will occur for arguments within this neighbourhood. As this 
is the general rule, let us consider it first. 

Let us consider what may happen as the argument approaches 
some value a from below. That is to say, we wish to consider 
what happens for arguments contained in the interval from 
a e to a, where e is some number which, in important cases, 
will be very small. 

The values of the function for arguments from a e to a (a 
excluded) will be a set of real numbers which will define a certain 
section of the set of real numbers, namely, the section consisting 
of those numbers that are not greater than all the values for 
arguments from a e to a. Given any number in this section, 
there are values at least as great as this number for arguments 
between a e and #, i.e. for arguments that fall very little short 

1 A number is said to be " numerically less " than e when it lies between 
e and +e. 



Limits and Continuity of Functions in 

of a (if c is very small). Let us take all possible e's and all 
possible corresponding sections. The common part of all these 
sections we will call the " ultimate section " as the argument 
approaches a. To say that a number z belongs to the ultimate 
section is to say that, however small we may make e, there are 
arguments between a e and a for which the value of the function 
is not less than z. 

We may apply exactly the same process to upper sections, 
i.e. to sections that go from some point up to the top, instead of 
from the bottom up to some point. Here we take those numbers 
that are not less than all the values for arguments from a e 
to a ; this defines an upper section which will vary as e varies. 
Taking the common part of all such sections for all possible e's, 
we obtain the " ultimate upper section." To say that a number 
z belongs to the ultimate upper section is to say that, however 
small we make e, there are arguments between a e and a for 
which the value of the function is not greater than z. 

If a term z belongs both to the ultimate section and to the 
ultimate upper section, we shall say that it belongs to the 
" ultimate oscillation." We may illustrate the matter by con 
sidering once more the function sin i/x as x approaches the 
value o. We shall assume, in order to fit in with the above 
definitions, that this value is approached from below. 

Let us begin with the " ultimate section." Between e 
and o, whatever e may be, the function will assume the value 
I for certain arguments, but will never assume any greater value. 
Hence the ultimate section consists of all real numbers, positive 
and negative, up to and including I ; i.e. it consists of all negative 
numbers together with o, together with the positive numbers 
up to and including I. 

Similarly the " ultimate upper section " consists of all positive 
numbers together with o, together with the negative numbers 
down to and including I. 

Thus the " ultimate oscillation " consists of all real numbers 
from I to i, both included. 



112 Introduction to Mathematical Philosophy 

We may say generally that the " ultimate oscillation " of 
a function as the argument approaches a from below consists 
of all those numbers x which are such that, however near we 
come to a y we shall still find values as great as x and values as 
small as x. 

The ultimate oscillation may contain no terms, or one term, 
or many terms. In the first two cases the function has a definite 
limit for approaches from below. If the ultimate oscillation 
has one term, this is fairly obvious. It is equally true if it has 
none ; for it is not difficult to prove that, if the ultimate oscilla 
tion is null, the boundary of the ultimate section is the same as 
that of the ultimate upper section, and may be defined as the 
limit of the function for approaches from below. But if the 
ultimate oscillation has many terms, there is no definite limit to 
the function for approaches from below. In this case we can 
take the lower and upper boundaries of the ultimate oscillation 
(i.e. the lower boundary of the ultimate upper section and the 
upper boundary of the ultimate section) as the lower and upper 
limits of its " ultimate " values for approaches from below. 
Similarly we obtain lower and upper limits of the " ultimate " 
values for approaches from above. Thus we have, in the general 
case,/owr limits to a function for approaches to a given argument. 
The limit for a given argument a only exists when all these four 
are equal, and is then their common value. If it is also the 
value for the argument a, the function is continuous for this 
argument. This may be taken as defining continuity : it is 
equivalent to our former definition. 

We can define the limit of a function for a given argument 
(if it exists) without passing through the ultimate oscillation 
and the four limits of the general case. The definition proceeds, 
in that case, just as the earlier definition of continuity proceeded. 
Let us define the limit for approaches from below. If there is to 
be a definite limit for approaches to a from below, it is necessary 
and sufficient that, given any small number cr, two values for 
arguments sufficiently near to a (but both less than a) will differ 



Limits and Continuity of Functions 113 

by less than cr ; i.e. if e is sufficiently small, and our arguments 
both lie between a e and a (a excluded), then the difference 
between the values for these arguments will be less than cr. 
This is to hold for any cr, however small ; in that case the 
function has a limit for approaches from below. Similarly 
we define the case when there is a limit for approaches from 
above. These two limits, even when both exist, need not be 
identical ; and if they are identical, they still need not be identical 
with the value for the argument a. It is only in this last case 
that we call the function continuous for the argument a. 

A function is called " continuous " (without qualification) 
when it is continuous for every argument. 

Another slightly different method of reaching the definition 
of continuity is the following : 

Let us say that a function " ultimately converges into a 
class a " if there is some real number such that, for this argument 
and all arguments greater than this, the value of the function 
is a member of the class a. Similarly we shall say that a function 
" converges into a as the argument approaches x from below " 
if there is some argument y less than x such that throughout 
the interval from y (included) to x (excluded) the function has 
values which are members of a. We may now say that a 
function is continuous for the argument a, for which it has the 
value fa, if it satisfies four conditions, namely : 

(1) Given any real number less than /#, the function con 
verges into the successors of this number as the argument 
approaches a from below ; 

(2) Given any real number greater than /, the function con 
verges into the predecessors of this number as the argument 
approaches a from below ; 

(3) and (4) Similar conditions for approaches to a from above. 
The advantages of this form of definition is that it analyses 

the conditions of continuity into four, derived from considering 
arguments and values respectively greater or less than the 
argument and value for which continuity is to be defined. 

8 



H4 Introduction to Mathematical Philosophy 

We may now generalise our definitions so as to apply to series 
which are not numerical or known to be numerically measurable. 
The case of motion is a convenient one to bear in mind. There 
is a story by H. G. Wells which will illustrate, from the case of 
motion, the difference between the limit of a function for a given 
argument and its value for the same argument. The hero of 
the story, who possessed, without his knowledge, the power of 
realising his wishes, was being attacked by a policeman, but on 

ejaculating "Go to " he found that the policeman disappeared. 

If f(t) was the policeman's position at time t, and t the moment 
of the ejaculation, the limit of the policeman's positions as t 
approached to t from below would be in contact with the hero, 
whereas the value for the argument t was . But such occur 
rences are supposed to be rare in the real world, and it is assumed, 
though without adequate evidence, that all motions are continu 
ous, i.e. that, given any body, if /(*) is its position at time t,f(t) 
is a continuous function of t. It is the meaning of " continuity " 
involved in such statements which we now wish to define as 
simply as possible. 

The definitions given for the case of functions where argument 
and value are real numbers can readily be adapted for more 
general use. 

Let P and Q be two relations, which it is well to imagine 
serial, though it is not necessary to our definitions that they 
should be so. Let R be a one-many relation whose domain 
is contained in the field of P, while its converse domain is con 
tained in the field of Q. Then R is (in a generalised sense) a 
function, whose arguments belong to the field of Q, while its 
values belong to the field of P. Suppose, for example, that we 
are dealing with a particle moving on a line : let Q be the time- 
series, P the series of points on our line from left to right, R the 
relation of the position of our particle on the line at time a to 
the time a, so that " the R of a " is its position at time a. This 
illustration may be borne in mind throughout our definitions. 

We shall say that the function R is continuous for the argument 



Limits and Continuity of Functions 115 

a if, given any interval a on the P-series containing the value 
of the function for the argument #, there is an interval on the 
Q-series containing a not as an end-point and such that, through 
out this interval, the function has values which are members 
of a. (We mean by an " interval " all the terms between any 
two ; i.e. if x and y are two members of the field of P, and x has 
the relation P to y, we shall mean by the " P-interval x to y " 
all terms z such that x has the relation P to x and z has the rela 
tion P to y together, when so stated, with x or y themselves.) 

We can easily define the " ultimate section " and the " ulti 
mate oscillation." To define the " ultimate section " for 
approaches to the argument a from below, take any argument 
y which precedes a (i.e. has the relation Q to a), take the values 
of the function for all arguments up to and including y, and 
form the section of P defined by these values, i.e. those members 
of the P-series which are earlier than or identical with some of 
these values. Form all such sections for all y's that precede a, 
and take their common part ; this will be the ultimate section. 
The ultimate upper section and the ultimate oscillation are then 
defined exactly as in the previous case. 

The adaptation of the definition of convergence and the 
resulting alternative definition of continuity offers no difficulty 
of any kind. 

We say that a function R is " ultimately Q-convergent into 
a " if there is a member y of the converse domain of R and the 
field of Q such that the value of the function for the argument 
y and for any argument to which y has the relation Q is a member 
of a. We say that R " Q-converges into a as the argument 
approaches a given argument a " if there is a term y having 
the relation Q to a and belonging to the converse domain of R 
and such that the value of the function for any argument in the 
Q-interval from y (inclusive) to a (exclusive) belongs to a. 

Of the four conditions that a function must fulfil in order 
to be continuous for the argument a, the first is, putting b for 
the value for the argument a : 



1 1 6 Introduction to Mathematical Philosophy 

Given any term having the relation P to b, R Q-converges 
into the successors of b (with respect to P) as the argument 
approaches a from below. 

The second condition is obtained by replacing P by its 
converse ; the third and fourth are obtained from the first and 
second by replacing Q by its converse. 

There is thus nothing, in the notions of the limit of a function 
or the continuity of a function, that essentially involves number. 
Both can be defined generally, and many propositions about 
them can be proved for any two series (one being the argument- 
series and the other the value-series). It will be seen that the 
definitions do not involve infinitesimals. They involve infinite 
classes of intervals, growing smaller without any limit short of 
zero, but they do not involve any intervals that are not finite. 
This is analogous to the fact that if a line an inch long be halved, 
then halved again, and so on indefinitely, we never reach infini 
tesimals in this 'way : after n bisections, the length of our bit is 

of an inch ; and this is finite whatever finite number n may 

2 n 

be. The process of successive bisection does not lead to 
divisions whose ordinal number is infinite, since it is essentially 
a one-by-one process. Thus infinitesimals are not to be reached 
in this way. Confusions on such topics have had much to do 
with the difficulties which have been found in the discussion of 
infinity and continuity. 



CHAPTER XII 

SELECTIONS AND THE MULTIPLICATIVE AXIOM 

IN this chapter we have to consider an axiom which can be 
enunciated, but not proved, in terms of logic, and which is con 
venient, though not indispensable, in certain portions of mathe 
matics. It is convenient, in the sense that many interesting 
propositions, which it seems natural to suppose true, cannot 
be proved without its help ; but it is not indispensable, because 
even without those propositions the subjects in which they 
occur still exist, though in a somewhat mutilated form. 

Before enunciating the multiplicative axiom, we must first 
explain the theory of selections, and the definition of multi 
plication when the number of factors may be infinite. 

In defining the arithmetical operations, the only correct pro 
cedure is to construct an actual class (or relation, in the case 
of relation-numbers) having the required number of terms. 
This sometimes demands a certain amount of ingenuity, but 
it is essential in order to prove the existence of the number 
defined. Take, as the simplest example, the case of addition. 
Suppose we are given a cardinal number ^, and a class a which 
has fji terms. How shall we define ju.+/z ? For this purpose 
we must have two classes having //, terms, and they must not 
overlap. We can construct such classes from a in various ways, 
of which the following is perhaps the simplest : Form first all 
the ordered couples whose first term is a class consisting of a 
single member of a, and whose second term is the null-class ; 
then, secondly, form all the ordered couples whose first term is 

117 



n8 Introduction to Mathematical Philosophy 

the null-class and whose second term is a class consisting of a 
single member of a. These two classes of couples have no 
member in common, and the logical sum of the two classes will 
have /z-f/*- terms. Exactly analogously we can define p,-\-v, 
given that /z, is the number of some class a and v is the number 
of some class j3. 

Such definitions, as a rule, are merely a question of a suitable 
technical device. But in the case of multiplication, where the 
number of factors may be infinite, important problems arise out 
of the definition. 

Multiplication when the number of factors is finite offers no 
difficulty. Given two classes a and j8, of which the first has 
ju, terms and the second v terms, we can define fix v as the number 
of ordered couples that can be formed by choosing the first term 
out of a and the second out of ]3. It will be seen that this de 
finition does not require that a and j3 should not overlap ; it 
even remains adequate when a and jS are identical. For example, 
let a be the class whose members are x l9 # 2 , # 3 . Then the class 
which is used to define the product /x X p, is the class of couples : 

(*i, *i), (*i, *a)> (*i> *B) 5 (**> *i) (*2> * 2 )> (*2> *a) > (*3> *i), 
(# 3 > *s) (*s> *a) 

This definition remains applicable when \i or v or both are 
infinite, and it can be extended step by step to three or four or 
any finite number of factors. No difficulty arises as regards 
this definition, except that it cannot be extended to an infinite 
number of factors. 

The problem of multiplication when the number of factors 
may be infinite arises in this way : Suppose we have a class K 
consisting of classes ; suppose the number of terms in each of 
these classes is given. How shall we define the product of all 
these numbers ? If we can frame our definition generally, it 
will be applicable whether K is finite or infinite. It is to be 
observed that the problem is to be able to deal with the case 
when K is infinite, not with the case when its members are. If 



Selections and the Multiplicative Axiom 119 

K is not infinite, the method defined above is just as applicable 
when its members are infinite as when they are finite. It is 
the case when K is infinite, even though its members may be 
finite, that we have to find a way of dealing with. 

The following method of defining multiplication generally is 
due to Dr Whitehead. It is explained and treated at length in 
Principia Mathematics*, vol. i. * 80 ff., and vol. ii. * 114. 

Let us suppose to begin with that K is a class of classes no two 
of which overlap say the constituencies in a country where 
there is no plural voting, each constituency being considered 
as a class of voters. Let us now set to work to choose one term 
out of each class to be its representative, as constituencies do 
when they elect members of Parliament, assuming that by law 
each constituency has to elect a man who is a voter in that 
constituency. We thus arrive at a class of representatives, who 
make up our Parliament, one being selected out of each con 
stituency. How many different possible ways of choosing a 
Parliament are there ? Each constituency can select any one 
of its voters, and therefore if there are p voters in a constituency, 
it can make JLC choices. The choices of the different constituencies 
are independent ; thus it is obvious that, when the total number 
of constituencies is finite, the number of possible Parliaments 
is obtained by multiplying together the numbers of voters in the 
various constituencies. When we do not know whether the 
number of constituencies is finite or infinite, we may take the 
number of possible Parliaments as defining the product of the 
numbers of the separate constituencies. This is the method 
by which infinite products are defined. We must now drop our 
illustration, and proceed to exact statements. 

Let K be a class of classes, and let us assume to begin with that 
no two members of ic overlap, i.e. that if a and j3 are two different 
members of K, then no member of the one is a member of the 
other. We shall call a class a " selection " from K when it con 
sists of just one term from each member of K ; i.e. p, is a " selec 
tion " from K if every member of JJL belongs to some member 



I2O Introduction to Mathematical Philosophy 

of K, and if a be any member of K, /i and a have exactly one term 
in common. The class of all " selections " from K we shall call 
the " multiplicative class " of K. The number of terms in the 
multiplicative class of /c, i.e. the number of possible selections 
from K, is defined as the product of the numbers of the members 
of K. This definition is equally applicable whether K is finite 
or infinite. 

Before we can be wholly satisfied with these definitions, we 
must remove the restriction that no two members of K are to 
overlap. For this purpose, instead of defining first a class 
called a " selection," we will define first a relation which we will 
call a " selector." A relation R will be called a " selector " 
from K if, from every member of /c, it picks out one term as the 
representative of that member, i.e. if, given any member a of /c, 
there is just one term x which is a member of a and has the 
relation R to a ; and this is to be all that R does. The formal 
definition is : 

A " selector " from a class of classes K is a one-many relation, 
having K for its converse domain, and such that, if x has the 
relation to a, then x is a member of a. 

If R is a selector from /c, and a is a member of K, and x is the 
term which has the relation R to a, we call x the " representative " 
of a in respect of the relation R. 

A " selection " from K will now be defined as the domain of a 
selector ; and the multiplicative class, as before, will be the class 
of selections. 

But when the members of K overlap, there may be more selectors 
than selections, since a term x which belongs to two classes a 
and j8 may be selected once to represent a and once to represent j3, 
giving rise to different selectors in the two cases, but to the same 
selection. For purposes of defining multiplication, it is the 
selectors we require rather than the selections. Thus we define : 

" The product of the numbers of the members of a class of 
classes K " is the number of selectors from /c. 

We can define exponentiation by an adaptation of the above 



Selections and the Multiplicative Axiom 121 

plan. We might, of course, define /A" as the number of selectors 
from v classes, each of which has ju, terms. But there are 
objections to this definition, derived from the fact that the 
multiplicative axiom (of which we shall speak shortly) is unneces 
sarily involved if it is adopted. We adopt instead the following 
construction : 

Let a be a class having (JL terms, and j3 a class having v terms. 

Let y be a member of j3, and form the class of all ordered 
couples that have y for their second term and a member of a for 
their first term. There will be p such couples for a given y, since 
any member of a may be chosen for the first term, and a has /z 
members. If we now form all the classes of this sort that result 
from varying y, we obtain altogether v classes, since y may be 
any member of j8, and j8 has v members. These v classes are each 
of them a class of couples, namely, all the couples that can be 
formed of a variable member of a and a fixed member of j8. We 
define \L V as the number of selectors from the class consisting of 
these v classes. Or we may equally well define ju," as the number of 
selections, for, since our classes of couples are mutually exclusive, 
the number of selectors is the same as the number of selections. 
A selection from our class of classes will be a set of ordered couples, 
of which there will be exactly one having any given member of jS 
for its second term, and the first term may be any member of a. 
Thus ju," is defined by the selectors from a certain set of v classes 
each having p, terms, but the set is one having a certain structure 
and a more manageable composition than is the case in general. 
The relevance of this to the multiplicative axiom will appear 
shortly. 

What applies to exponentiation applies also to the product of 
two cardinals. We might define "jz.Xi'" as the sum of the 
numbers of v classes each having JJL terms, but we prefer to define 
it as the number of ordered couples to be formed consisting of a 
member of a followed by a member of j5, where a has ^ terms 
and j8 has v terms. This definition, also, is designed to evade the 
necessity of assuming the multiplicative axiom. 



122 Introduction to Mathematical Philosophy 

With our definitions, we can prove the usual formal laws of 
multiplication and exponentiation. But there is one thing we 
cannot prove : we cannot prove that a product is only zero when 
one of its factors is zero. We can prove this when the number 
of factors is finite, but not when it is infinite. In other words, 
we cannot prove that, given a class of classes none of which is 
null, there must be selectors from them ; or that, given a class 
of mutually exclusive classes, there must be at least one class 
consisting of one term out of each of the given classes. These 
things cannot be proved ; and although, at first sight, they seem 
obviously true, yet reflection brings gradually increasing doubt, 
until at last we become content to register the assumption and 
its consequences, as we register the axiom of parallels, without 
assuming that we can know whether it is true or false. The 
assumption, loosely worded, is that selectors and selections exist 
when we should expect them. There are many equivalent ways 
of stating it precisely. We may begin with the following : 

" Given any class of mutually exclusive classes, of which none 
is null, there is at least one class which has exactly one term in 
common with each of the given classes." 

This proposition we will call the " multiplicative axiom." 1 
We will first give various equivalent forms of the proposition, 
and then consider certain ways in which its truth or falsehood 
is of interest to mathematics. 

The multiplicative axiom is equivalent to the proposition that 
a product is only zero when at least one of its factors is zero ; 
i.e. that, if any number of cardinal numbers be multiplied together, 
the result cannot be o unless one of the numbers concerned is o. 

The multiplicative axiom is equivalent to the proposition that, 
if R be any relation, and K any class contained in the converse 
domain of R, then there is at least one one-many relation implying 
R and having K for its converse domain. 

The multiplicative axiom is equivalent to the assumption that 
if a be any class, and K all the sub-classes of a with the exception 

1 See Principia Mathematica, vol. i. * 88. Also vol. iii. * 257-258. 



Selections and the Multiplicative Axiom 123 

of the null-class, then there is at least one selector from K. This 
is the form in which the axiom was first brought to the notice of 
the learned world by Zermelo, in his " Beweis, dass jede Menge 
wohlgeordnet werden kann." 1 Zermelo regards the axiom as an 
unquestionable truth. It must be confessed that, until he made 
it explicit, mathematicians had used it without a qualm ; but it 
would seem that they had done so unconsciously. And the credit 
due to Zermelo for having made it explicit is entirely independent 
of the question whether it is true or false. 

The multiplicative axiom has been shown by Zermelo, in the 
above-mentioned proof, to be equivalent to the proposition that 
every class can be well-ordered, i.e. can be arranged in a series in 
which every sub-class has a first term (except, of course, the null- 
class). The full proof of this proposition is difficult, but it is not 
difficult to see the general principle upon which it proceeds. It 
uses the form which we call " Zermelo's axiom," i.e. it assumes 
that, given any class a, there is at least one one-many relation R 
whose converse domain consists of all existent sub-classes of a 
and which is such that, if x has the relation R to f , then x is a 
member of f . Such a relation picks out a " representative " 
from each sub-class ; of course, it will often happen that two 
sub-classes have the same representative. What Zermelo does, 
in effect, is to count off the members of a, one by one, by means 
of R and transfinite induction. We put first the representative 
of a; call it x r Then take the representative of the class consisting 
of all of a except x 1 ; call it x 2 . It must be different from x l9 
because every representative is a member of its class, and x is 
shut out from this class. Proceed similarly to take away X 2 , and 
let # 3 be the representative of what is left. In this way we first 
obtain a progression x^ X 2 , . . . x m . . ., assuming that a is not 
finite. We then take away the whole progression ; let # w be the 
representative of what is left of a. In this way we can go on 
until nothing is left. The successive representatives will form a 

1 Mathematische Annalen, vol. lix. pp. 514-6. In this form we shall 
speak of it as Zermelo's axiom. 






124 Introduction to Mathematical Philosophy 

well-ordered series containing all the members of a. (The above 
is, of course, only a hint of the general lines of the proof.) This 
proposition is called " Zermelo's theorem." 

The multiplicative axiom is also equivalent to the assumption 
that of any two cardinals which are not equal, one must be the 
greater. If the axiom is false, there will be cardinals p and v 
such that ju- is neither less than, equal to, nor greater than v. We 
have seen that Nj and 2 No possibly form an instance of such a pair. 

Many other forms of the axiom might be given, but the above 
are the most important of the forms known at present. As to 
the truth or falsehood of the axiom in any of its forms, nothing 
is known at present. 

The propositions that depend upon the axiom, without being 
known to be equivalent to it, are numerous and important. Take 
first the connection of addition and multiplication. We naturally 
think that the sum of v mutually exclusive classes, each having 
jit terms, must have p,Xv terms. When v is finite, this can be 
proved. But when v is infinite, it cannot be proved without the 
multiplicative axiom, except where, owing to some special cir 
cumstance, the existence of certain selectors can be proved. The 
way the multiplicative axiom enters in is as follows : Suppose 
we have two sets of v mutually exclusive classes, each having ^ 
terms, and we wish to prove that the sum of one set has as many 
terms as the sum of the other. In order to prove this, we must 
establish a one-one relation. Now, since there are in each case 
v classes, there is some one-one relation between the two sets of 
classes ; but what we want is a one-one relation between their 
terms. Let us consider some one-one relation S between the 
classes. Then if K and A are the two sets of classes, and a is some 
member of K, there will be a member j3 of A which will be the 
correlate of a with respect to S. Now a and j3 each have /x terms, 
and are therefore similar. There are, accordingly, one-one cor 
relations of a and jS. The trouble is that there are so many. In 
order to obtain a one-one correlation of the sum of K with the 
sum of A, we have to pick out one selection from a set of classes 



Selections and the Multiplicative Axiom 125 

of correlators, one class of the set being all the one-one correlators 
of a with j3. If K and A are infinite, we cannot in general know 
that such a selection exists, unless we can know that the multi 
plicative axiom is true. Hence we cannot establish the usual 
kind of connection between addition and multiplication. 

This fact has various curious consequences. To begin with, 
we know that N 2 =N x = N o- ^ * s commonly inferred from 
this that the sum of N classes each having N members must 
itself have N members, but this inference is fallacious, since we 
do not know that the number of terms in such a sum is N X N 
nor consequently that it is N . This has a bearing upon the theory 
of transfinite ordinals. It is easy to prove that an ordinal which 
has NO predecessors must be one of what Cantor calls the " second 
class," i.e. such that a series having this ordinal number will have 
N terms in its field. It is also easy to see that, if we take any 
progression of ordinals of the second class, the predecessors of 
their limit form at most the sum of N classes each having N 
terms. It is inferred thence fallaciously, unless the multi 
plicative axiom is true that the predecessors of the limit are N 
in number, and therefore that the limit is a number of the " second 
class." That is to say, it is supposed to be proved that any pro 
gression of ordinals of the second class has a limit which is again 
an ordinal of the second class. This proposition, with the corol 
lary that a} (the smallest ordinal of the third class) is not the 
limit of any progression, is involved in most of the recognised 
theory of ordinals of the second class. In view of the way in 
which the multiplicative axiom is involved, the proposition and 
its corollary cannot be regarded as proved. They may be true, 
or they may not. All that can be said at present is that we do 
not know. Thus the greater part of the theory of ordinals of 
the second class must be regarded as unproved. 

Another illustration may help to make the point clearer. We 
know that 2XN =N . Hence we might suppose that the sum 
of N pairs must have N terms. But this, though we can prove 
that it is sometimes the case, cannot be proved to happen always 



126 Introduction to Mathematical Philosophy 

unless we assume the multiplicative axiom. This is illustrated 
by the millionaire who bought a pair of socks whenever he bought 
a pair of boots, and never at any other time, and who had such 
a passion for buying both that at last he had N pairs of boots 
and N O pairs of socks. The problem is : How many boots had 
he, and how many socks ? One would naturally suppose that 
he had twice as many boots and twice as many socks as he had 
pairs of each, and that therefore he had N of each, since that 
number is not increased by doubling. But this is an instance of 
the difficulty, already noted, of connecting the sum of v classes 
each having p terms with fjiXv. Sometimes this can be done, 
sometimes it cannot. In our case it can be done with the boots, 
but not with the socks, except by some very artificial device. 
The reason for the difference is this : Among boots we can dis 
tinguish right and left, and therefore we can make a selection of 
one out of each pair, namely, we can choose all the right boots or 
all the left boots ; but with socks no such principle of selection 
suggests itself, and we cannot be sure, unless we assume the 
multiplicative axiom, that there is any class consisting of one 
sock out of each pair. Hence the problem. 

We may put the matter in another way. To prove that a 
class has N terms, it is necessary and sufficient to find some way 
of arranging its terms in a progression. There is no difficulty in 
doing this with the boots. The pairs are given as forming an N O , 
and therefore as the field of a progression. Within each pair, 
take the left boot first and the right second, keeping the order 
of the pair unchanged ; in this way we obtain a progression of 
all the boots. But with the socks we shall have to choose arbi 
trarily, with each pair, which to put first ; and an infinite number 
of arbitrary choices is an impossibility. Unless we can find a 
rule for selecting, i.e. a relation which is a selector, we do not know 
that a selection is even theoretically possible. Of course, in the 
case of objects in space, like socks, we always can find some 
principle of selection. For example, take the centres of mass 
of the socks : there will be points p in space such that, with any 



Selections and the Multiplicative Axiom 127 

pair, the centres of mass of the two socks are not both at exactly 
the same distance from p ; thus we can choose, from each pair, 
that sock which has its centre of mass nearer to p. But there is 
no theoretical reason why a method of selection such as this 
should always be possible, and the case of the socks, with a little 
goodwill on the part of the reader, may serve to show how a 
selection might be impossible. 

It is to be observed that, if it were impossible to select one out 
of each pair of socks, it would follow that the socks could not be 
arranged in a progression, and therefore that there were not N 
of them. This case illustrates that, if fj, is an infinite number, 
one set of p pairs may not contain the same number of terms as 
another set of p, pairs ; for, given N pairs of boots, there are 
certainly N boots, but we cannot be sure of this in the case of 
the socks unless we assume the multiplicative axiom or fall back 
upon some fortuitous geometrical method of selection such as 
the above. 

Another important problem involving the multiplicative 
axiom is the relation of reflexiveness to non-inductiveness. It 
will be remembered that in Chapter VIII. we pointed out that a 
reflexive number must be non-inductive, but that the converse 
(so far as is known at present) can only be proved if we assume 
the multiplicative axiom. The way in which this comes about 
is as follows : 

It is easy to prove that a reflexive class is one which contains 
sub-classes having N terms. (The class may, of course, itself 
have N terms.) Thus we have to prove, if we can, that, given 
any non-inductive class, it is possible to choose a progression 
out of its terms. Now there is no difficulty in showing that 
a non-inductive class must contain more terms than any inductive 
class, or, what comes to the same thing, that if a is a non-induc 
tive class and v is any inductive number, there are sub-classes 
of a that have v terms. Thus we can form sets of finite sub 
classes of a : First one class having no terms, then classes having 
I term (as many as there are members of a), then classes having 



128 Introduction to Mathematical Philosophy 

2 terms, and so on. We thus get a progression of sets of sub 
classes, each set consisting of all those that have a certain given 
finite number of terms. So far we have not used the multiplica 
tive axiom, but we have only proved that the number of collec 
tions of sub-classes of a is a reflexive number, i.e. that, if p is 
the number of members of a, so that 2* is the number of sub 
classes of a and 2 2 ^ is the number of collections of sub-classes, 
then, provided JLC is not inductive, 2 2f * must be reflexive. But 
this is a long way from what we set out to prove. 

In order to advance beyond this point, we must employ the 
multiplicative axiom. From each set of sub-classes let us 
choose out one, omitting the sub-class consisting of the null- 
class alone. That is to say, we select one sub-class containing 
one term, 04, say ; one containing two terms, a 2 , say ; one con 
taining three, a 3 , say ; and so on. (We can do this if the multipli 
cative axiom is assumed ; otherwise, we do not know whether 
we can always do it or not.) We have now a progression 
a i a 2> a s> f sub-classes of a, instead of a progression of 
collections of sub-classes ; thus we are one step nearer to our 
goal. We now know that, assuming the multiplicative axiom, 
if ju, is a non-inductive number, 2* must be a reflexive number. 

The next step is to notice that, although we cannot be sure 
that new members of a come in at any one specified stage in the 
progression a x , a 2 , a 3 , . . . we can be sure that new members 
keep on coming in from time to time. Let us illustrate. 
The class c^, which consists of one term, is a new beginning; 
let the one term be x v The class a 2 , consisting of two terms, 
may or may not contain x 1 ; if it does, it introduces one new 
term ; and if it does not, it must introduce two new terms, say 
# 2 , x z . In this case it is possible that a 3 consists of x l9 # 2 , x st 
and so introduces no new terms, but in that case a 4 must introduce 
a new term. The first v classes a ly a 2 , a 3 , . . . a v contain, at 
the very most, 1+2+3+ +" terms, i.e. j/(v+i)/2 terms; 
thus it would be possible, if there were no repetitions in the 
first v classes, to go on with only repetitions from the 



Selections and the Multiplicative Axiom 129 

class to the v(v+i)/2 th class. But by that time the old terms 
would no longer be sufficiently numerous to form a next class 
with the right number of members, i.e. v(i/-|-i)/2-[-i, therefore 
new terms must come in at this point if not sooner. It 
follows that, if we omit from our progression 04, a 2 , a 3 , , . , all 
those classes that are composed entirely of members that have 
occurred in previous classes, we shall still have a progression. 
Let our new progression be called fi l9 j8 2 , j8 3 . . . . (We shall 
have a>i=pi and a 2 =j3 2 , because a x and a 2 must introduce new 
terms. We may or may not have a 3 =j8 3 , but, speaking generally, 
p^ will be a,, where v is some number greater than p ; i.e. the 
j8's are some of the a's.) Now these jS's are such that any one 
of them, say jS^, contains members which have not occurred in 
any of the previous j8's. Let y^ be the part of /^ which consists 
of new members. Thus we get a new progression y l9 y 2 , y 3 , . . . 
(Again y 5 will be identical with j8j and with c^ ; if a 2 does not 
contain the one member of a l9 we shall have y 2 =j3 2 =a 2 , but if 
a 2 does contain this one member, y 2 will consist of the other 
member of a 2 .) This new progression of y's consists of mutually 
exclusive classes. Hence a selection from them will be a pro 
gression ; i.e. if x l is the member of y l9 x 2 is a member of y a , x s 
is a member of y s , and so on ; then x l9 # 2 , # 3 , . . . is a progression, 
and is a sub-class of a. Assuming the multiplicative axiom, 
such a selection can be made. Thus by twice using this axiom 
we can prove that, if the axiom is true, every non-inductive 
cardinal must be reflexive. This could also be deduced from 
Zermelo's theorem, that, if the axiom is true, every class can be 
well ordered ; for a well-ordered series must have either a finite 
or a reflexive number of terms in its field. 

There is one advantage in the above direct argument, as 
against deduction from Zermelo's theorem, that the above 
argument does not demand the universal truth of the multi 
plicative axiom, but only its truth as applied to a set of N classes. 
It may happen that the axiom holds for N classes, though not 
for larger numbers of classes. For this reason it is better, when 

9 



130 Introduction to Mathematical Philosophy 

it is possible, to content ourselves with the more restricted 
assumption. The assumption made in the above direct argu 
ment is that a product of N factors is never zero unless one of 
the factors is zero. We may state this assumption in the form : 
" N is a multipliable number," where a number v is defined as 
" multipliable " when a product of v factors is never zero unless 
one of the factors is zero. We can prove that a finite number is 
always multipliable, but we cannot prove that any infinite number 
is so. The multiplicative axiom is equivalent to the assumption 
that all cardinal numbers are multipliable. But in order to 
identify the reflexive with the non-inductive, or to deal with the 
problem of the boots and socks, or to show that any progression 
of numbers of the second class is of the second class, we only 
need the very much smaller assumption that N is multipliable. 

It is not improbable that there is much to be discovered 
in regard to the topics discussed in the present chapter. Cases 
may be found where propositions which seem to involve the 
multiplicative axiom can be proved without it. It is conceivable 
that the multiplicative axiom in its general form may be shown 
to be false. From this point of view, Zermelo's theorem offers 
the best hope : the continuum or some still more dense series 
might be proved to be incapable of having its terms well ordered, 
which would prove the multiplicative axiom false, in virtue of 
Zermelo's theorem. But so far, no method of obtaining such 
results has been discovered, and the subject remains wrapped in 
obscurity. 



CHAPTER XIII 

THE AXIOM OF INFINITY AND LOGICAL TYPES 

THE axiom of infinity is an assumption which may be enunciated 
as follows : 

" If n be any inductive cardinal number, there is at least one 
class of individuals having n terms." 

If this is true, it follows, of course, that there are many classes 
of individuals having n terms, and that the total number of 
individuals in the world is not an inductive number. For, by 
the axiom, there is at least one class having n-f- 1 terms, from which 
it follows that there are many classes of n terms and that n is 
not the number of individuals in the world. Since n is any 
inductive number, it follows that the number of individuals 
in the world must (if our axiom be true) exceed any inductive 
number. In view of what we found in the preceding chapter, 
about the possibility of cardinals which are neither inductive 
nor reflexive, we cannot infer from our axiom that there are at 
least N individuals, unless we assume the multiplicative axiom. 
But we do know that there are at least N classes of classes, 
since the inductive cardinals are classes of classes, and form a 
progression if our axiom is true. The way in which the need 
for this axiom arises may be explained as follows : One of 
Peano's assumptions is that no two inductive cardinals have the 
same successor, i.e. that we shall not have ra-f !=-{- 1 unless 
m=n, if m and n are inductive cardinals. In Chapter VIII. we 
had occasion to use what is virtually the same as the above 
assumption of Peano's, namely, that, if n is an inductive cardinal, 



132 Introduction to Mathematical Philosophy 

n is not equal to w-f-i. It might be thought that this could be 
proved. We can prove that, if a is an inductive class, and n 
is the number of members of a, then n is not equal to +i. 
This proposition is easily proved by induction, and might be 
thought to imply the other. But in fact it does not, since there 
might be no such class as a. What it does imply is this : If 
n is an inductive cardinal such that there is at least one class 
having n members, then n is not equal to n-\-i. The axiom of 
infinity assures us (whether truly or falsely) that there are classes 
having n members, and thus enables us to assert that n is not 
equal to +i. But without this axiom we should be left with 
the possibility that n and n-\-i might both be the null-class. 

Let us illustrate this possibility by an example : Suppose 
there were exactly nine individuals in the world. (As to what 
is meant by the word " individual," I must ask the reader to 
be patient.) Then the inductive cardinals from o up to 9 would 
be such as we expect, but 10 (defined as 9+ 1 ) would be the 
null-class. It will be remembered that n-\-i may be defined as 
follows : tt-j- I is the collection of all those classes which have a 
term x such that, when x is taken away, there remains a class 
of n terms. Now applying this definition, we see that, in the 
case supposed, 9+1 is a class consisting of no classes, i.e. it is 
the null-class. The same will be true of 9+2, or generally of 
9+w, unless n is zero. Thus 10 and all subsequent inductive 
cardinals will all be identical, since they will all be the null-class. 
In such a case the inductive cardinals will not form a progression, 
nor will it be true that no two have the same successor, for 9 
and 10 will both be succeeded by the null-class (10 being itself 
the null-class). It is in order to prevent such arithmetical 
catastrophes that we require the axiom of infinity. 

As a matter of fact, so long as we are content with the arith 
metic of finite integers, and do not introduce either infinite 
integers or infinite classes or series of finite integers or ratios, 
it is possible to obtain all desired results without the axiom of 
infinity. That is to say, we can deal with the addition, multi- 



The Axiom of Infinity and Logical Types 133 

plication, and exponentiation of finite integers and of ratios, 
but we cannot deal with infinite integers or with irrationals. 
Thus the theory of the transfinite and the theory of real numbers 
fails us. How these various results come about must now be 
explained. 

Assuming that the number of individuals in the world is n, 
the number of classes of individuals will be 2 n . This is in virtue 
of the general proposition mentioned in Chapter VIII. that the 
number of classes contained in a class which has n members 
is 2 n . Now 2 n is always greater than n. Hence the number 
of classes in the world is greater than the number of individuals. 
If, now, we suppose the number of individuals to be 9, as we did 
just now, the number of classes will be 2 9 , i.e. 512. Thus if we 
take our numbers as being applied to the counting of classes 
instead of to the counting of individuals, our arithmetic will 
be normal until we reach 512 : the first number to be null will 
be 513. And if we advance to classes of classes we shall do still 
better : the number of them will be 2 512 , a number which is so 
large as to stagger imagination, since it has about 153 digits. 
And if we advance to classes of classes of classes, we shall obtain 
a number represented by 2 raised to a power which has about 
153 digits ; the number of digits in this number will be about 
three times io 152 . In a time of paper shortage it is undesirable 
to write out this number, and if we want larger ones we can 
obtain them by travelling further along the logical hierarchy. 
In this way any assigned inductive cardinal can be made to 
find its place among numbers which are not null, merely by 
travelling along the hierarchy for a sufficient distance. 1 

As regards ratios, we have a very similar state of affairs. 
If a ratio p,/v is to have the expected properties, there must 
be enough objects of whatever sort is being counted to insure 
that the null-class does not suddenly obtrude itself. But this 
can be insured, for any given ratio JJL/V, without the axiom of 

1 On this subject see Principia Mathematica, vol. ii. * 120 ff. On the 
corresponding problems as regards ratio, see ibid., vol. iii. * 303 ff. 



134 Introduction to Mathematical Philosophy 

infinity, by merely travelling up the hierarchy a sufficient distance. 
If we cannot succeed by counting individuals, we can try counting 
classes of individuals ; if we still do not succeed, we can try 
classes of classes, and so on. Ultimately, however few indi 
viduals there may be in the world, we shall reach a stage where 
there are many more than /x objects, whatever inductive number 
p may be. Even if there were no individuals at all, this would 
still be true, for there would then be one class, namely, the null- 
class, 2 classes of classes (namely, the null-class of classes and the 
class whose only member is the null-class of individuals), 4 classes 
of classes of classes, 16 at the next stage, 65,536 at the next 
stage, and so on. Thus no such assumption as the axiom of 
infinity is required in order to reach any given ratio or any given 
inductive cardinal. 

It is when we wish to deal with the whole class or series of 
inductive cardinals or of ratios that the axiom is required. We 
need the whole class of inductive cardinals in order to establish 
the existence of N , and the whole series in order to establish 
the existence of progressions : for these results, it is necessary 
that we should be able to make a single class or series in which 
no inductive cardinal is null. We need the whole series of ratios 
in order of magnitude in order to define real numbers as segments : 
this definition will not give the desired result unless the series 
of ratios is compact, which it cannot be if the total number of 
ratios, at the stage concerned, is finite. 

It would be natural to suppose as I supposed myself in former 
days that, by means of constructions such as we have been 
considering, the axiom of infinity could be proved. It may be 
said : Let us assume that the number of individuals is n, where 
n may be o without spoiling our argument ; then if we form the 
complete set of individuals, classes, classes of classes, etc., all 
taken together, the number of terms in our whole set will be 



which is N . Thus taking all kinds of objects together, and not 






The Axiom of Infinity and Logical Types 135 

confining ourselves to objects of any one type, we shall certainly 
obtain an infinite class, and shall therefore not need the axiom 
of infinity. So it might be said. 

Now, before going into this argument, the first thing to observe 
is that there is an air of hocus-pocus about it : something reminds 
one of the conjurer who brings things out of the hat. The man 
who has lent his hat is quite sure there wasn't a live rabbit in it 
before, but he is at a loss to say how the rabbit got there. So 
the reader, if he has a robust sense of reality, will feel convinced 
that it is impossible to manufacture an infinite collection out of 
a finite collection of individuals, though he may be unable to 
say where the flaw is in the above construction. It would be a 
mistake to lay too much stress on such feelings of hocus-pocus ; 
like other emotions, they may easily lead us astray. But they 
afford a prima facie ground for scrutinising very closely any 
argument which arouses them. And when the above argument 
is scrutinised it will, in my opinion, be found to be fallacious, 
though the fallacy is a subtle one and by no means easy to avoid 
consistently. 

The fallacy involved is the fallacy which may be called " con 
fusion of types." To explain the subject of " types " fully would 
require a whole volume ; moreover, it is the purpose of this book 
to avoid those parts of the subjects which are still obscure and 
controversial, isolating, for the convenience of beginners, those 
parts which can be accepted as embodying mathematically ascer 
tained truths. Now the theory of types emphatically does not 
belong to the finished and certain part of our subject : much of 
this theory is still inchoate, confused, and obscure. But the need 
of some doctrine of types is less doubtful than the precise form 
the doctrine should take ; and in connection with the axiom of 
infinity it is particular'y easy to see the necessity of some such 
doctrine. 

This necessity results, for example, from the " contradiction of 
the greatest cardinal." We saw in Chapter VIII. that the number 
of classes contained in a given class is always greater than the 



136 Introduction to Mathematical Philosophy 

number of members of the class, and we inferred that there is 
no greatest cardinal number. But if we could, as we suggested 
a moment ago, add together into one class the individuals, classes 
of individuals, classes of classes of individuals, etc., we should 
obtain a class of which its own sub-classes would be members. 
The class "consisting of all objects that can be counted, of whatever 
sort, must, if there be such a class, have a cardinal number which 
is the greatest possible. Since all its sub-classes will be members 
of it, there cannot be more of them than there are members. 
Hence we arrive at a contradiction. 

When I first came upon this contradiction, in the year 1901, 
I attempted to discover some flaw in Cantor's proof that there is 
no greatest cardinal, which we gave in Chapter VIII. Apply 
ing this proof to the supposed class of all imaginable objects, 
I was led to a new and simpler contradiction, namely, the 
following : 

The comprehensive class we are considering, which is to embrace 
everything, must embrace itself as one of its members. In other 
words, if there is such a thing as " everything," then " every 
thing " is something, and is a member of the class " everything." 
But normally a class is not a member of itself. Mankind, for 
example, is not a man. Form now the assemblage of all classes 
which are not members of themselves. This is a class : is it a 
member of itself or not ? If it is, it is one of those classes that 
are not members of themselves, i.e. it is not a member of itself. 
If it is not, it is not one of those classes that are not members of 
themselves, i.e. it is a member of itself. Thus of the two hypo 
theses that it is, and that it is not, a member of itself each 
implies its contradictory. This is a contradiction. 

There is no difficulty in manufacturing similar contradictions 
ad lib. The solution of such contradictions by the theory of 
types is set forth fully in Principia Mathematical and also, more 
briefly, in articles by the present author in the American Journal 

1 Vol. i., Introduction, chap, ii., # 12 and * 20; vol ii., Prefatory 
Statement. 



The Axiom of Infinity and Logical Types 137 

of Mathematics 1 and in the Revue de Metaphysique et de Morale? 
For the present an outline of the solution must suffice. 

The fallacy consists in the formation of what we may call 
" impure " classes, i.e. classes which are not pure as to " type." 
As we shall see in a later chapter, classes are logical fictions, and 
a statement which appears to be about a class will only be signi 
ficant if it is capable of translation into a form in which no mention 
is made of the class. This places a limitation upon the ways in 
which what are nominally, though not really, names for classes 
can occur significantly : a sentence or set of symbols in which 
such pseudo-names occur in wrong ways is not false, but strictly 
devoid of meaning. The supposition that a class is, or that it 
is not, a member of itself is meaningless in just this way. And 
more generally, to suppose that one class of individuals is a 
member, or is not a member, of another class of individuals 
will be to suppose nonsense ; and to construct symbolically any 
class whose members are not all of the same grade in the logical 
hierarchy is to use symbols in a way which makes them no 
longer symbolise anything. 

Thus if there are n individuals in the world, and 2 n classes of 
individuals, we cannot form a new class, consisting of both 
individuals and classes and having w-f-2 n members. In this way 
the attempt to escape from the need for the axiom of infinity 
breaks down. I do not pretend to have explained the doctrine 
of types, or done more than indicate, in rough outline, why there 
is need of such a doctrine. I have aimed only at saying just 
so much as was required in order to show that we cannot 'prove 
the existence of infinite numbers and classes by such conjurer's 
methods as we have been examining. There remain, however, 
certain other possible methods which must be considered. 

Various arguments professing to prove the existence of infinite 
classes are given in the Principles of Mathematics, 339 (p. 357). 

1 " Mathematical Logic as based on the Theory of Types," vol. xxx., 
1908, pp. 222-262. 

" Les paradoxes de la logique," 1906, pp. 627-650. 



138 Introduction to Mathematical Philosophy 

In so far as these arguments assume that, if n is an inductive 
cardinal, n is not equal to n-\-i, they have been already dealt 
with. There is an argument, suggested by a passage in Plato's 
ParmfnidlSy to the effect that, if there is such a number as I, 
then I has being ; but I is not identical with being, and therefore 
I and being are two, and therefore there is such a number as 2, 
and 2 together with I and being gives a class of three terms, and 
so on. This argument is fallacious, partly because " being " is 
not a term having any definite meaning, and still more because, 
if a definite meaning were invented for it, it would be found that 
numbers do not have being they are, in fact, what are called 
" logical fictions,'' as we shall see when we come to consider 
the definition of classes. 

The argument that the number of numbers from o to n (both 
inclusive) is n-\-i depends upon the assumption that up to and 
including n no number is equal to its successor, which, as we have 
seen, will not be always true if the axiom of infinity is false. It 
must be understood that the equation n=n-\-i, which might be 
true for a finite n\in exceeded the total number of individuals 
in the world, is quite different from the same equation as applied 
to a reflexive number. As applied to a reflexive number, it 
means that, given a class of n terms, this class is " similar " to 
that obtained by adding another term. But as applied to a 
number which is too great for the actual world, it merely means 
that there is no class of n individuals, and no class of n-\-\ indi 
viduals ; it does not mean that, if we mount the hierarchy of 
types sufficiently far to secure the existence of a class of n terms, 
we shall then find this class " similar " to one of n-\- 1 terms, for 
if n is inductive this will not be the case, quite independently of 
the truth or falsehood of the axiom of infinity. 

There is an argument employed by both Bolzano 1 and Dede- 
kind 2 to prove the existence of reflexive classes. The argument, 
in brief, is this : An object is not identical with the idea of the 

1 Bolzano, Paradoxien des Unendlichen, 13. 

1 Dedekind, Was sind und was sollen die Zahlen ? No. 66. 



The Axiom of Infinity and Logical Types 139 

object, but there is (at least in the realm of being) an idea of any 
object. The relation of an object to the idea of it is one-one, and 
ideas are only some among objects. Hence the relation " idea 
of " constitutes a reflexion of the whole class of objects into a 
part of itself, namely, into that part which consists of ideas. 
Accordingly, the class of objects and the class of ideas are both 
infinite. This argument is interesting, not only on its own 
account, but because the mistakes in it (or what I judge to be 
mistakes) are of a kind which it is instructive to note. The 
main error consists in assuming that there is an idea of every 
object. It is, of course, exceedingly difficult to decide what is 
meant by an " idea " ; but let us assume that we know. We are 
then to suppose that, starting (say) with Socrates, there is the 
idea of Socrates, and so on ad inf. Now it is plain that this is not 
the case in the sense that all these ideas have actual empirical 
existence in people's minds. Beyond the third or fourth stage 
they become mythical. If the argument is to be upheld, the 
" ideas " intended must be Platonic ideas laid up in heaven, for 
certainly they are not on earth. But then it at once becomes 
doubtful whether there are such ideas. If we are to know that 
there are, it must be on the basis of some logical theory, proving 
that it is necessary to a thing that there should be an idea of it. 
We certainly cannot obtain this result empirically, or apply it, 
as Dedekind does, to " meine Gedankenwelt " the world of my 
thoughts. 

If we were concerned to examine fully the relation of idea and 
object, we should have to enter upon a number of psychological 
and logical inquiries, which are not relevant to our main purpose. 
But a few further points should be noted. If " idea " is to be 
understood logically, it may be identical with the object, or it 
may stand for a description (in the sense to be explained in a 
subsequent chapter). In the former case the argument fails, 
because it was essential to the proof of reflexiveness that object 
and idea should be distinct. In the second case the argument 
also fails, because the relation of object and description is not 



140 Introduction to Mathematical Philosophy 

one-one : there are innumerable correct descriptions of any given 
object. Socrates (e.g) may be described as " the master of 
Plato," or as " the philosopher who drank the hemlock," or as 
" the husband of Xantippe." If to take up the remaining 
hypothesis " idea " is to be interpreted psychologically, it must 
be maintained that there is not any one definite psychological 
entity which could be called the idea of the object : there are in 
numerable beliefs and attitudes, each of which could be called an 
idea of the object in the sense in which we might say " my idea 
of Socrates is quite different from yours," but there is not any 
central entity (except Socrates himself) to bind together various 
" ideas of Socrates," and thus there is not any such one-one rela 
tion of idea and object as the argument supposes. Nor, of course, 
as we have already noted, is it true psychologically that there are 
ideas (in however extended a sense) of more than a tiny proportion 
of the things in the world. For all these reasons, the above 
argument in favour of the logical existence of reflexive classes 
must be rejected. 

It might be thought that, whatever may be said of logical 
arguments, the empirical arguments derivable from space and 
time, the diversity of colours, etc., are quite sufficient to prove 
the actual existence of an infinite number of particulars. I do 
not believe this. We have no reason except prejudice for believ 
ing in the infinite extent of space and time, at any rate in the sense 
in which space and time are physical facts, not mathematical 
fictions. We naturally regard space and time as continuous, or, 
at least, as compact ; but this again is mainly prejudice. The 
theory of " quanta " in physics, whether true or false, illustrates 
the fact that physics can never afford proof of continuity, though 
it might quite possibly afford disproof. The senses are not 
sufficiently exact to distinguish between continuous motion and 
rapid discrete succession, as anyone may discover in a cinema. 
A world in which all motion consisted of a series of small finite 
jerks would be empirically indistinguishable from one in which 
motion was continuous. It would take up too much space to 



The Axiom of Infinity and Logical Types 141 

defend these theses adequately ; for the present I am merely 
suggesting them for the reader's consideration. If they are valid, 
it follows that there is no empirical reason for believing the 
number of particulars in the world to be infinite, and that there 
never can be ; also that there is at present no empirical reason 
to believe the number to be finite, though it is theoretically 
conceivable that some day there might be evidence pointing, 
though not conclusively, in that direction. 

From the fact that the infinite is not self-contradictory, but is 
also not demonstrable logically, we must conclude that nothing 
can be known a priori as to whether the number of things 
in the world is finite or infinite. The conclusion is, therefore, 
to adopt a Leibnizian phraseology, that some of the possible 
worlds are finite, some infinite, and we have no means of 
knowing to which of these two kinds our actual world belongs. 
The axiom of infinity will be true in some possible worlds 
and false in others ; whether it is true or false in this world, 
we cannot tell. 

Throughout this chapter the synonyms " individual " and 
" particular " have been used without explanation. It would be 
impossible to explain them adequately without a longer disquisi 
tion on the theory of types than would be appropriate to the 
present work, but a few words before we leave this topic may 
do something to diminish the obscurity which would otherwise 
envelop the meaning of these words. 

In an ordinary statement we can distinguish a verb, expressing 
an attribute or relation, from the substantives which express the 
subject of the attribute or the terms of the relation. " Caesar 
lived " ascribes an attribute to Caesar ; " Brutus killed Caesar " 
expresses a relation between Brutus and Caesar. Using the word 
"subject" in a generalised sense, we may call both Brutus and 
Caesar subjects of this proposition : the fact that Brutus is gram 
matically subject and Caesar object is logically irrelevant, since 
the same occurrence may be expressed in the words " Caesar was 
killed by Brutus," where Caesar is the grammatical subject. 



142 Introduction to Mathematical Philosophy 

Thus in the simpler sort of proposition we shall have an attribute 
or relation holding of or between one, two or more " subjects " 
in the extended sense. (A relation may have more than two 
terms : e.g. " A gives B to C " is a relation of three terms.) Now 
it often happens that, on a closer scrutiny, the apparent subjects 
are found to be not really subjects, but to be capable of analysis ; 
the only result of this, however, is that new subjects take their 
places. It also happens that the verb may grammatically be 
made subject : e.g. we may say, " Killing is a relation which 
holds between Brutus and Caesar." But in such cases the 
grammar is misleading, and in a straightforward statement, 
following the rules that should guide philosophical grammar, 
Brutus and Cssar will appear as the subjects and killing 
as the verb. 

We are thus led to the conception of terms which, when they 
occur in propositions, can only occur as subjects, and never in 
any other way. This is part of the old scholastic definition 
of substance ; but persistence through time, which belonged to 
that notion, forms no part of the notion with which we are con 
cerned. We shall define " proper names " as those terms which 
can only occur as subjects in propositions (using " subject " 
in the extended sense just explained). We shall further define 
" individuals " or " particulars " as the objects that can be 
named by proper names. (It would be better to define them 
directly, rather than by means of the kind of symbols by which 
they are symbolised ; but in order to do that we should have 
to plunge deeper into metaphysics than is desirable here.) It 
is, of course, possible that there is an endless regress : that 
whatever appears as a particular is really, on closer scrutiny, 
a class or some kind of complex. If this be the case, the axiom 
of infinity must of course be true. But if it be not the case, 
it must be theoretically possible for analysis to reach ultimate 
subjects, and it is these that give the meaning of " particulars " 
or " individuals." It is to the number of these that the axiom 
of infinity is assumed to apply. If it is true of them, it is true 



The Axiom of Infinity and Logical Types 143 

of classes of them, and classes of classes of them, and so on ; 
similarly if it is false of them, it is false throughout this hierarchy. 
Hence it is natural to enunciate the axiom concerning them rather 
than concerning any other stage in the hierarchy. But whether 
the axiom is true or false, there seems no known method of 
discovering. 



CHAPTER XIV 

INCOMPATIBILITY AND THE THEORY OF DEDUCTION 

WE have now explored, somewhat hastily it is true, that part 
of the philosophy of mathematics which does not demand a 
critical examination of the idea of class. In the preceding 
chapter, however, we found ourselves confronted by problems 
which make such an examination imperative. Before we can 
undertake it, we must consider certain other parts of the philos 
ophy of mathematics, which we have hitherto ignored. In a 
synthetic treatment, the parts which we shall now be concerned 
with come first : they are more fundamental than anything 
that we have discussed hitherto. Three topics will concern us 
before we reach the theory of classes, namely : (i) the theory 
of deduction, (2) prepositional functions, (3) descriptions. Of 
these, the third is not logically presupposed in the theory of 
classes, but it is a simpler example of the kind of theory that 
is needed in dealing with classes. It is the first topic, the theory 
of deduction, that will concern us in the present chapter. 

Mathematics is a deductive science : starting from certain 
premisses, it arrives, by a strict process of deduction, at the 
various theorems which constitute it. It is true that, in the past, 
mathematical deductions were often greatly lacking in rigour ; 
it is true also that perfect rigour is a scarcely attainable ideal. 
Nevertheless, in so far as rigour is lacking in a mathematical 
proof, the proof is defective ; it is no defence to urge that common 
sense shows the result to be correct, for if we were to rely upon 
that, it would be better to dispense with argument altogether, 

144 



Incompatibility and the Theory of Deduction 145 

rather than bring fallacy to the rescue of common sense. No 
appeal to common sense, or " intuition," or anything except strict 
deductive logic, ought to be needed in mathematics after the 
premisses have been laid down. 

Kant, having observed that the geometers of his day could 
not prove their theorems by unaided argument, but required 
an appeal to the figure, invented a theory of mathematical 
reasoning according to which the inference is never strictly 
logical, but always requires the support of what is called 
" intuition." The whole trend of modern mathematics, with 
its increased pursuit of rigour, has been against this Kantian 
theory. The things in the mathematics of Kant's day which 
cannot be proved, cannot be known for example, the axiom of 
parallels. What can be known, in mathematics and by mathe 
matical methods, is what can be deduced from pure logic. What 
else is to belong to human knowledge must be ascertained other 
wise empirically, through the senses or through experience in 
some form, but not a priori. The positive grounds for this 
thesis are to be found in Principia Mathematica, passim ; a 
controversial defence of it is given in the Principles of Mathe 
matics. We cannot here do more than refer the reader to those 
works, since the subject is too vast for hasty treatment. Mean 
while, we shall assume that all mathematics is deductive, and 
proceed to inquire as to what is involved in deduction. 

In deduction, we have one or more propositions called pre 
misses, from which we infer a proposition called the conclusion. 
For our purposes, it will be convenient, when there are originally 
several premisses, to amalgamate them into a single proposition, 
so as to be able to speak of the premiss as well as of the con 
clusion. Thus we may regard deduction as a process by which 
we pass from knowledge of a certain proposition, the premiss, 
to knowledge of a certain other proposition, the conclusion. 
But we shall not regard such a process as logical deduction unless 
it is correct, i.e. unless there is such a relation between premiss 
and conclusion that we have a right to believe the conclusion 

10 



146 Introduction to Mathematical Philosophy 

if we know the premiss to be true. It is this relation that is 
chiefly of interest in the logical theory of deduction. 

In order to be able validly to infer the truth of a proposition, 
we must know that some other proposition is true, and that 
there is between the two a relation of the sort called "implication," 
i.e. that (as we say) the premiss " implies " the conclusion. (We 
shall define this relation shortly.) Or we may know that a certain 
other proposition is false, and that there is a relation between 
the two of the sort called " disjunction," expressed by " p or ^," 1 
so that the knowledge that the one is false allows us to infer 
that the other is true. Again, what we wish to infer may be 
the falsehood of some proposition, not its truth. This may be 
inferred from the truth of another proposition, provided we know 
that the two are " incompatible," i.e. that if one is true, the other 
is false. It may also be inferred from the falsehood of another 
proposition, in just the same circumstances in which the truth 
of the other might have been inferred from the truth of the one ; 
i.e. from the falsehood of p we may infer the falsehood of q, when 
q implies p. All these four are cases of inference. When our 
minds are fixed upon inference, it seems natural to take " impli 
cation " as the primitive fundamental relation, since this is the 
relation which must hold between p and q if we are to be able 
to infer the truth of q from the truth of p. But for technical 
reasons this is not the best primitive idea to choose. Before 
proceeding to primitive ideas and definitions, let us consider 
further the various functions of propositions suggested by the 
above-mentioned relations of propositions. 

The simplest of such functions is the negative, " not-^>." 
This is that function of p which is true when p is false, and false 
when p is true. It is convenient to speak of the truth of a pro 
position, or its falsehood, as its " truth-value " 2 ; i.e. truth is 
the " truth-value " of a true proposition, and falsehood of a false 
one. Thus not- has the opposite truth-value to p. 

1 We shall use the letters p, q, r, s, t to denote variable propositions. 

2 This term is due to Frege. 



Incompatibility and the Theory of Deduction 147 

We may take next disjunction, " p or <?." This is a function 
whose truth-value is truth when p is true and also when q is true, 
but is falsehood when both p and q are false. 

Next we may take conjunction, " p and q" This has truth 
for its truth-value when p and q are both true ; otherwise it 
has falsehood for its truth-value. 

Take next incompatibility, i.e. " p and q are not both true." 
This is the negation of conjunction ; it is also the disjunction 
of the negations of p and q, i.e. it is " not-/) or not-y." Its truth- 
value is truth when p is false and likewise when q is false ; its 
truth-value is falsehood when p and q are both true. 

Last take implication, i.e. " p implies q," or " if p, then <?." 
This is to be understood in the widest sense that will allow us 
to infer the truth of q if we know the truth of p. Thus we inter 
pret it as meaning : " Unless p is false, q is true," or " either 
p is false or q is true." (The fact that " implies " is capable 
of other meanings does not concern us ; this is the meaning which 
is convenient for us.) That is to say, " p implies q " is to mean 
" not-/> or q " : its truth-value is to be truth if p is false, likewise 
if q is true, and is to be falsehood if p is true and q is false. 

We have thus five functions: negation, disjunction, conjunction, 
incompatibility, and implication. We might have added others, 
for example, joint falsehood, " not-p and not-^," but the above 
five will suffice. Negation differs from the other four in being 
a function of one proposition, whereas the others are functions 
of two. But all five agree in this, that their truth-value depends 
only upon that of the propositions which are their arguments. 
Given the truth or falsehood of p, or of p and q (as the case may 
be), we are given the truth or falsehood of the negation, disjunc 
tion, conjunction, incompatibility, or implication. A function of 
propositions which has this property is called a " truth-function." 

The whole meaning of a truth-function is exhausted by the 
statement of the circumstances under which it is true or false. 
" Not-/)," for example, is simply that function of p which is true 
when p is false, and false when p is true : there is no further 



148 Introduction to Mathematical Philosophy 

meaning to be assigned to it. The same applies to " p or q " 
and the rest. It follows that two truth-functions which have 
the same truth-value for all values of the argument are indis 
tinguishable. For example, " p and q " is the negation of 
" not-/) or not-^ " and vice versa ; thus either of these may be 
defined as the negation of the other. There is no further meaning 
in a truth-function over and above the conditions under which 
it is true or false. 

It is clear that the above five truth-functions are not all inde 
pendent. We can define some of them in terms of others. There 
is no great difficulty in reducing the number to two ; the two 
chosen in Principia Mathematica are negation and disjunction. 
Implication is then defined as " not-/) or q " ; incompatibility 
as " not-/> or not-q " ; conjunction as the negation of incompati 
bility. But it has been shown by Sheffer * that we can be content 
with one primitive idea for all five, and by Nicod 2 that this enables 
us to reduce the primitive propositions required in the theory 
of deduction to two non-formal principles and one formal one. 
For this purpose, we may take as our one indefinable either 
incompatibility or joint falsehood. We will choose the former. 

Our primitive idea, now, is a certain truth-function called 
" incompatibility," which we will denote by p/q. Negation 
can be at once defined as the incompatibility of a proposition 
with itself, i.e. " not-/) " is defined as " />//>." Disjunction is 
the incompatibility of not-/) and not-<?, i.e. it is (p/p)\(q/q). 
Implication is the incompatibility of p and not-^, i.e. p\(q/q)> 
Conjunction is the negation of incompatibility, i.e. it is (p/q) \ 
(p/q)' Thus all our four other functions are defined in terms 
of incompatibility. 

It is obvious that there is no limit to the manufacture of truth- 
functions, either by introducing more arguments or by repeating 
arguments. What we are concerned with is the connection of 
this subject with inference. 

1 Trans. Am. Math. Soc., vol. xiv. pp. 481-488. 

2 Proc. Camb. Phil. Soc., vol. xix., i., January 1917. 



Incompatibility and the Theory of Deduction 149 

If we know that p is true and that p implies q, we can proceed 
to assert q. There is always unavoidably something psycho 
logical about inference : inference is a method by which we arrive 
at new knowledge, and what is not psychological about it is the 
relation which allows us to infer correctly ; but the actual passage 
from the assertion of p to the assertion of q is a psychological 
process, and we must not seek to represent it in purely logical 
terms. 

In mathematical practice, when we infer, we have always 
some expression containing variable propositions, say p and q y 
which is known, in virtue of its form, to be true for all values 
of p and q ; we have also some other expression, part of the former, 
which is also known to be true for all values of p and q ; and in 
virtue of the principles of inference, we are able to drop this part 
of our original expression, and assert what is left. This somewhat 
abstract account may be made clearer by a few examples. 

Let us assume that we know the five formal principles of 
deduction enumerated in Principia Matbematica. (M. Nicod has 
reduced these to one, but as it is a complicated proposition, 
we will begin with the five.) These five propositions are as 
follows : 

(1) " p or p " implies p i.e. if either p is true or p is true, 
then p is true. 

(2) q implies " p or q " i.e. the disjunction " p or q " is true 
when one of its alternatives is true. 

(3) " p or q " implies " q or />." This would not be required 
if we had a theoretically more perfect notation, since in the 
conception of disjunction there is no order involved, so that 
" p or q " and " q or p " should be identical. But since our 
symbols, in any convenient form, inevitably introduce an order, 
we need suitable assumptions for showing that the order is 
irrelevant. 

(4) If either p is true or " q or r " is true, then either q is true 
or " p or r " is true. (The twist in this proposition serves to 
increase its deductive power.) 



150 Introduction to Mathematica Philosophy 

(5) If q implies r, then " p or q " implies " p or r." 
These are the formal principles of deduction employed in 
Principia Mathematica. A formal principle of deduction has a 
double use, and it is in order to make this clear that we have 
cited the above five propositions. It has a use as the premiss 
of an inference, and a use as establishing the fact that the pre 
miss implies the conclusion. In the schema of an inference 
we have a proposition p, and a proposition " p implies ," from 
which we infer q. Now when we are concerned with the princi 
ples of deduction, our apparatus of primitive propositions has 
to yield both the p and the " p implies q " of our inferences. 
That is to say, our rules of deduction are to be used, not only as 
rules, which is their use for establishing " p implies q" but also 
as substantive premisses, i.e. as the p of our schema. Suppose, 
for example, we wish to prove that if p implies q, then if q 
implies r it follows that p implies r. We have here a relation of 
three propositions which state implications. Put 

pi=p implies q, p 2 =q implies r, and p 3 =p implies r. 

Then we have to prove that p implies that p z implies p s . Now 
take the fifth of our above principles, substitute not-/> for p, 
and remember that " not-/) or q " is by definition the same as 
" p implies q." Thus our fifth principle yields : 

" If q implies r, then ' p implies q ' implies ' p implies r,' ' 
i.e. " p 2 implies that p^ implies p 3 ." Call this propo 
sition A. 

But the fourth of our principles, when we substitute not-/), 
not-, for p and q y and remember the definition of implication, 
becomes : 

" If p implies that q implies r, then q implies that p implies r." 

Writing p 2 in place of p, p in place of q, and p 3 in place of r y this 
becomes : 

" If p z implies that p 1 implies /> 3 , then p l implies that p 2 implies 
1>" Call this B. 



Incompatibility and the Theory of Deduction 1 5 1 

Now we proved by means of our fifth principle that 

" p 2 implies that p^ implies p 3 " which was what we called A. 

Thus we have here an instance of the schema of inference, 
since A represents the p of our scheme, and B represents the 
" p implies q." Hence we arrive at q, namely, 

" p l implies that p z implies p 3 " 

which was the proposition to be proved. In this proof, the 
adaptation of our fifth principle, which yields A, occurs as a 
substantive premiss ; while the adaptation of our fourth principle, 
which yields B, is used to give the form of the inference. The 
formal and material employments of premisses in the theory 
of deduction are closely intertwined, and it is not very important 
to keep them separated, provided we realise that they are in 
theory distinct. 

The earliest method of arriving at new results from a premiss 
is one which is illustrated in the above deduction, but which 
itself can hardly be called deduction. The primitive propositions, 
whatever they may be, are to be regarded as asserted for all 
possible values of the variable propositions p, q, r which occur 
in them. We may therefore substitute for (say) p any expression 
whose value is always a proposition, e.g. not-p, " s implies t," 
and so on. By means of such substitutions we really obtain 
sets of special cases of our original proposition, but from a prac 
tical point of view we obtain what are virtually new propositions. 
The legitimacy of substitutions of this kind has to be insured by 
means of a non-formal principle of inference. 1 

We may now state the one formal principle of inference to 
which M. Nicod has reduced the five given above. For this 
purpose we will first show how certain truth-functions can be 
defined in terms of incompatibility. We saw already that 

p | (q/q) means " p implies q" 

1 No such principle is enunciated in Pnncipia Mathematics, or in M. 
Nicod's article mentioned above. But this would seem to be an omission, 



152 Introduction to Mathematical Philosophy 

We now observe that 

p | (q/r) means " p implies both q and r" 

For this expression means " p is incompatible with the incom 
patibility of q and r," i.e. " p implies that q and r are not incom 
patible," i.e. " p implies that q and r are both true " for, as 
we saw, the conjunction of q and r is the negation of their 
incompatibility. 

Observe next that t \ (t/t) means " t implies itself." This is a 
particular case of p \ (q/q). 

Let us write p for the negation of p ; thus p/s will mean the 
negation of p/s, i.e. it will mean the conjunction of p and s. It 
follows that 

<V?)f?A 

expresses the incompatibility of s/q with the conjunction of 
p and s ; in other words, it states that if p and s are both true, 
s/q is false, i.e. s and q are both true ; in still simpler words, 
it states that p and s jointly imply s and q jointly. 

Now, put P=p | (q/r), 



Q=(s/q)\p/s. 
Then M. Nicod's sole formal principle of deduction is 

Pk/Q, 

in other words, P implies both TT and Q. 

He employs in addition one non-formal principle belonging 
to the theory of types (which need not concern us), and one 
corresponding to the principle that, given p, and given that 
p implies q, we can assert q. This principle is : 

"If p | (r/q) is true, and p is true, then q is true." From 
this apparatus the whole theory of deduction follows, except 
in so far as we are concerned with deduction from or to the 
existence or the universal truth of " prepositional functions," 
which we shall consider in the next chapter. 

There is ? if I am not mistaken, a certain confusion in the 



Incompatibility and the Theory of Deauction 153 

minds of some authors as to the relation, between propositions, 
in virtue of which an inference is valid. In order that it may 
be valid to infer q from />, it is only necessary that p should be 
true and that the proposition " not-/> or q " should be true. 
Whenever this is the case, it is clear that q must be true. But 
inference will only in fact take place when the proposition " not-/> 
or q " is known otherwise than through knowledge of not-/) or 
knowledge of q. Whenever p is false, " not-/> or q " is true, 
but is useless for inference, which requires that p should be true. 
Whenever q is already known to be true, " not-/) or q " is of 
course also known to be true, but is again useless for inference, 
since q is already known, and therefore does not need to be 
inferred. In fact, inference only arises when " not-/) or q " 
can be known without our knowing already which of the two 
alternatives it is that makes the disjunction true. Now, the 
circumstances under which this occurs are those in which certain 
relations of form exist between p and q. For example, we know 
that if r implies the negation of s, then s implies the negation 
of r . Between " r implies not-5 " and " s implies not-r " there 
is a formal relation which enables us to know that the first implies 
the second, without having first to know that the first is false 
or to know that the second is true. It is under such circum 
stances that the relation of implication is practically useful for 
drawing inferences. 

But this formal relation is only required in order that we may 
be able to know that either the premiss is false or the conclusion 
is true. It is the truth of " not-/) or q " that is required for 
the validity of the inference ; what is required further is only 
required for the practical feasibility of the inference. Professor 
C. I. Lewis * has especially studied the narrower, formal relation 
which we may call " formal deducibility." He urges that the 
wider relation, that expressed by " not-/> or q" should not be 
called " implication." That is, however, a matter of words. 

1 See Mind, vol. xxi., 1912, pp. 522-531 ; and vol. xxiii., 1914, pp. 
240-247. 



154 Introduction to Mathematical Philosophy 

Provided our use of words is consistent, it matters little how we 
define them. The essential point of difference between the 
theory which I advocate and the theory advocated by Professor 
Lewis is this : He maintains that, when one proposition q is 
" formally deducible " from another p, the relation which we 
perceive between them is one which he calls " strict implication," 
which is not the relation expressed by " not-p or q " but a narrower 
relation, holding only when there are certain formal connections 
between p and q. I maintain that, whether or not there be 
such a relation as he speaks of, it is in any case one that mathe 
matics does not need, and therefore one that, on general grounds 
of economy, ought not to be admitted into our apparatus of 
fundamental notions ; that, whenever the relation of " formal 
deducibility " holds between two propositions, it is the case that 
we can see that either the first is false or the second true, and that 
nothing beyond this fact is necessary to be admitted into our 
premisses ; and that, finally, the reasons of detail which Professor 
Lewis adduces against the view which I advocate can all be met 
in detail, and depend for their plausibility upon a covert and 
unconscious assumption of the point of view which I reject. 
I conclude, therefore, that there is no need to admit as a funda 
mental notion any form of implication not expressible as a 
truth-function. 



CHAPTER XV 

PROPOSITIONAL FUNCTIONS 

WHEN, in the preceding chapter, we were discussing propositions, 
we did not attempt to give a definition of the word " proposition." 
But although the word cannot be formally defined, it is necessary 
to say something as to its meaning, in order to avoid the very 
common confusion with " prepositional functions," which are to 
be the topic of the present chapter. 

We mean by a " proposition " primarily a form of words which 
expresses what is either true or false. I say " primarily," 
because I do not wish to exclude other than verbal symbols, or 
even mere thoughts if they have a symbolic character. But I 
think the word " proposition " should be limited to what may, 
in some sense, be called " symbols," and further to such symbols 
as give expression to truth and falsehood. Thus " two and two 
are four " and " two and two are five " will be propositions, 
and so will " Socrates is a man " and " Socrates is not a man." 
The statement : " Whatever numbers a and b may be, 
a*+2ab+b 2 " is a proposition ; but the bare formula " 
a?-\-2ab-\-b 2 " alone is not, since it asserts nothing definite unless 
we are further told, or led to suppose, that a and b are to have 
all possible values, or are to have such-and-such values. The 
former of these is tacitly assumed, as a rule, in the enunciation 
of mathematical formulae, which thus become propositions ; 
but if no such assumption were made, they would be " preposi 
tional functions." A " prepositional function," in fact, is an 
expression containing one or more undetermined constituents, 



156 Introduction to Mathematical Philosophy 

such that, when values are assigned to these constituents, the 
expression becomes a proposition. In other words, it is a function 
whose values are propositions. But this latter definition must 
be used with caution. A descriptive function, e.g. " the hardest 
proposition in A's mathematical treatise," will not be a pro- 
positional function, although its values are propositions. But in 
such a case the propositions are only described : in a proposi- 
tional function, the values must actually enunciate propositions. 

Examples of prepositional functions are easy to give : " x 
is human " is a prepositional function ; so long as x remains 
undetermined, it is neither true nor false, but when a value 
is assigned to x it becomes a true or false proposition. Any 
mathematical equation is a prepositional function. So long as 
the variables have no definite value, the equation is merely an 
expression awaiting determination in order to become a true or 
false proposition. If it is an equation containing one variable, 
it becomes true when the variable is made equal to a root 
of the equation, otherwise it becomes false ; but if it is an 
" identity " it will be true when the variable is any number. 
The equation to a curve in a plane or to a surface in space is a 
propositional function, true for values of the co-ordinates belong 
ing to points on the curve or surface, false for other values. 
Expressions of traditional logic such as " all A is B " are pro- 
positional functions : A and B have to be determined as definite 
classes before such expressions become true or false. 

The notion of " cases " or " instances " depends upon pro- 
positional functions. Consider, for example, the kind of process 
suggested by what is called " generalisation," and let us take 
some very primitive example, say, " lightning is followed by 
thunder." We have a number of " instances " of this, i.e. a 
number of propositions such as : " this is a flash of lightning 
and is followed by thunder." What are these occurrences 
" instances " of ? They are instances of the propositional 
function : " If x is a flash of lightning, x is followed by thunder." 
The process of generalisation (with whose validity we are fortun- 



Prepositional Functions 157 

ately not concerned) consists in passing from a number of such 
instances to the universal truth of the prepositional function : 
" If x is a flash of lightning, x is followed by thunder." It will 
be found that, in an analogous way, prepositional functions 
are always involved whenever we talk of instances or cases or 
examples. 

We do not need to ask, or attempt to answer, the question : 
" What is a prepositional function ? " A prepositional function 
standing all alone may be taken to be a mere schema, a mere 
shell, an empty receptacle for meaning, not something already 
significant. We are concerned with prepositional functions, 
broadly speaking, in two ways : first, as involved in the notions 
" true in all cases " and " true in some cases " ; secondly, as 
involved in the theory of classes and relations. The second of 
these topics we will postpone to a later chapter ; the first must 
occupy us now. 

When we say that something is " always true " or " true in 
all cases," it is clear that the " something " involved cannot be 
a proposition. A proposition is just true or false, and there 
is an end of the matter. There are no instances or cases of 
" Socrates is a man " or " Napoleon died at St Helena." These 
are propositions, and it would be meaningless to speak of their 
being true " in all cases." This phrase is only applicable to 
prepositional functions. Take, for example, the sort of thing 
that is often said when causation is being discussed. (We are 
net concerned with the truth or falsehood of what is said, but 
only with its logical analysis.) We are told that A is, in every 
instance, followed by B. Now if there are " instances " of A, 
A must be some general concept of which it is significant to say 
" #! is A," " x 2 is A," " # 3 is A," and so on, where x l9 x 2 , x 3 are 
particulars which are not identical one with another. This 
applies, e.g., to our previous case of lightning. We say that 
lightning (A) is followed by thunder (B). But the separate 
flashes are particulars, not identical, but sharing the common 
property of being lightning. The only way of expressing a 



158 Introduction to Mathematical Philosophy 

common property generally is to say that a common property 
of a number of objects is a prepositional function which becomes 
true when any one of these objects is taken as the value of the 
variable. In this case all the objects are " instances " of the 
truth of the prepositional function for a prepositional function, 
though it cannot itself be true or false, is true in certain instances 
and false in certain others, unless it is " always true " or " always 
false." When, to return to our example, we say that A is in 
every instance followed by B, we mean that, whatever x may be, 
if x is an A, it is followed by a B ; that is, we are asserting that 
a certain propositional function is " always true." 

Sentences involving such words as " all," " every," " a," 
" the," " some " require propositional functions for their inter 
pretation. The way in which propositional functions occur 
can be explained by means of two of the above words, namely, 
" all " and " some." 

There are, in the last analysis, only two things that can be 
done with a propositional function : one is to assert that it is 
true in all cases, the other to assert that it is true in at least one 
case, or in some cases (as we shall say, assuming that there is 
to be no necessary implication of a plurality of cases). All the 
other uses of propositional functions can be reduced to these two. 
When we say that a propositional function is true " in all cases," 
or " always " (as we shall also say, without any temporal sugges 
tion), we mean that all its values are true. If " fa " is the 
function, and a is the right sort of object to be an argument to 
" fa," then (f>a is to be true, however a may have been chosen. 
For example, " if a is human, a is mortal " is true whether a 
is human or not ; in fact, every proposition of this form is true. 
Thus the propositional function " if x is human, x is mortal " 
is " always true," or " true in all cases." Or, again, the state 
ment " there are no unicorns " is the same as the statement 
" the propositional function * x is not a unicorn ' is true in all 
cases." The assertions in the preceding chapter about pro 
positions, e.g. " ' p or q ' implies * q or p, " are really assertions 



Prepositional Functions 159 

that certain prepositional functions are true in all cases. We do 
not assert the above principle, for example, as being true only 
of this or that particular p or q, but as being true of any p or q 
concerning which it can be made significantly. The condition 
that a function is to be significant for a given argument is the same 
as the condition that it shall have a value for that argument, 
either true or false. The study of the conditions of significance 
belongs to the doctrine of types, which we shall not pursue 
beyond the sketch given in the preceding chapter. 

Not only the principles of deduction, but all the primitive 
propositions of logic, consist of assertions that certain preposi 
tional functions are always true. If this were not the case, they 
would have to mention particular things or concepts Socrates, 
or redness, or east and west, or what not, and clearly it is not 
the province of logic to make assertions which are true concerning 
one such thing or concept but not concerning another. It is 
part of the definition of logic (but not the whole of its definition) 
that all its propositions are completely general, i.e. they all 
consist of the assertion that some propositional function con 
taining no constant terms is always true. We shall return in 
our final chapter to the discussion of propositional functions 
containing no constant terms. For the present we will proceed 
to the other thing that is to be done with a propositional function, 
namely, the assertion that it is " sometimes true," i.e. true in at 
least one instance. 

When we say " there are men," that means that the pro- 
positional function " x is a man " is sometimes true. When we 
say " some men are Greeks," that means that the propositional 
function " x is a man and a Greek " is sometimes true. When we 
say " cannibals still exist in Africa," that means that the pro- 
positional function " x is a cannibal now in Africa " is sometimes 
true, i.e. is true for some values of x. To say " there are at least 
n individuals in the world " is to say that the propositional 
function " a is a class of individuals and a member of the cardinal 
number n " is sometimes true, or, as we may say, is true for certain 



160 Introduction to Mathematical Philosophy 

values of a. This form of expression is more convenient when it 
is necessary to indicate which is the variable constituent which 
we are taking as the argument to our prepositional function. 
For example, the above prepositional function, which we may 
shorten to " a, is a class of n individuals," contains two variables, 
a and n. The axiom of infinity, in the language of prepositional 
functions, is : " The prepositional function * if n is an inductive 
number, it is true for some values of a that a is a class of n indi 
viduals ' is true for all possible values of ." Here there is a 
subordinate function, " a is a class of n individuals," which is 
said to be, in respect of a, sometimes true ; and the assertion 
that this happens if n is an inductive number is said to be, in 
respect of , always true. 

The statement that a function fa is always true is the negation 
of the statement that not- fa is sometimes true, and the state 
ment that fa is sometimes true is the negation of the state 
ment that Tiot-fa is always true. Thus the statement " all 
men are mortals " is the negation of the statement that the 
function " x is an immortal man " is sometimes true. And the 
statement " there are unicorns " is the negation of the state 
ment that the function " x is not a unicorn " is always true. 1 
We say that fa is " never true " or " always false " if not-fa is 
always true. We can, if we choose, take one of the pair " always," 
" sometimes " as a primitive idea, and define the other by means 
of the one and negation. Thus if we choose " sometimes " as 
our primitive idea, we can define : " ' (f>x is always true ' is to 
mean * it is false that not- fa is sometimes true.' " 2 But for 
reasons connected with the theory of types it seems more correct 
to take both " always " and " sometimes " as primitive ideas, 
and define by their means the negation of propositions in which 
they occur. That is to say, assuming that we have already 

1 The method of deduction is given in Principia Mathematica, 
vol. i. * 9. 

2 For linguistic reasons, to avoid suggesting either the plural or the 
singular, it is often convenient to say " yx is not always false " rather 
than " cpx sometimes " or " <px is sometimes true." 



Prepositional Functions 161 

defined (or adopted as a primitive idea) the negation of pro 
positions of the type to which x belongs, we define : " The 
negation of ' </>x always ' is * not-0# sometimes ' ; and the nega 
tion of ' (j>x sometimes ' is * not-<# always.' ' In like manner 
we can re-define disjunction and the other truth-functions, 
as applied to propositions containing apparent variables, in 
terms of the . definitions and primitive ideas for propositions 
containing no apparent variables. Propositions containing no 
apparent variables are called " elementary propositions." From 
these we can mount up step by step, using such methods as have 
just been indicated, to the theory of truth-functions as applied 
to propositions containing one, two, three . . . variables, or any 
number up to n, where n is any assigned finite number. 

The forms which are taken as simplest in traditional formal 
logic are really far from being so, and all involve the assertion 
of all values or some values of a compound prepositional function. 
Take, to begin with, " all S is P." We will take it that S is 
defined by a prepositional function </>x, and P by a prepositional 
function i/jx. E.g., if S is men, <j)X will be " x is human " ; if P is 
mortals, t/jx will be " there is a time at which x dies." Then 
" all S is P " means : " ' <f>x implies i/jx ' is always true." It is 
to be observed that " all S is P " does not apply only to those 
terms that actually are S's ; it says something equally about 
terms which are not S's. Suppose we come across an x of which 
we do not know whether it is an S or not ; still, our statement 
" all S is P " tells us something about x, namely, that if x is an S, 
then x is a P. And this is every bit as true when x is not an S as 
when x is an S. If it were not equally true in both cases, the 
reductio ad absurdum would not be a valid method ; for the 
essence of this method consists in using implications in cases 
where (as it afterwards turns out) the hypothesis is false. We may 
put the matter another way. In order to understand " all S is P," 
it is not necessary to be able to enumerate what terms are S's ; 
provided we know what is meant by being an S and what by 
being a P, we can understand completely what is actually affirmed 

II 



1 62 Introduction to Mathematical Philosophy 

by " all S is P," however little we may know of actual instances 
of either. This shows that it is not merely the actual terms that 
are S's that are relevant in the statement " all S is P," but all the 
terms concerning which the supposition that they are S's is 
significant, i.e. all the terms that are S's, together with all the 
terms that are not S's i.e. the whole of the appropriate logical 
" type." What applies to statements about all applies also to 
statements about some. " There are men," e.g., means that 
" x is human " is true for some values of x. Here all values of x 
(i.e. all values for which " x is human " is significant, whether 
true or false) are relevant, and not only those that in fact are 
human. (This becomes obvious if we consider how we could 
prove such a statement to be false.) Every assertion about 
" all " or " some " thus involves not only the arguments that 
make a certain function true, but all that make it significant, 
i.e. all for which it has a value at all, whether true or false. 

We may now proceed with our interpretation of the traditional 
forms of the old-fashioned formal logic. We assume that S 
is those terms x for which fa is true, and P is those for which fa 
is true. (As we shall see in a later chapter, all classes are derived 
in this way from prepositional functions.) Then : 

" All S is P " means " ' fa implies fa ' is always true." 
" Some S is P " means " * fa and fa ' is sometimes true." 
" No S is P " means " ' fa implies not-fa ' is always true." 
" Some S is not P " means " ' fa and not-fa ' is sometimes 
true." 

It will be observed that the propositional functions which are 
here asserted for all or some values are not fa and fa them 
selves, but truth-functions of fa and fa for the same argument 
x. The easiest way to conceive of the sort of thing that is 
intended is to start not from fa and fa in general, but from 
(j>a and ipa, where a is some constant. Suppose we are consider 
ing all " men are mortal " : we will begin with 

" If Socrates is human, Socrates is mortal," 



Prepositional Functions 163 

and then we will regard " Socrates " as replaced by a variable x 
wherever " Socrates " occurs. The object to be secured is that, 
although x remains a variable, without any definite value, yet 
it is to have the same value in " fa " as in " fa " when we are 
asserting that " fa implies fa " is always true. This requires 
that we shall start with a function whose values are such as 
" cf>a implies $a" rather than with two separate functions fa 
and fa ; for if we start with two separate functions we can 
never secure that the x, while remaining undetermined, shall 
have the same value in both. 

For brevity we say " fa always implies iftx " when we 
mean that " (j>x implies fa " is always true. Propositions 
of the form " fa always implies fa " are called " formal 
implications " ; this name is given equally if there are several 
variables. 

The above definitions show how far removed from the simplest 
forms are such propositions as " all S is P," with which tradi 
tional logic begins. It is typical of the lack of analysis involved 
that traditional logic treats "all S is P " as a proposition of 
the same form as " x is P " e.g., it treats " all men are mortal " 
as of the same form as " Socrates is mortal." As we have just 
seen, the first is of the form " fa always implies fa" while the 
second is of the form " fa" The emphatic separation of these 
two forms, which was effected by Peano and Frege, was a very 
vital advance in symbolic logic. 

It will be seen that " all S is P " and " no S is P " do not 
really differ in form, except by the substitution of not-iffx for fa, 
and that the same applies to " some S is P " and " some S is 
not P." It should also be observed that the traditional rules 
of conversion are faulty, if we adopt the view, which is the only 
technically tolerable one, that such propositions as " all S is P " 
do not involve the " existence " of S's, i.e. do not require that 
there should be terms which are S's. The above definitions 
lead to the result that, if fa is always false, i.e. if there are no 
S's, then " all S is P " and no S is P " will both be true, what- 



164 Introduction to Mathematical Philosophy 

ever P may be. For, according to the definition in the last 
chapter, " fa implies fa " means " not- fa or fa" which is 
always true if not-<# is always true. At the first moment, 
this result might lead the reader to desire different definitions, 
but a little practical experience soon shows that any different 
definitions would be inconvenient and would conceal the important 
ideas. The proposition " fa always implies fa, and fa 
is sometimes true " is essentially composite, and it would be 
very awkward to give this as the definition of "all S is P," 
for then we should have no language left for " fa always implies 
fa," which is needed a hundred times for once that the other is 
needed. But, with our definitions, " all S is P " does not imply 
" some S is P," since the first allows the non-existence of S and 
the second does not; thus conversion per accidens becomes 
invalid, and some moods of the syllogism are fallacious, e.g. 
Darapti : " All M is S, all M is P, therefore some S is P," which 
fails if there is no M. 

The notion of " existence " has several forms, one of which 
will occupy us in the next chapter ; but the fundamental form 
is that which is derived immediately from the notion of " some 
times true." We say that an argument a (< satisfies " a function 
fa if <f>a is true ; this is the same sense in which the roots of an 
equation are said to satisfy the equation. Now if fa is sometimes 
true, we may say there are #'s for which it is true, or we may say 
" arguments satisfying fa exist" This is the fundamental mean 
ing of the word " existence." Other meanings are either derived 
from this, or embody mere confusion of thought. We may 
correctly say " men exist," meaning that " x is a man " is some 
times true. But if we make a pseudo-syllogism : " Men exist, 
Socrates is a man, therefore Socrates exists," we are talking 
nonsense, since " Socrates " is not, like " men," merely an un 
determined argument to a given prepositional function. The 
fallacy is closely analogous to that of the argument : " Men are 
numerous, Socrates is a man, therefore Socrates is numerous." 
In this case it is obvious that the conclusion is nonsensical, but 



Prepositional Functions 165 

in the case of existence it is not obvious, for reasons which will 
appear more fully in the next chapter. For the present let us 
merely note the fact that, though it is correct to say " men exist," 
it is incorrect, or rather meaningless, to ascribe existence to a 
given particular x who happens to be a man. Generally, " terms 
satisfying fa exist" means "fa is sometimes true"; but "a 
exists " (where a is a term satisfying fa) is a mere noise or shape, 
devoid of significance. It will be found that by bearing in mind 
this simple fallacy we can solve many ancient philosophical 
puzzles concerning the meaning of existence. 

Another set of notions as to which philosophy has allowed 
itself to fall into hopeless confusions through not sufficiently 
separating propositions and prepositional functions are the 
notions of " modality " : necessary, possible, and impossible. 
(Sometimes contingent or assertoric is used instead of possible) 
The traditional view was that, among true propositions, some 
were necessary, while others were merely contingent or assertoric ; 
while among false propositions some were impossible, namely, 
those whose contradictories were necessary, while others merely 
happened not to be true. In fact, however, there was never 
any clear account of what was added to truth by the conception 
of necessity. In the case of prepositional functions, the three 
fold division is obvious. If " fa " is an undetermined value of a 
certain prepositional function, it will be necessary if the function 
is always true, possible if it is sometimes true, and impossible if 
it is never true. This sort of situation arises in regard to prob 
ability, for example. Suppose a ball x is drawn from a bag 
which contains a number of balls : if all the balls are white, 
" x is white " is necessary ; if some are white, it is possible ; 
if none, it is impossible. Here all that is known about x is that 
it satisfies a certain prepositional function, namely, " x was a 
ball in the bag." This is a situation which is general in prob 
ability problems and not uncommon in practical life e.g. when 
a person calls of whom we know nothing except that he brings 
a letter of introduction from our friend so-and-so. In all such 



1 66 Introduction to Mathematical Philosophy 

cases, as in regard to modality in general, the prepositional 
function is relevant. For clear thinking, in many very diverse 
directions, the habit of keeping prepositional functions sharply 
separated from propositions is of the utmost importance, and 
the failure to do so in the past has been a disgrace to 
philosophy. 



CHAPTER XVI 

DESCRIPTIONS 

WE dealt in the preceding chapter with the words all and some ; 
in this chapter we shall consider the word the in the singular, 
and in the next chapter we shall consider the word the in the 
plural. It may be thought excessive to devote two chapters 
to one word, but to the philosophical mathematician it is a 
word of very great importance : like Browning's Grammarian 
with the enclitic Se, I would give the doctrine of this word if I 
were " dead from the waist down " and not merely in a prison. 

We have already had occasion to mention " descriptive 
functions," i.e. such expressions as " the father of x " or " the sine 
of x" These are to be defined by first defining " descriptions." 

A " description " may be of two sorts, definite and indefinite 
(or ambiguous). An indefinite description is a phrase of the 
form " a so-and-so," and a definite description is a phrase of 
the form " the so-and-so " (in the singular). Let us begin with 
the former. 

" Who did you meet ? " "I met a man." " That is a very 
indefinite description." We are therefore not departing from 
usage in our terminology. Our question is : What do I really 
assert when I assert " I met a man " ? Let us assume, for the 
moment, that my assertion is true, and that in fact I met Jones. 
It is clear that what I assert is not " I met Jones." I may say 
" I met a man, but it was not Jones " ; in that case, though I lie, 
I do not contradict myself, as I should do if when I say I met a 

167 



1 68 Introduction to Mathematical Philosophy 

man I really mean that I met Jones. It is clear also that the 
person to whom I am speaking can understand what I say, even 
if he is a foreigner and has never heard of Jones. 

But we may go further : not only Jones, but no actual man, 
enters into my statement. This becomes obvious when the state 
ment is false, since then there is no more reason why Jones 
should be supposed to enter into the proposition than why any 
one else should. Indeed the statement would remain significant, 
though it could not possibly be true, even if there were no man 
at all. " I met a unicorn " or " I met a sea-serpent " is a 
perfectly significant assertion, if we know what it would be to 
be a unicorn or a sea-serpent, i.e. what is the definition of these 
fabulous monsters. Thus it is only what we may call the concept 
that enters into the proposition. In the case of " unicorn," 
for example, there is only the concept : there is not also, some 
where among the shades, something unreal which may be called 
" a unicorn." Therefore, since it is significant (though false) 
to say " I met a unicorn," it is clear that this proposition, rightly 
analysed, does not contain a constituent " a unicorn," though 
it does contain the concept " unicorn." 

The question of " unreality," which confronts us at this 
point, is a very important one. Misled by grammar, the great 
majority of those logicians who have dealt with this question 
have dealt with it on mistaken lines. They have regarded 
grammatical form as a surer guide in analysis than, in fact, 
it is. And they have not known what differences in gram 
matical form are important. " I met Jones " and " I met a 
man " would count traditionally as propositions of the same form, 
but in actual fact they are of quite different forms : the first 
names an actual person, Jones ; while the second involves a 
prepositional function, and becomes, when made explicit : " The 
function ' I met x and x is human ' is sometimes true." (It 
will be remembered that we adopted the convention of using 
" sometimes " as not implying more than once.) This proposi 
tion is obviously not of the form " I met x," which accounts 



Descriptions 1 69 

for the existence of the proposition " I met a unicorn " in spite 
of the fact that there is no such thing as " a unicorn." 

For want of the apparatus of prepositional functions, many 
logicians have been driven to the conclusion that there are 
unreal objects. It is argued, e.g. by Meinong, 1 that we can 
speak about " the golden mountain," " the round square," 
and so on ; we can make true propositions of which these are 
the subjects ; hence they must have some kind of logical being, 
since otherwise the propositions in which they occur would be 
meaningless. In such theories, it seems to me, there is a failure 
of that feeling for reality which ought to be preserved even in 
the most abstract studies. Logic, I should maintain, must no 
more admit a unicorn than zoology can ; for logic is concerned 
with the real world just as truly as zoology, though with its 
more abstract and general features. To say that unicorns have 
an existence in heraldry, or in literature, or in imagination, 
is a most pitiful and paltry evasion. What exists in heraldry 
is not an animal, rrlade of flesh and blood, moving and breathing 
of its own initiative. What exists is a picture, or a description 
in words. Similarly, to maintain that Hamlet, for example, 
exists in his own world, namely, in the world of Shakespeare's 
imagination, just as truly as (say) Napoleon existed in the 
ordinary world, is to say something deliberately confusing, or 
else confused to a degree which is scarcely credible. There is 
only one world, the " real " world : Shakespeare's imagination 
is part of it, and the thoughts that he had in writing Hamlet 
are real. So are the thoughts that we have in reading the play. 
But it is of the very essence of fiction that only the thoughts, 
feelings, etc., in Shakespeare and his readers are real, and that 
there is not, in addition to them, an objective Hamlet. When 
you have taken account of all the feelings roused by Napoleon 
in writers and readers of history, you have not touched the actual 
man ; but in the case of Hamlet you have come to the end of 
him. If no one thought about Hamlet, there would be nothing 
1 Untersuchungen zur Gegenstandstheorie und Psychologic, 1904. 



170 Introduction to Mathematical Philosophy 

left of him ; if no one had thought about Napoleon, he would 
have soon seen to it that some one did. The sense of reality is 
vital in logic, and whoever juggles with it by pretending that 
Hamlet has another kind of reality is doing a disservice to 
thought. A robust sense of reality is very necessary in framing 
a correct analysis of propositions about unicorns, golden moun 
tains, round squares, and other such pseudo-objects. 

In obedience to the feeling of reality, we shall insist that, 
in the analysis of propositions, nothing " unreal " is to be 
admitted. But, after all, if there is nothing unreal, how, it 
may be asked, could we admit anything unreal ? The reply 
is that, in dealing with propositions, we are dealing in the first 
instance with symbols, and if we attribute significance to groups 
of symbols which have no significance, we shall fall into the 
error of admitting unrealities, in the only sense in which this is 
possible, namely, as objects described. In the proposition 
" I met a unicorn," the whole four words together make a signi 
ficant proposition, and the word " unicorn " by itself is significant, 
in just the same sense as the word " man." But the two words 
" a unicorn " do not form a subordinate group having a meaning 
of its own. Thus if we falsely attribute meaning to these two 
words, we find ourselves saddled with " a unicorn," and with 
the problem how there can be such a thing in a world where 
there are no unicorns. " A unicorn " is an indefinite descrip 
tion which describes nothing. It is not an indefinite description 
which describes something unreal. Such a proposition as 
" x is unreal " only has meaning when " x " is a description, 
definite or indefinite ; in that case the proposition will be true 
if " x " is a description which describes nothing. But whether 
the description " x " describes something or describes nothing, 
it is in any case not a constituent of the proposition in which it 
occurs ; like " a unicorn " just now, it is not a subordinate group 
having a meaning of its own. All this results from the fact that, 
when " x " is a description, " x is unreal " or " x does not exist " 
is not nonsense, but is always significant and sometimes true. 



Descriptions 171 

We may now proceed to define generally the meaning of 
propositions which contain ambiguous descriptions. Suppose 
we wish to make some statement about " a so-and-so," where 
"so-and-so's" are those objects that have a certain property 
<, i.e. those objects x for which the prepositional function (fax is 
true. (E.g. if we take " a man " as our instance of " a so-and-so," 
t/)X will be " x is human.") Let us now wish to assert the property 
ifj of " a so-and-so," i.e. we wish to assert that " a so-and-so " has 
that property which x has when i/jx is true. (E.g. in the case 
of " I met a man," ifix will be " I met #.") Now the proposition 
that " a so-and-so " has the property ift is not a proposition of 
the form " 0#." If it were, " a so-and-so " would have to be 
identical with x for a suitable x ; and although (in a sense) this 
may be true in some cases, it is certainly not true in such a case 
as " a unicorn." It is just this fact, that the statement that a 
so-and-so has the property ijj is not of the form ifrx, which makes 
it possible for " a so-and-so " to be, in a certain clearly definable 
sense, " unreal." The definition is as follows : 

The statement that " an object having the property ^ has 

the property ift " 
means : 

" The joint assertion of <f>x and i/ix is not always false." 

So far as logic goes, this is the same proposition as might 
be expressed by " some <'s are ^'s " ; but rhetorically there is 
a difference, because in the one case there is a suggestion of 
singularity, and in the other case of plurality. This, however, 
is not the important point. The important point is that, when 
rightly analysed, propositions verbally about " a so-and-so " 
are found to contain no constituent represented by this phrase. 
And that is why such propositions can be significant even when 
there is no such thing as a so-and-so. 

The definition of existence, as applied to ambiguous descrip 
tions, results from what was said at the end of the preceding 
chapter. We say that " men exist " or " a man exists " if the 



172 Introduction to Mathematical Philosophy 

prepositional function " x is human " is sometimes true ; and 
generally " a so-and-so " exists if " x is so-and-so " is sometimes 
true. We may put this in other language. The proposition 
" Socrates is a man " is no doubt equivalent to " Socrates is 
human," but it is not the very same proposition. The is of 
" Socrates is human " expresses the relation of subject and 
predicate ; the is of " Socrates is a man " expresses identity. 
It is a disgrace to the human race that it has chosen to employ 
the same word " is " for these two entirely different ideas a 
disgrace which a symbolic logical language of course remedies. 
The identity in " Socrates is a man " is identity between an 
object named (accepting " Socrates " as a name, subject to 
qualifications explained later) and an object ambiguously 
described. An object ambiguously described will " exist " when 
at least one such proposition is true, i.e. when there is at least 
one true proposition of the form " x is a so-and-so," where " x " 
is a name. It is characteristic of ambiguous (as opposed to 
definite) descriptions that there may be any number of true 
propositions of the above form Socrates is a man, Plato is a 
man, etc. Thus " a man exists " follows from Socrates, or 
Plato, or anyone else. With definite descriptions, on the other 
hand, the corresponding form of proposition, namely, " x is the 
so-and-so " (where " x " is a name), can only be true for one 
value of x at most. This brings us to the subject of definite 
descriptions, which are to be defined in a way analogous to 
that employed for ambiguous descriptions, but rather more 
complicated. 

We come now to the main subject of the present chapter, 
namely, the definition of the word the (in the singular). One 
very important point about the definition of " a so-and-so " 
applies equally to " the so-and-so " ; the definition to be sought 
is a definition of propositions in which this phrase occurs, not a 
definition of the phrase itself in isolation. In the case of " a 
so-and-so," this is fairly obvious : no one could suppose that 
" a man " was a definite object, which could be defined by itself. 



Descriptions 173 

Socrates is a man, Plato is a man, Aristotle is a man, but we 
cannot infer that " a man " means the same as " Socrates " 
means and also the same as " Plato " means and also the same 
as " Aristotle " means, since these three names have different 
meanings. Nevertheless, when we have enumerated all the 
men in the world, there is nothing left of which we can say, 
" This is a man, and not only so, but it is the ' a man,' the quintes 
sential entity that is just an indefinite man without being any 
body in particular." It is of course quite clear that whatever 
there is in the world is definite : if it is a man it is one definite 
man and not any other. Thus there cannot be such an entity 
as " a man " to be found in the world, as opposed to specific 
man. And accordingly it is natural that we do not define " a 
man " itself, but only the propositions in which it occurs. 

In the case of " the so-and-so " this is equally true, though 
at first sight less obvious. We may demonstrate that this must 
be the case, by a consideration of the difference between a name 
and a definite description. Take the proposition, " Scott is the 
author of Waverley" We have here a name, " Scott," and a 
description, " the author of Waverley" which are asserted to 
apply to the same person. The distinction between a name and 
all other symbols may be explained as follows : 

A name is a simple symbol whose meaning is something that 
can only occur as subject, i.e. something of the kind that, in 
Chapter XIII., we defined as an " individual " or a " particular." 
And a " simple " symbol is one which has no parts that are 
symbols. Thus " Scott " is a simple symbol, because, though it 
has parts (namely, separate letters), these parts are not symbols. 
On the other hand, " the author of Waverley " is not a simple 
symbol, because the separate words that compose the phrase 
are parts which are symbols. If, as may be the case, whatever 
seems to be an " individual " is really capable of further analysis, 
we shall have to content ourselves with what may be called 
" relative individuals," which will be terms that, throughout 
the context in question, are never analysed and never occur 



174 Introduction to Mathematical Philosophy 

otherwise than as subjects. And in that case we shall have 
correspondingly to content ourselves with " relative names." 
From the standpoint of our present problem, namely, the defini 
tion of descriptions, this problem, whether these are absolute 
names or only relative names, may be ignored, since it con 
cerns different stages in the hierarchy of " types," whereas we 
have to compare such couples as " Scott " and " the author of 
Waverley" which both apply to the same object, and do not 
raise the problem of types. We may, therefore, for the moment, 
treat names as capable of being absolute ; nothing that we shall 
have to say will depend upon this assumption, but the wording 
may be a little shortened by it. 

We have, then, two things to compare : (i) a name, which 
is a simple symbol, directly designating an individual which 
is its meaning, and having this meaning in its own right, in 
dependently of the meanings of all other words ; (2) a description, 
which consists of several words, whose meanings are already 
fixed, and from which results whatever is to be taken as the 
" meaning " of the description. 

A proposition containing a description is not identical with 
what that proposition becomes when a name is substituted, 
even if the name names the same object as the description 
describes. " Scott is the author of Waverley " is obviously a 
different proposition from " Scott is Scott " : the first is a fact 
in literary history, the second a trivial truism. And if we put 
anyone other than Scott in place of " the author of Waverley" 
our proposition would become false, and would therefore certainly 
no longer be the same proposition. But, it may be said, our 
proposition is essentially of the same form as (say) " Scott is 
Sir Walter," in which two names are said to apply to the same 
person. The reply is that, if " Scott is Sir Walter " really means 
" the person named e Scott ' is the person named ' Sir Walter,' ' 
then the names are being used as descriptions : i.e. the individual, 
instead of being named, is being described as the person having 
that name. This is a way in which names are frequently used 



Descriptions 175 

in practice, and there will, as a rule, be nothing in the phraseology 
to show whether they are being used in this way or as names. 
When a name is used directly, merely to indicate what we are 
speaking about, it is no part of the fact asserted, or of the falsehood 
if our assertion happens to be false : it is merely part of the 
symbolism by which we express our thought. What we want 
to express is something which might (for example) be translated 
into a foreign language ; it is something for which the actual 
words are a vehicle, but of which they are no part. On the other 
hand, when we make a proposition about " the person called 
' Scott,' " the actual name " Scott " enters into what we are 
asserting, and not merely into the language used in making the 
assertion. Our proposition will now be a different one if we 
substitute " the person called ' Sir Walter.' ' But so long as 
we are using names as names, whether we say " Scott " or whether 
we say " Sir Walter " is as irrelevant to what we are asserting 
as whether we speak English or French. Thus so long as names 
are used as names, " Scott is Sir Walter " is the same trivial 
proposition as " Scott is Scott." This completes the proof that 
" Scott is the author of Waverley " is not the same proposition 
as results from substituting a name for " the author of Waverley" 
no matter what name may be substituted. 

When we use a variable, and speak of a propositional function, 
(/>x say, the process of applying general statements about x to 
particular cases will consist in substituting a name for the letter 
" x" assuming that ^ is a function which has individuals for its 
arguments. Suppose, for example, that <j>x is " always true " ; 
let it be, say, the " law of identity," x=x. Then we may sub 
stitute for " x " any name we choose, and we shall obtain a true 
proposition. Assuming for the moment that " Socrates," 
" Plato," and " Aristotle " are names (a very rash assumption), 
we can infer from the law of identity that Socrates is Socrates, 
Plato is Plato, and Aristotle is Aristotle. But we shall commit 
a fallacy if we attempt to infer, without further premisses, that 
the author of Waverley is the author of Waverley. This results 



176 Introduction to Mathematical Philosophy 

from what we have just proved, that, if we substitute a name for 
" the author of Waverley " in a proposition, the proposition 
we obtain is a different one. That is to say, applying the result 
to our present case : If " x " is a name, " x=x " is not the same 
proposition as " the author of Waverley is the author of Waverley" 
no matter what name " x " may be. Thus from the fact that 
all propositions of the form " x=x " are true we cannot infer, 
without more ado, that the author of Waverley is the author of 
Waverley. In fact, propositions of the form " the so-and-so 
is the so-and-so " are not always true : it is necessary that the 
so-and-so should exist (a term which will be explained shortly). 
It is false that the present King of France is the present King of 
France, or that the round square is the round square. When we 
substitute a description for a name, prepositional functions 
which are " always true " may become false, if the description 
describes nothing. There is no mystery in this as soon as we 
realise (what was proved in the preceding paragraph) that when 
we substitute a description the result is not a value of the 
propositional function in question. 

We are now in a position to define propositions in which a 
definite description occurs. The only thing that distinguishes 
" the so-and-so " from " a so-and-so " is the implication of 
uniqueness. We cannot speak of " the inhabitant of London," 
because inhabiting London is an attribute which is not unique. 
We cannot speak about " the present King of France," because 
there is none ; but we can speak about " the present King of 
England." Thus propositions about " the so-and-so " always 
imply the corresponding propositions about " a so-and-so," 
with the addendum that there is not more than one so-and-so. 
Such a proposition as " Scott is the author of Waverley " could 
not be true if Waverley had never been written, or if several 
people had written it ; and no more could any other proposition 
resulting from a propositional function x by the substitution 
of " the author of Waverley " for " x." We may say that " the 
author of Waverley " means " the value of x for which ( x wrote 



Descriptions 177 

Waverley ' is true." Thus the proposition " the author of 
Waverley was Scotch," for example, involves : 

(1) " x wrote Waverley " is not always false ; 

(2) " if x and y wrote Waverley, x and y are identical " is 

always true ; 

(3) " if x wrote Waverley, x was Scotch " is always true. 

These three propositions, translated into ordinary language, 
state : 

(1) at least one person wrote Waverley ; 

(2) at most one person wrote Waverley ; 

(3) whoever wrote Waverley was Scotch. 

All these three are implied by " the author of Waverley was 
Scotch." Conversely, the three together (but no two of them) 
imply that the author of Waverley was Scotch. Hence the 
three together may be taken as defining what is meant by the 
proposition " the author of Waverley was Scotch." 

We may somewhat simplify these three propositions. The 
first and second together are equivalent to : " There is a term 
c such that ' x wrote Waverley ' is true when x is c and is false 
when x is not c ." In other words, " There is a term c such that 
* x wrote Waverley ' is always equivalent to * x is c. 9 " (Two 
propositions are " equivalent " when both are true or both are 
false.) We have here, to begin with, two functions of x, " x 
wrote Waverley " and " x is r," and we form a function of c by 
considering the equivalence of these two functions of x for all 
values of x ; we then proceed to assert that the resulting function 
of c is " sometimes true," i.e. that it is true for at least one value 
of c. (It obviously cannot be true for more than one value of c .) 
These two conditions together are defined as giving the meaning 
of " the author of Waverley exists." 

We may now define " the term satisfying the function <f>x 
exists." This is the general form of which the above is a par 
ticular case. " The author of Waverley " is " the term satisfying 
the function ' x wrote Waverley' " And " the so-and-so " will 

12 



178 Introduction to Mathematical Philosophy 

always involve reference to some prepositional function, namely, 
that which defines the property that makes a thing a so-and-so. 
Our definition is as follows : 

" The term satisfying the function fa exists " means : 
" There is a term c such that fa is always equivalent to ' x is c? ' 
In order to define " the author of Waverley was Scotch," 
we have still to take account of the third of our three proposi 
tions, namely, " Whoever wrote Waverley was Scotch." This 
will be satisfied by merely adding that the c in question is to 
be Scotch. Thus " the author of Waverley was Scotch " is : 

" There is a term c such that (i) * x wrote Waverley 9 is always 
equivalent to ' x is cj (2) c is Scotch." 

And generally : " the term satisfying </>x satisfies fa " is 
defined as meaning : 

" There is a term c such that (i) <{>x is always equivalent to 
' x is c, 9 (2) ific is true." 

This is the definition of propositions in which descriptions occur. 
It is possible to have much knowledge concerning a term 
described, i.e. to know many propositions concerning " the so- 
and-so," without actually knowing what the so-and-so is, i.e. 
without knowing any proposition of the form " x is the so-and-so," 
where " x " is a name. In a detective story propositions about 
" the man who did the deed " are accumulated, in the hope 
that ultimately they will suffice to demonstrate that it was 
A who did the deed. We may even go so far as to say that, 
in all such knowledge as can be expressed in words with the 
exception of " this " and " that " and a few other words of 
which the meaning varies on different occasions no names, 
in the strict sense, occur, but what seem like names are really 
descriptions. We may inquire significantly whether Homer 
existed, which we could not do if " Homer " were a name. The 
proposition " the so-and-so exists " is significant, whether 
true or false ; but if a is the so-and-so (where " a " is a name), 
the words " a exists " are meaningless. It is only of descriptions 



Descriptions 179 

definite or indefinite that existence can be significantly 
asserted ; for, if "a " is a name, it must name something : what 
does not name anything is not a name, and therefore, if intended 
to be a name, is a symbol devoid of meaning, whereas a descrip 
tion, like " the present King of France," does not become in 
capable of occurring significantly merely on the ground that it 
describes nothing, the reason being that it is a complex symbol, 
of which the meaning is derived from that of its constituent 
symbols. And so, when we ask whether Homer existed, we are 
using the word " Homer " as an abbreviated description : we 
may replace it by (say) " the author of the Iliad and the Odyssey" 
The same considerations apply to almost all uses of what look 
like proper names. 

When descriptions occur in propositions, it is necessary to 
distinguish what may be called " primary " and " secondary " 
occurrences. The abstract distinction is as follows. A descrip 
tion has a " primary " occurrence when the proposition in 
which it occurs results from substituting the description for 
" x " in some prepositional function (/>x ; a description has a 
" secondary " occurrence when the result of substituting the 
description for x in <j>x gives only part of the proposition con 
cerned. An instance will make this clearer. Consider " the 
present King of France is bald." Here " the present King of 
France " has a primary occurrence, and the proposition is false. 
Every proposition in which a description which describes nothing 
has a primary occurrence is false. But now consider " the 
present King of France is not bald." This is ambiguous. If 
we are first to take " x is bald," then substitute " the present 
King of France " for " x" and then deny the result, the occurrence 
of " the present King of France " is secondary and our proposition 
is true ; but if we are to take " x is not bald " and substitute 
" the present King of France " for " x" then " the present 
King of France " has a primary occurrence and the proposition 
is false. Confusion of primary and secondary occurrences is a 
ready source of fallacies where descriptions are concerned. 



i8o Introduction to Mathematical Philosophy 

Descriptions occur in mathematics chiefly in the form of 
descriptive functions, i.e. " the term having the relation R to 
y," or " the R of y " as we may say, on the analogy of " the 
father of y " and similar phrases. To say " the father of y is 
rich," for example, is to say that the following prepositional 
function of c : " c is rich, and ' x begat y ' is always equivalent 
to ' x is cj " is " sometimes true," i.e. is true for at least one 
value of c. It obviously cannot be true for more than one 
value. 

The theory of descriptions, briefly outlined in the present 
chapter, is of the utmost importance both in logic and in theory 
of knowledge. But for purposes of mathematics, the more 
philosophical parts of the theory are not essential, and have 
therefore been omitted in the above account, which has confined 
itself to the barest mathematical requisites. 



CHAPTER XVIi 

CLASSES 

IN the present chapter we shall be concerned with the in the 
plural : the inhabitants of London, the sons of rich men, and 
so on. In other words, we shall be concerned with classes. We 
saw in Chapter II. that a cardinal number is to be defined as a 
class of classes, and in Chapter III. that the number I is to be 
defined as the class of all unit classes, i.e. of all that have just 
one member, as we should say but for the vicious circle. Of 
course, when the number I is defined as the class of all unit 
classes, " unit classes " must be defined so as not to assume 
that we know what is meant by " one " ; in fact, they are defined 
in a way closely analogous to that used for descriptions, namely : 
A class a is said to be a " unit " class if the prepositional function 
" * x is an a ' is always equivalent to ' x is c 9 " (regarded as a 
function of c) is not always false, i.e., in more ordinary language, 
if there is a term c such that x will be a member of a when x is c 
but not otherwise. This gives us a definition of a unit class if we 
already know what a class is in general. Hitherto we have, in 
dealing with arithmetic, treated " class " as a primitive idea. 
But, for the reasons set forth in Chapter XIII., if for no others, 
we cannot accept " class " as a primitive idea. We must seek a 
definition on the same lines as the definition of descriptions, 
i.e. a definition which will assign a meaning to propositions in 
whose verbal or symbolic expression words or symbols apparently 
representing classes occur, but which will assign a meaning that 
altogether eliminates all mention of classes from a right analysis 

181 



1 82 Introduction to Mathematical Philosophy 

of such propositions. We shall then be able to say that the 
symbols for classes are mere conveniences, not representing 
objects called " classes," and that classes are in fact, like descrip 
tions, logical fictions, or (as we say) " incomplete symbols." 

The theory of classes is less complete than the theory of descrip 
tions, and there are reasons (which we shall give in outline) 
for regarding the definition of classes that will be suggested as 
not finally satisfactory. Some further subtlety appears to be 
required ; but the reasons for regarding the definition which 
will be offered as being approximately correct and on the right 
lines are overwhelming. 

The first thing is to realise why classes cannot be regarded 
as part of the ultimate furniture of the world. It is difficult 
to explain precisely what one means by this statement, but one 
consequence which it implies may be used to elucidate its meaning. 
If we had a complete symbolic language, with a definition for 
everything definable, and an undefined symbol for everything 
indefinable, the undefined symbols in this language would repre 
sent symbolically what I mean by " the ultimate furniture of 
the world." I am maintaining that no symbols either for " class " 
in general or for particular classes would be included in this 
apparatus of undefined symbols. On the other hand, all the 
particular things there are in the world would have to have 
names which would be included among undefined symbols. 
We might try to avoid this conclusion by the use of descriptions. 
Take (say) " the last thing Cassar saw before he died." This 
is a description of some particular ; we might use it as (in one 
perfectly legitimate sense) a definition of that particular. But 
if " a " is a name for the same particular, a proposition in which 
" a " occurs is not (as we saw in the preceding chapter) identical 
with what this proposition becomes when for " a " we substitute 
" the last thing Caesar saw before he died." If our language 
does not contain the name " a" or some other name for the same 
particular, we shall have no means of expressing the proposition 
which we expressed by means of " a " as opposed to the one that 



Classes 183 

we expressed by means of the description. Thus descriptions 
would not enable a perfect language to dispense with names for 
all particulars. In this respect, we are maintaining, classes 
differ from particulars, and need not be represented by undefined 
symbols. Our first business is to give the reasons for this opinion. 

We have already seen that classes cannot be regarded as a 
species of individuals, on account of the contradiction about 
classes which are not members of themselves (explained in 
Chapter XIIL), and because we can prove that the number of 
classes is greater than the number of individuals. 

We cannot take classes in the pure extensional way as simply 
heaps or conglomerations. If we were to attempt to do that, 
we should find it impossible to understand how there can be such 
a class as the null-class, which has no members at all and cannot 
be regarded as a " heap " ; we should also find it very hard to 
understand how it comes about that a class which has only one 
member is not identical with that one member. I do not mean 
to assert, or to deny, that there are such entities as " heaps." 
As a mathematical logician, I am not called upon to have an 
opinion on this point. All that I am maintaining is that, if there 
are such things as heaps, we cannot identify them with the classes 
composed of their constituents. 

We shall come much nearer to a satisfactory theory if we 
try to identify classes with prepositional functions. Every 
class, as we explained in Chapter II., is defined by some pro- 
positional function which is true of the members of the class 
and false of other things. But if a class can be defined by one 
prepositional function, it can equally well be defined by any 
other which is true whenever the first is true and false when 
ever the first is false. For this reason the class cannot be identi 
fied with any one such prepositional function rather than with 
any other and given a prepositional function, there are always 
many others which are true when it is true and false when it is 
false. We say that two prepositional functions are " formally 
equivalent " when this happens. Two propositions are " equiva- 



184 Introduction to Mathematical Philosophy 

lent " when both are true or both false ; two prepositional 
functions <f>x, ifjx are " formally equivalent " when <frx is always 
equivalent to iftx. It is the fact that there are other functions 
formally equivalent to a given function that makes it impossible 
to identify a class with a function ; for we wish classes to be such 
that no two distinct classes have exactly the same members, 
and therefore two formally equivalent functions will have to 
determine the same class. 

When we have decided that classes cannot be things of the 
same sort as their members, that they cannot be just heaps or 
aggregates, and also that they cannot be identified with pro- 
positional functions, it becomes very difficult to see what they 
can be, if they are to be more than symbolic fictions. And if 
we can find any way of dealing with them as symbolic fictions, 
we increase the logical security of our position, since we avoid 
the need of assuming that there are classes without being com 
pelled to make the opposite assumption that there are no classes. 
We merely abstain from both assumptions. This is an example 
of Occam's razor, namely, " entities are not to be multiplied 
without necessity." But when we refuse to assert that there 
are classes, we must not be supposed to be asserting dogmatically 
that there are none. We are merely agnostic as regards them : 
like Laplace, we can say, " je n'ai pas besoin de cette hypotbese." 

Let us set forth the conditions that a symbol must fulfil if 
it is to serve as a class. I think the following conditions will 
be found necessary and sufficient : 

(i) Every prepositional function must determine a class, 
consisting of those arguments for which the function is true. 
Given any proposition (true or false), say about Socrates, we 
can imagine Socrates replaced by Plato or Aristotle or a gorilla 
or the man in the moon or any other individual in the world. 
In general, some of these substitutions will give a true proposition 
and some a false one. The class determined will consist of all 
those substitutions that give a true one. Of course, we have 
still to decide what we mean by " all those which, etc." All that 



Classes 1 8 5 

we are observing at present is that a class is rendered determinate 
by a prepositional function, and that every propositional function 
determines an appropriate class. 

(2) Two formally equivalent propositional functions must 
determine the same class, and two which are not formally equiva 
lent must determine different classes. That is, a class is deter 
mined by its membership, and no two different classes can have 
the same membership. (If a class is determined by a function 
<f>x, we say that a is a " member " of the class if c/>a is true.) 

(3) We must find some way of defining not only classes, but 
classes of classes. We saw in Chapter II. that cardinal numbers 
are to be defined as classes of classes. The ordinary phrase 
of elementary mathematics, " The combinations of n things 
m at a time " represents a class of classes, namely, the class of 
all classes of m terms that can be selected out of a given class 
of n terms. Without some symbolic method of dealing with 
classes of classes, mathematical logic would break down. 

(4) It must under all circumstances be meaningless (not false) 
to suppose a class a member of itself or not a member of itself. 
This results from the contradiction which we discussed in 
Chapter XIII. 

(5) Lastly and this is the condition which is most difficult 
of fulfilment, it must be possible to make propositions about 
all the classes that are composed of individuals, or about all the 
classes that are composed of objects of any one logical " type." 
If this were not the case, many uses of classes would go astray 
for example, mathematical induction. In defining the posterity 
of a given term, we need to be able to say that a member of the 
posterity belongs to all hereditary classes to which the given 
term belongs, and this requires the sort of totality that is in 
question. The reason there is a difficulty about this condition 
is that it can be proved to be impossible to speak of all the pro- 
positional functions that can have arguments of a given type. 

We will, to begin with, ignore this last condition and the 
problems which it raises. The first two conditions may be 



1 86 Introduction to Mathematical Philosophy 

taken together. They state that there is to be one class, no 
more and no less, for each group of formally equivalent pro- 
positional functions ; e.g. the class of men is to be the same as 
that of featherless bipeds or rational animals or Yahoos or what 
ever other characteristic may be preferred for defining a human 
being. Now, when we say that two formally equivalent pro- 
positional functions may be not identical, although they define 
the same class, we may prove the truth of the assertion by point 
ing out that a statement may be true of the one function and 
false of the other ; e.g. " I believe that all men are mortal " 
may be true, while " I believe that all rational animals are 
mortal " may be false, since I may believe falsely that the 
Phoenix is an immortal rational animal. Thus we are led to 
consider statements about functions, or (more correctly) functions 
of functions. 

Some of the things that may be said about a function may 
be regarded as said about the class defined by the function, 
whereas others cannot. The statement " all men are mortal " 
involves the functions " x is human " and " x is mortal " ; or, 
if we choose, we can say that it involves the classes men and 
mortals. We can interpret the statement in either way, because 
its truth-value is unchanged if we substitute for " x is human " 
or for " x is mortal " any formally equivalent function. But, 
as we have just seen, the statement " I believe that all men are 
mortal " cannot be regarded as being about the class determined 
by either function, because its truth-value may be changed 
by the substitution of a formally equivalent function (which 
leaves the class unchanged). We will call a statement involving 
a function <frx an " extensional " function of the function <#, if 
it is like " all men are mortal," i.e. if its truth-value is unchanged 
by the substitution of any formally equivalent function ; and 
when a function of a function is not extensional, we will call it 
" intensional," so that " I believe that all men are mortal " 
is an intensional function of " x is human " or " x is mortal." 
Thus extensional functions of a function x may, for practical 



Classes 187 

purposes, be regarded as functions of the class determined by 
x, while intensional functions cannot be so regarded. 

It is to be observed that all the specific functions of functions 
that we have occasion to introduce in mathematical logic are 
extensional. Thus, for example, the two fundamental functions 
of functions are : " </>x is always true " and " <j>x is sometimes 
true." Each of these has its truth-value unchanged if any 
formally equivalent function is substituted for </>x. In the 
language of classes, if a is the class determined by </>x 9 " (f>x is 
always true " is equivalent to " everything is a member of a," 
and " </>x is sometimes true " is equivalent to " a has members " 
or (better) " a has at least one member." Take, again, the 
condition, dealt with in the preceding chapter, for the existence 
of " the term satisfying <#." The condition is that there is a 
term c such that $x is always equivalent to " x is c" This 
is obviously extensional. It is equivalent to the assertion 
that the class defined by the function (f>x is a unit class, i.e. a 
class having one member; in other words, a class which is a 
member of I. 

Given a function of a function which may or may not be 
extensional, we can always derive from it a connected and 
certainly extensional function of the same function, by the 
following plan : Let our original function of a function be one 
which attributes to <j>x the property f\ then consider the asser 
tion " there is a function having the property / and formally 
equivalent to <#." This is an extensional function of <f>x ; it 
is true when our original statement is true, and it is formally 
equivalent to the original function of </>x if this original function 
is extensional ; but when the original function is intensional, 
the new one is more often true than the old one. For example, 
consider again " I believe that all men are mortal," regarded 
as a function of " x is human." The derived extensional function 
is : " There is a function formally equivalent to * x is human ' 
and such that I believe that whatever satisfies it is mortal." 
This remains true when we substitute " x is a rational animal " 



1 88 Introduction to Mathematical Philosophy 

for " x is human," even if I believe falsely that the Phoenix is 
rational and immortal. 

We give the name of " derived extensional function " to the 
function constructed as above, namely, to the function : " There 
is a function having the property / and formally equivalent to 
$x," where the original function was " the function j>x has 
the property/." 

We may regard the derived extensional function as having 
for its argument the class determined by the function <f>x, and 
as asserting/ of this class. This may be taken as the definition 
of a proposition about a class. I.e. we may define : 

To assert that " the class determined by the function <f>x 
has the property/" is to assert that <j>x satisfies the extensional 
function derived from/. 

This gives a meaning to any statement about a class which 
can be made significantly about a function ; and it will be 
found that technically it yields the results which are required 
in order to make a theory symbolically satisfactory. 1 

What we have said just now as regards the definition of 
classes is sufficient to satisfy our first four conditions. The 
way in which it secures the third and fourth, namely, the possi 
bility of classes of classes, and the impossibility of a class being 
or not being a member of itself, is somewhat technical ; it is 
explained in Principia Mathematics but may be taken for 
granted here. It results that, but for our fifth condition, we 
might regard our task as completed. But this condition at 
once the most important and the most difficult is not fulfilled 
in virtue of anything we have said as yet. The difficulty is 
connected with the theory of types, and must be briefly discussed. 2 

We saw in Chapter XIII. that there is a hierarchy of logical 
types, and that it is a fallacy to allow an object belonging to 
one of these to be substituted for an object belonging to another. 

1 See Principia Mathematica, vol. i. pp. 75-84 and * 20. 

2 The reader who desires a fuller discussion should consult Principia 
Mathematica, Introduction, chap, ii.; also * 12. 



Classes 189 

Now it is not difficult to show that the various functions which 
can take a given object a as argument are not all of one type. 
Let us call them all ^-functions. We may take first those among 
them which do not involve reference to any collection of functions ; 
these we will call " predicative ^-functions." If we now proceed 
to functions involving reference to the totality of predicative 
^-functions, we shall incur a fallacy if we regard these as of the 
same type as the predicative ^-functions. Take such an every 
day statement as " a is a typical Frenchman." How shall 
we define a " typical " Frenchman ? We may define him as 
one " possessing all qualities that are possessed by most French 
men." But unless we confine " all qualities " to such as do not 
involve a reference to any totality of qualities, we shall have to 
observe that most Frenchmen are not typical in the above sense, 
and therefore the definition shows that to be not typical is 
essential to a typical Frenchman. This is not a logical contra 
diction, since there is no reason why there should be any typical 
Frenchmen; but it illustrates the need for separating off 
qualities that involve reference to a totality of qualities from 
those that do not. 

Whenever, by statements about " all " or " some " of the 
values that a variable can significantly take, we generate a 
new object, this new object must not be among the values which 
our previous variable could take, since, if it were, the totality 
of values over which the variable could range would only be 
definable in terms of itself, and we should be involved in a vicious 
circle. For example, if I say "Napoleon had all the qualities 
that make a great general," I must define " qualities " in such a 
way that it will not include what I am now saying, i.e. " having 
all the qualities that make a great general " must not be itself a 
quality in the sense supposed. This is fairly obvious, and is 
the principle which leads to the theory of types by which vicious- 
circle paradoxes are avoided. As applied to ^-functions, we 
may suppose that " qualities " is to mean " predicative functions." 
Then when I say " Napoleon had all the qualities, etc.," I mean 



190 Introduction to Mathematical Philosophy 

" Napoleon satisfied all the predicative functions, etc." This 
statement attributes a property to Napoleon, but not a pre 
dicative property ; thus we escape the vicious circle. But 
wherever " all functions which " occurs, the functions in question 
must be limited to one type if a vicious circle is to be avoided ; 
and, as Napoleon and the typical Frenchman have shown, the 
type is not rendered determinate by that of the argument. It 
would require a much fuller discussion to set forth this point 
fully, but what has been said may suffice to make it clear that 
the functions which can take a given argument are of an infinite 
series of types. We could, by various technical devices, con 
struct a variable which would run through the first n of these 
types, where n is finite, but we cannot construct a variable which 
will run through them all, and, if we could, that mere fact would 
at once generate a new type of function with the same arguments, 
and would set the whole process going again. 

We call predicative ^-functions the first type of ^-functions ; 
^-functions involving reference to the totality of the first type 
we call the second, type ; and so on. No variable ^-function 
can run through all these different types : it must stop short at 
some definite one. 

These considerations are relevant to our definition of the 
derived extensional function. We there spoke of " a function 
formally equivalent to fa" It is necessary to decide upon 
the type of our function. Any decision will do, but some decision 
is unavoidable. Let us call the supposed formally equivalent 
function 0. Then ^ appears as a variable, and must be of 
some determinate type. All that we know necessarily about 
the type of (/> is that it takes arguments of a given type that 
it is (say) an ^-function. But this, as we have just seen, does 
not determine its type. If we are to be able (as our fifth requisite 
demands) to deal with all classes whose members are of the same 
type as a, we must be able to define all such classes by means of 
functions of some one type ; that is to say, there must be some 
type of ^-function, say the n ih 9 such that any ^-function is formally 



Classes 191 

equivalent to some ^-function of the n th type. If this is the case, 
then any extensional function which holds of all ^-functions 
of the n th type will hold of any ^-function whatever. It is chiefly 
as a technical means of embodying an assumption leading to 
this result that classes are useful. The assumption is called the 
" axiom of reducibility," and may be stated as follows : 

" There is a type (r say) of ^-functions such that, given any 
tf-f unction, it is formally equivalent to some function of the type 
in question." 

If this axiom is assumed, we use functions of this type in 
defining our associated extensional function. Statements about 
all ^-classes (i.e. all classes defined by ^-functions) can be reduced 
to statements about all ^-functions of the type r. So long as 
only extensional functions of functions are involved, this gives 
us in practice results which would otherwise have required the 
impossible notion of " all ^-functions." One particular region 
where this is vital is mathematical induction. 

The axiom of reducibility involves all that is really essential 
in the theory of classes. It is therefore worth while to ask 
whether there is any reason to suppose it true. 

This axiom, like the multiplicative axiom and the axiom 
of infinity, is necessary for certain results, but not for the bare 
existence of deductive reasoning. The theory of deduction, 
as explained in Chapter XIV., and the laws for propositions 
involving " all " and " some," are of the very texture of mathe 
matical reasoning : without them, or something like them, 
we should not merely not obtain the same results, but we should 
not obtain any results at all. We cannot use them as hypo 
theses, and deduce hypothetical consequences, for they are 
rules of deduction as well as premisses. They must be absolutely 
true, or else what we deduce according to them does not even 
follow from the premisses. On the other hand, the axiom of 
reducibility, like our two previous mathematical axioms, could 
perfectly well be stated as an hypothesis whenever it is used, 
instead of being assumed to be actually true. We can deduce 



192 Introduction to Mathematical Philosophy 

its consequences hypothetically ; we can also deduce the con 
sequences of supposing it false. It is therefore only convenient, 
not necessary. And in view of the complication of the theory 
of types, and of the uncertainty of all except its most general 
principles, it is impossible as yet to say whether there may 
not be some way of dispensing with the axiom of reducibility 
altogether. However, assuming the correctness of the theory 
outlined above, what can we say as to the truth or falsehood of 
the axiom ? 

The axiom, we may observe, is a generalised form of Leibniz's 
identity of indiscernibles. Leibniz assumed, as a logical principle, 
that two different subjects must differ as to predicates. Now 
predicates are only some among what we called " predicative 
functions," which will include also relations to given terms, 
and various properties not to be reckoned as predicates. Thus 
Leibniz's assumption is a much stricter and narrower one than 
ours. (Not, of course, according to his logic, which regarded 
all propositions as reducible to the subject-predicate form.) 
But there is no good reason for believing his form, so far as I can 
see. There might quite well, as a matter of abstract logical 
possibility, be two things which had exactly the same predicates, 
in the narrow sense in which we have been using the word " pre 
dicate." How does our axiom look when we pass beyond pre 
dicates in this narrow sense ? In the actual world there seems 
no way of doubting its empirical truth as regards particulars, 
owing to spatio-temporal differentiation : no two particulars 
have exactly the same spatial and temporal relations to all other 
particulars. But this is, as it were, an accident, a fact about 
the world in which we happen to find ourselves. Pure logic, 
and pure mathematics (which is the same thing), aims at being 
true, in Leibnizian phraseology, in all possible worlds, not only 
in this higgledy-piggledy job-lot of a world in which chance has 
imprisoned us. There is a certain lordliness which the logician 
should preserve : he must not condescend to derive arguments 
from the things he sees about him. 



Classes 193 

Viewed from this strictly logical point of view, I do not see 
any reason to believe that the axiom of reducibility is logically 
necessary, which is what would be meant by saying that it is 
true in all possible worlds. The admission of this axiom into 
a system of logic is therefore a defect, even if the axiom is empir 
ically true. It is for this reason that the theory of classes cannot 
be regarded as being as complete as the theory of descriptions. 
There is need of further work on the theory of types, in the hope 
of arriving at a doctrine of classes which does not require such a 
dubious assumption. But it is reasonable to regard the theory 
outlined in the present chapter as right in its main lines, i.e. in 
its reduction of propositions nominally about classes to pro 
positions about their defining functions. The avoidance of 
classes as entities by this method must, it would seem, be sound 
in principle, however the detail may still require adjustment. 
It is because this seems indubitable that we have included the 
theory of classes, in spite of our desire to exclude, as far as possible, 
whatever seemed open to serious doubt. 

The theory of classes, as above outlined, reduces itself to one 
axiom and one definition. For the sake of definiteness, we will 
here repeat them. The axiom is : 

Ther e is a type r such that if $ is a function which can take a 
given object a as argument, then there is a Junction $ of the type 
r which is formally equivalent to <j>. 

The definition is : 

If </) i s a function which can take a given object a as argument, 
and r the type mentioned in the above axiom, then to say that 
the class determined by <j> has the property f is to say that there 
is a function of type T, formally equivalent to <, and having the 
property f. 



CHAPTER XVIII 

MATHEMATICS AND LOGIC 

MATHEMATICS and logic, historically speaking, have been entirely 
distinct studies. Mathematics has been connected with science, 
logic with Greek. But both have developed in modern times : 
logic has become more mathematical and mathematics has 
become more logical. The consequence is that it has now become 
wholly impossible to draw a line between the two ; in fact, the 
two are one. They differ as boy and man : logic is the youth 
of mathematics and mathematics is the manhood of logic. This 
view is resented by logicians who, having spent their time in 
the study of classical texts, are incapable of following a piece 
of symbolic reasoning, and by mathematicians who have learnt 
a technique without troubling to inquire into its meaning or 
justification. Both types are now fortunately growing rarer. 
So much of modern mathematical work is obviously on the 
border-line of logic, so much of modern logic is symbolic and 
formal, that the very close relationship of logic and mathematics 
has become obvious to every instructed student. The proof 
of their identity is, of course, a matter of detail : starting with 
premisses which would be universally admitted to belong to 
logic, and arriving by deduction at results which as obviously 
belong to mathematics, we find that there is no point at which 
a sharp line can be drawn, with logic to the left and mathe 
matics to the right. If there are still those who do not admit 
the identity of logic and mathematics, we may challenge them 
to indicate at what point, in the successive definitions and 

194 



Mathematics and Logic 195 

deductions of Principia Maihematica, they consider that logic 
ends and mathematics begins. It will then be obvious that any 
answer must be quite arbitrary. 

In the earlier chapters of this book, starting from the natural 
numbers, we have first defined " cardinal number " and shown 
how to generalise the conception of number, and have then 
analysed the conceptions involved in the definition, until we found 
ourselves dealing with the fundamentals of logic. In a synthetic, 
deductive treatment these fundamentals come first, and the 
natural numbers are only reached after a long journey. Such 
treatment, though formally more correct than that which we 
have adopted, is more difficult for the reader, because the ultimate 
logical concepts and propositions with which it starts are remote 
and unfamiliar as compared with the natural numbers. Also 
they represent the present frontier of knowledge, beyond which 
is the still unknown ; and the dominion of knowledge over them 
is not as yet very secure. 

It used to be said that mathematics is the science of " quantity." 
" Quantity " is a vague word, but for the sake of argument 
we may replace it by the word " number." The statement 
that mathematics is the science of number would be untrue 
in two different ways. On the one hand, there are recognised 
branches of mathematics which have nothing to do with number 
all geometry that does not use co-ordinates or measurement, 
for example : projective and descriptive geometry, down to 
the point at which co-ordinates are introduced, does not have 
to do with number, or even with quantity in the sense of greater 
and less. On the other hand, through the definition of cardinals, 
through the theory of induction and ancestral relations, through 
the general theory of series, and through the definitions of the 
arithmetical operations, it has become possible to generalise much 
that used to be proved only in connection with numbers. The 
result is that what was formerly the single study of Arithmetic 
has now become divided into numbers of separate studies, no 
one of which is specially concerned with numbers. The most 



196 Introduction to Mathematical Philosophy 

elementary properties of numbers are concerned with one-one 
relations, and similarity between classes. Addition is concerned 
with the construction of mutually exclusive classes respectively 
similar to a set of classes which are not known to be mutually 
exclusive. Multiplication is merged in the theory of " selec 
tions," i.e. of a certain kind of one-many relations. Finitude 
is merged in the general study of ancestral relations, which yields 
the whole theory of mathematical induction. The ordinal 
properties of the various kinds of number-series, and the elements 
of the theory of continuity of functions and the limits of functions, 
can be generalised so as no longer to involve any essential reference 
to numbers. It is a principle, in all formal reasoning, to generalise 
to the utmost, since we thereby secure that a given process of 
deduction shall have more widely applicable results ; we are, 
therefore, in thus generalising the reasoning of arithmetic, 
merely following a precept which is universally admitted in 
mathematics. And in thus generalising we have, in effect, 
created a set of new deductive systems, in which traditional 
arithmetic is at once dissolved and enlarged ; but whether any 
one of these new deductive systems for example, the theory of 
selections is to be said to belong to logic or to arithmetic is 
entirely arbitrary, and incapable of being decided rationally. 

We are thus brought face to face with the question : What 
is this subject, which may be called indifferently either mathe 
matics or logic ? Is there any way in which we can define it ? 

Certain characteristics of the subject are clear. To begin 
with, we do not, in this subject, deal with particular things or 
particular properties : we deal formally with what can be said 
about any thing or any property. We are prepared to say that 
one and one are two, but not that Socrates and Plato are two, 
because, in our capacity of logicians or pure mathematicians, 
we have never heard of Socrates and Plato. A world in which 
there were no such individuals would still be a world in which 
one and one are two. It is not open to us, as pure mathematicians 
or logicians, to mention anything at all, because, if we do so, 



Mathematics and Logic 197 

we introduce something irrelevant and not formal. We may 
make this clear by applying it to the case of the syllogism. 
Traditional logic says : " All men are mortal, Socrates is a man, 
therefore Socrates is mortal." Now it is clear that what we 
mean to assert, to begin with, is only that the premisses imply 
the conclusion, not that premisses and conclusion are actually 
true ; even the most traditional logic points out that the actual 
truth of the premisses is irrelevant to logic. Thus the first 
change to be made in the above traditional syllogism is to state 
it in the form : " If all men are mortal and Socrates is a man, 
then Socrates is mortal." We may now observe that it is intended 
to convey that this argument is valid in virtue of its form, not 
in virtue of the particular terms occurring in it. If we had 
omitted " Socrates is a man " from our premisses, we should 
have had a non-formal argument, only admissible because 
Socrates is in fact a man ; in that case we could not have general 
ised the argument. But when, as above, the argument is formal, 
nothing depends upon the terms that occur in it. Thus we may 
substitute a for men, j8 for mortals, and x for Socrates, where 
and j3 are any classes whatever, and x is any individual. We 
then arrive at the statement : " No matter what possible values 
x and a and j3 may have, if all a's are j8's and x is an a, then x 
is a j8 " ; in other words, " the prepositional function ' if all a's 
are ]8 and x is an a, then x is a j8 ' is always true." Here at last 
we have a proposition of logic the one which is only suggested by 
the traditional statement about Socrates and men and mortals. 

It is clear that, if formal reasoning is what we are aiming at, 
we shall always arrive ultimately at statements like the above, 
in which no actual things or properties are mentioned ; this 
will happen through the mere desire not to waste our time proving 
in a particular case what can be proved generally. It would be 
ridiculous to go through a long argument about Socrates, and then 
go through precisely the same argument again about Plato. If 
our argument is one (say) which holds of all men, we shall prove 
it concerning " x" with the hypothesis " if x is a man." With 



198 Introduction to Mathematical Philosophy 

this hypothesis, the argument will retain its hypothetical validity 
even when x is not a man. But now we shall find that our argu 
ment would still be valid if, instead of supposing x to be a man, 
we were to suppose him to be a monkey or a goose or a Prime 
Minister. We shall therefore not waste our time taking as our 
premiss " x is a man " but shall take " x is an a," where a is any 
class of individuals, or " (f>x " where </) is any prepositional 
function of some assigned type. Thus the absence of all mention 
of particular things or properties in logic or pure mathematics 
is a necessary result of the fact that this study is, as we say, 
" purely formal." 

At this point we find ourselves faced with a problem which 
is easier to state than to solve. The problem is : " What are 
the constituents of a logical proposition ? " I do not know the 
answer, but I propose to explain how the problem arises. 

Take (say) the proposition " Socrates was before Aristotle." 
Here it seems obvious that we have a relation between two terms, 
and that the constituents of the proposition (as well as of the 
corresponding fact) are simply the two terms and the relation, 
i.e. Socrates, Aristotle, and before. (I ignore the fact that 
Socrates and Aristotle are not simple ; also the fact that what 
appear to be their names are really truncated descriptions. 
Neither of these facts is relevant to the present issue.) We may 
represent the general form of such propositions by " x R y," 
which may be read " x has the relation R to y." This general 
form may occur in logical propositions, but no particular instance 
of it can occur. Are we to infer that the general form itself is a 
constituent of such logical propositions ? 

Given a proposition, such as " Socrates is before Aristotle," 
we have certain constituents and also a certain form. But the 
form is not itself a new constituent ; if it were, we should need a 
new form to embrace both it and the other constituents. We 
can, in fact, turn all the constituents of a proposition into 
variables, while keeping the form unchanged. This is what we 
do when we use such a schema as " x R y," which stands for any 



Mathematics and Logic 199 

one of a certain class of propositions, namely, those asserting 
relations between two terms. We can proceed to general asser 
tions, such as " x R y is sometimes true " i.e. there are cases 
where dual relations hold. This assertion will belong to logic 
(or mathematics) in the sense in which we are using the word. 
But in this assertion we do not mention any particular things 
or particular relations ; no particular things or relations can 
ever enter into a proposition of pure logic. We are left with pure 
forms as the only possible constituents of logical propositions. 

I do not wish to assert positively that pure forms e.g. the 
form " x R y " do actually enter into propositions of the kind 
we are considering. The question of the analysis of such pro 
positions is a difficult one, with conflicting considerations on the 
one side and on the other. We cannot embark upon this question 
now, but we may accept, as a first approximation, the view 
that forms are what enter into logical propositions as their 
constituents. And we may explain (though not formally define) 
what we mean by the " form " of a proposition as follows : 

The " form " of a proposition is that, in it, that remains un 
changed when every constituent of the proposition is replaced 
by another. 

Thus " Socrates is earlier than Aristotle " has the same form 
as " Napoleon is greater than Wellington," though every con 
stituent of the two propositions is different. 

We may thus lay down, as a necessary (though not sufficient) 
characteristic of logical or mathematical propositions, that they 
are to be such as can be obtained from a proposition containing 
no variables (i.e. no such words as all, some, a, the, etc.) by turning 
every constituent into a variable and asserting that the result 
is always true or sometimes true, or that it is always true in 
respect of some of the variables that the result is sometimes true 
in respect of the others, or any variant of these forms. And 
another way of stating the same thing is to say that logic (or 
mathematics) is concerned only with forms, and is concerned 
with them only in the way of stating that they are always or 



2oo Introduction to Mathematical Philosophy 

sometimes true with all the permutations of " always " and 
" sometimes " that may occur. 

There are in every language some words whose sole function is 
to indicate form. These words, broadly speaking, are commonest 
in languages having fewest inflections. Take " Socrates is 
human." Here " is " is not a constituent of the proposition, 
but merely indicates the subject-predicate form. Similarly 
in " Socrates is earlier than Aristotle," " is " and " than " 
merely indicate form ; the proposition is the same as " Socrates 
precedes Aristotle," in which these words have disappeared 
and the form is otherwise indicated. Form, as a rule, can be 
indicated otherwise than by specific words : the order of the 
words can do most of what is wanted. But this principle 
must not be pressed. For example, it is difficult to see how we 
could conveniently express molecular forms of propositions 
(i.e. what we call " truth-functions ") without any word at all. 
We saw in Chapter XIV. that one word or symbol is enough for 
this purpose, namely, a word or symbol expressing incompati 
bility. But without even one we should find ourselves in diffi 
culties. This, however, is not the point that is important for 
our present purpose. What is important for us is to observe 
that form may be the one concern of a general proposition, 
even when no word or symbol in that proposition designates 
the form. If we wish to speak about the form itself, we must 
have a word for it ; but if, as in mathematics, we wish to speak 
about all propositions that have the form, a word for the form 
will usually be found not indispensable ; probably in theory it 
is never indispensable. 

Assuming as I think we may that the forms of propositions 
can be represented by the forms of the propositions in which 
they are expressed without any special word for forms, we should 
arrive at a language in which everything formal belonged to 
syntax and not to vocabulary. In such a language we could 
express all the propositions of mathematics even if we did not 
know one single word of the language. The language of mathe- 



Mathematics and Logic 201 

matical logic, if it were perfected, would be such a language. 
We should have symbols for variables, such as " x " and " R " 
and " y," arranged in various ways ; and the way of arrange 
ment would indicate that something was being said to be true of 
all values or some values of the variables. We should not need 
to know any words, because they would only be needed for giving 
values to the variables, which is the business of the applied 
mathematician, not of the pure mathematician or logician. 
It is one of the marks of a proposition of logic that, given a 
suitable language, such a proposition can be asserted in such a 
language by a person who knows the syntax without knowing 
a single word of the vocabulary. 

But, after all, there are words that express form, such as " is " 
and " than." And in every symbolism hitherto invented for 
mathematical logic there are symbols having constant formal 
meanings. We may take as an example the symbol for in 
compatibility which is employed in building up truth-functions. 
Such words or symbols may occur in logic. The question is : 
How are we to define them ? 

Such words or symbols express what are called " logical 
constants." Logical constants may be defined exactly as 
we denned forms ; in fact, they are in essence the same thing. 
A fundamental logical constant will be that which is in common 
among a number of propositions, any one of which can result 
from any other by substitution of terms one for another. For 
example, " Napoleon is greater than Wellington " results from 
" Socrates is earlier than Aristotle " by the substitution of 
"Napoleon" for "Socrates," "Wellington" for "Aristotle," 
and " greater " for " earlier." Some propositions can be obtained 
in this way from the prototype " Socrates is earlier than Aris 
totle " and some cannot ; those that can are those that are of 
the form " x R y," i.e. express dual relations. We cannot obtain 
from the above prototype by term-for-term substitution such 
propositions as " Socrates is human " or " the Athenians gave 
the hemlock to Socrates," because the first is of the subject- 



2O2 Introduction to Mathematical Philosophy 

predicate form and the second expresses a three-term relation. 
If we are to have any words in our pure logical language, they 
must be such as express " logical constants," and " logical 
constants " will always either be, or be derived from, what is in 
common among a group of propositions derivable from each 
other, in the above manner, by term-for-term substitution. And 
this which is in common is what we call " form." 

In this sense all the " constants " that occur in pure mathe 
matics are logical constants. The number I, for example, is 
derivative from propositions of the form : " There is a term c 
such that (f>x is true when, and only when, x is c" This is a 
function of ^, and various different propositions result from 
giving different values to <. We may (with a little omission 
of intermediate steps not relevant to our present purpose) take 
the above function of <f> as what is meant by " the class deter 
mined by ^ is a unit class " or " the class determined by <j> is a 
member of I " (i being a class of classes). In this way, proposi 
tions in which I occurs acquire a meaning which is derived from 
a certain constant logical form. And the same will be found 
to be the case with all mathematical constants : all are logical 
constants, or symbolic abbreviations whose full use in a proper 
context is defined by means of logical constants. 

But although all logical (or mathematical) propositions can 
be expressed wholly in terms of logical constants together with 
variables, it is not the case that, conversely, all propositions 
that can be expressed in this way are logical. We have found 
so far a necessary but not a sufficient criterion of mathematical 
propositions. We have sufficiently defined the character of the 
primitive ideas in terms of which all the ideas of mathematics 
can be defined, but not of the primitive propositions from which 
all the propositions of mathematics can be deduced. This is a 
more difficult matter, as to which it is not yet known what the 
full answer is. 

We may take the axiom of infinity as an example of a pro 
position which, though it can be enunciated in logical terms, 



Mathematics and Logic 203 

cannot be asserted by logic to be true. All the propositions of 
logic have a characteristic which used to be expressed by saying 
that they were analytic, or that their contradictories were self- 
contradictory. This mode of statement, however, is not satis 
factory. The law of contradiction is merely one among logical 
propositions ; it has no special pre-eminence ; and the proof 
that the contradictory of some proposition is self-contradictory 
is likely to require other principles of deduction besides the 
law of contradiction. Nevertheless, the characteristic of logical 
propositions that we are in search of is the one which was felt, 
and intended to be defined, by those who said that it consisted 
in deducibility from the law of contradiction. This character 
istic, which, for the moment, we may call tautology, obviously 
does not belong to the assertion that the number of individuals 
in the universe is , whatever number n may be. But for the 
diversity of types, it would be possible to prove logically that 
there are classes of n terms, where n is any finite integer ; or even 
that there are classes of N terms. But, owing to types, such 
proofs, as we saw in Chapter XIII., are fallacious. We are left 
to empirical observation to determine whether there are as many 
as n individuals in the world. Among " possible " worlds, 
in the Leibnizian sense, there will be worlds having one, two, 
three, . . . individuals. There does not even seem any logical 
necessity why there should be even one individual 1 why, in 
fact, there should be any world at all. The ontological proof 
of the existence of God, if it were valid, would establish the 
logical necessity of at least one individual. But it is generally 
recognised as invalid, and in fact rests upon a mistaken view of 
existence i.e. it fails to realise that existence can only be asserted 
of something described, not of something named, so that it is 
meaningless to argue from " this is the so-and-so " and " the 
so-and-so exists " to " this exists." If we reject the ontological 

1 The primitive propositions in Principia Mathematica are such as to 
allow the inference that at least one individual exists. But I now view 
this as a defect in logical purity. 



204 Introduction to Mathematical Philosophy 

argument, we seem driven to conclude that the existence of a 
world is an accident i.e. it is not logically necessary. If that 
be so, no principle of logic can assert " existence " except under 
a hypothesis, i.e. none can be of the form " the prepositional 
function so-and-so is sometimes true." Propositions of this 
form, when they occur in logic, will have to occur as hypotheses 
or consequences of hypotheses, not as complete asserted pro 
positions. The complete asserted propositions of logic will all 
be such as affirm that some prepositional function is always true. 
For example, it is always true that if p implies q and q implies 
r then p implies r, or that, if all a's are jS's and x is an a then 
x is a ]8. Such propositions may occur in logic, and their truth 
is independent of the existence of the universe. We may lay 
it down that, if there were no universe, all general propositions 
would be true ; for the contradictory of a general proposition 
(as we saw in Chapter XV.) is a proposition asserting existence, 
and would therefore always be false if no universe existed. 

Logical propositions are such as can be known a 'priori, without 
study of the actual world. We only know from a study of 
empirical facts that Socrates is a man, but we know the correct 
ness of the syllogism in its abstract form (i.e. when it is stated 
in terms of variables) without needing any appeal to experience. 
This is a characteristic, not of logical propositions in themselves, 
but of the way in which we know them. It has, however, a 
bearing upon the question what their nature may be, since there 
are some kinds of propositions which it would be very difficult 
to suppose we could know without experience. 

It is clear that the definition of " logic " or " mathematics " 
must be sought by trying to give a new definition of the old 
notion of " analytic " propositions. Although we can no longer 
be satisfied to define logical propositions as those that follow 
from the law of contradiction, we can and must still admit that 
they are a wholly different class of propositions from those that 
we come to know empirically. They all have the characteristic 
which, a moment ago, we agreed to call " tautology." This, 



Mathematics and Logic 205 

combined with the fact that they can be expressed wholly in terms 
of variables and logical constants (a logical constant being some 
thing which remains constant in a proposition even when all 
its constituents are changed) will give the definition of logic 
or pure mathematics. For the moment, I do not know how to 
define " tautology." 1 It would be easy to offer a definition 
which might seem satisfactory for a while ; but I know of none 
that I feel to be satisfactory, in spite of feeling thoroughly 
familiar with the characteristic of which a definition is wanted. 
At this point, therefore, for the moment, we reach the frontier 
of knowledge on our backward journey into the logical founda 
tions of mathematics. 

We have now come to an end of our somewhat summary intro 
duction to mathematical philosophy. It is impossible to convey 
adequately the ideas that are concerned in this subject so long 
as we abstain from the use of logical symbols. Since ordinary 
language has no words that naturally express exactly what we 
wish to express, it is necessary, so long as we adhere to ordinary 
language, to strain words into unusual meanings ; and the reader 
is sure, after a time if not at first, to lapse into attaching the usual 
meanings to words, thus arriving at wrong notions as to what is 
intended to be said. Moreover, ordinary grammar and syntax 
is extraordinarily misleading. This is the case, e.g., as regards 
numbers ; " ten men " is grammatically the same form as 
" white men," so that 10 might be thought to be an adjective 
qualifying " men." It is the case, again, wherever propositional 
functions are involved, and in particular as regards existence and 
descriptions. Because language is misleading, as well as because 
it is diffuse and inexact when applied to logic (for which it was 
never intended), logical symbolism is absolutely necessary to 
any exact or thorough treatment of our subject. Those readers, 

1 The importance of " tautology " for a definition of mathematics was 
pointed out to me by my former pupil Ludwig Wittgenstein, who was 
working on the problem. I do not know whether he has solved it, or even 
whether he is alive or dead. 



206 Introduction to Mathematical Philosophy 

therefore, who wish to acquire a mastery of the principles of 
mathematics, will, it is to be hoped, not shrink from the labour 
of mastering the symbols a labour which is, in fact, much less 
than might be thought. As the above hasty survey must have 
made evident, there are innumerable unsolved problems in the 
subject, and much work needs to be done. If any student is 
led into a serious study of mathematical logic by this little 
book, it will have served the chief purpose for which it has been 
written. 



INDEX 



Aggregates, 12. 

Alephs, 83, 92, 97, 125. 

Aliorelatives, 32. 

All, 158 &. 

Analysis, 4. 

Ancestors, 25, 33. 

Argument of a function, 47, 108. 

Arithmetising of mathematics, 4. 

Associative law, 58, 94. 

Axioms, i. 

Between, 38 ff., 58. 
Bolzano, 138 n. 
Boots and socks, 126. 
Boundary, 70, 98, 99. 

Cantor, Georg, 77, 79, 85 ., 86, 89, 
95, 102, 136. 

Classes, 12, 137, 181 ff. ; reflexive, 80, 
127, 138 ; similar, 15, 16. 

Clifford, W. K., 76. 

Collections, infinite, 13. 

Commutative law, 58, 94. 

Conjunction, 147. 

Consecutiveness, 37, 38, 81. 

Constants, 202. 

Construction, method of, 73. 

Continuity, 86, 97 ff. ; Cantorian, 102 
ff. ; Dedekindian, 101 ; in philos 
ophy, 105 ; of functions, 106 ff. 

Contradictions, 135 ff. 

Convergence, 115. 

Converse, 16, 32, 49. 

Correlators, 54. 

Counterparts, objective, 61. 

Counting, 14, 16. 

Dedekind, 69, 99, 138 n. 
Deduction, 144 ff. 

Definition, 3 ; extensional and inten 
sion al, 12. 
Derivatives, 100. 
Descriptions, 139, 144, 167 ff. 
Dimensions, 29. 
Disjunction, 147. 
Distributive law, 58, 94. 
Diversity, 87. 
Domain, 16, 32, 49. 



Equivalence, 183. 
Euclid, 67. 

Existence, 164, 171, 177. 
Exponentiation, 94, 120. 
Extension of a relation, 60. 

Fictions, logical, 14 n., 45, 137. 

Field of a relation, 32, 53. 

Finite, 27. 

Flux, 105. 

Form, 198. 

Fractions, 37, 64. 

Frege, 7, 10, 25 n., 77, 95, 146 . 

Functions, 46 ; descriptive, 46, 180 ; 

intensional and extensional, 186 ; 

predicative, 189 ; prepositional, 46, 

144, 155 ff. 

Gap, Dedekindian, 70 ff., 99. 

Generalisation, 156. 

Geometry, 29, 59, 67, 74, 100, 145 ; 

analytical, 4, 86. 
Greater and less, 65, 90. 

Hegel, 107. 

Hereditary properties, 21. 

Implication, 146, 153 ; formal, 163. 

Incommensurables, 4, 66. 

Incompatibility, 147 ff., 200. 

Incomplete symbols, 182. 

Indiscernibles, 192. 

Individuals, 132, 141, 173. 

Induction, mathematical, 20 ff., 87, 93, 
185. 

Inductive properties, 21. 

Inference, 148 ff. 

Infinite, 28 ; of ratipnals, 65 ; Can 
torian, 65 ; of cardinals, 77 ff. ; and 
series and ordinals, 89 ff. 

Infinity, axiom of, 66 n., 77, 131 ff., 
202. 

Instances, 156. 

Integers, positive and negative, 64. 

Intervals, 115. 

Intuition, 145. 

Irrationals, 66, 72. 



307 



208 Introduction to Mathematical Philosophy 



Kant, 145. 

Leibniz, 80, 107, 192. 

Lewis, C. I., 153, 154. 

Likeness, 52. 

Limit,29,69ff.,97ff.; of functions, 1 06 ff. 

Limiting points, 99. 

Logic, 159, 169, 194 ff. ; mathematical, 

v, 201, 206. 
Logicising of mathematics, 7. 

Maps, 52, 60 ff., 80. 
Mathematics, 194 ff. 
Maximum, 70, 98. 
Median class, 104. 
Meinong, 169. 
Method, vi. 
Minimum, 70, 98. 
Modality, 165. 
Multiplication, 118 ff. 
Multiplicative axiom, 92, 117 ff. 

Names, 173, 182. 

Necessity, 165. 

Neighbourhood, 109. 

Nicod, 148, 149, 151 H. 

Null-class, 23, 132. 

Number, cardinal, 10 ff., 56, 77 ff., 95 ; 
complex, 74 ff. ; finite, 20 ff. ; in 
ductive, 27, 78, 131 ; infinite, 77 ff. ; 
irrational, 66, 72 ; maximum ? 135 ; 
multipliable, 130 ; natural, 2 ff., 22 ; 
non-inductive, 88, 127 ; real, 66, 72, 
84 ; reflexive, 80, 127 ; relation, 56, 
94 ; serial, 57- 

Occam, 184. 

Occurrences, primary and secondary, 

179- 

Ontological proof, 203. 
Order, 29 ff. ; cyclic, 40. 
Oscillation, ultimate, in. 

Parmenides, 138. 
Particulars, 140 ff., 173. 
Peano, 5 ff., 23, 24, 78, 81, 131, 163. 
Peirce, 32 n. 
Permutations, 50. 
Philosophy, mathematical, v, i. 
Plato, 138. 
Plurality, 10. 
Poincare, 27. 
Points, 59. 

Posterity, 32 ff. ; proper, 36. 
Postulates, 71, 73. 
Precedent, 98. 
Premisses of arithmetic, 5. 
Primitive ideas and propositions, 5, 202. 
Progressions, 8, 81 ff. 
Propositions, 155 ; analytic, 204 ; ele 
mentary, 161. 
Pythagoras, 4, 67. 



Quantity, 97, 195. 

Ratios, 64, 71, 84, 133. 

Reducibility, axiom of, 191. 

Referent, 48. 

Relation numbers, 56 ff. 

Relations, asymmetrical, 31, 42 ; con 
nected, 32 ; many-one, 15 ; one- 
many, 15, 45; one-one, 15, 47, 79 J 
reflexive, 16 ; serial, 34 ; similar, 
52 ff ; squares of, 32 ; symmetrical, 
16, 44 ; transitive, 16, 32. 

Relatum, 48. 

Representatives, 120. 

Rigour, 144. 

Royce, 80. 

Section, Dedekindian, 69 ff. ; ultimate, 
in. 

Segments, 72, 98. 

Selections, 117 ff. 

Sequent, 98. 

Series, 29 ff. ; closed, 103 ; compact, 
66, 93, 100 ; condensed in itself, 
102 ; Dedekindian, 71, 73, 101 ; 
generation of, 41 ; infinite, 89 ff. ; 
perfect, 102, 103 ; well-ordered, 92, 
123. 

Sheffer, 148. 

Similarity, of classes, 15 ff. ; of rela 
tions, 52 ff., 83. 

Some, 158 ff. 

Space, 61, 86, 140. 

Structure, 60 ff. 

Sub-classes, 84 ff. 

Subjects, 142. 

Subtraction, 87. 

Successor of a number, 23, 35. 

Syllogism, 197. 

Tautology, 203, 205. 

The, 167, 172 ff. 

Time, 61, 86, 140. 

Truth-function, 147. 

Truth-value, 146. 

Types, logical, 53, 135 ff., 185, i&8. 

Unreality, 168. 

Value of a function, 47, 108. 
Variables, 10, 161, 199. 
Veblen, 58. 
Verbs, 141. 

Weierstrass, 97, 107. 
Wells, H. G., 114. 
Whitehead, 64, 76, 107, 119. 
Wittgenstein, 205 n. 

Zermelo, 123, 129. 
Zero, 65. 



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Introduction to 

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