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what is now a late stage in our analysis. But forthe moment the natural numbers seem to representwhat is easiest and most familiar in mathematics.But though familiar, they are not understood.Very few people are prepared with a definition ofwhat is meant by “number,” or “ ,” or “ .” It isnot very difficult to see that, starting from , anyother of the natural numbers can be reached by repeated additions of , but we shall have to definewhat we mean by “adding ,” and what we mean by“repeated.” These questions are by no means easy.It was believed until recently that some, at least, ofthese first notions of arithmetic must be acceptedas too simple and primitive to be defined. Since allterms that are defined are defined by means of otherterms, it is clear that human knowledge must always be content to accept some terms as intelligiblewithout definition, in order to have a starting-point for its definitions. It is not clear that there must beterms which are incapable of definition: it is possiblethat, however far back we go in defining, we alwaysmight go further still. On the other hand, it is alsopossible that, when analysis has been pushed farenough, 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 ourpurposes it is sufficient to observe that, since humanpowers are finite, the definitions known to us mustalways begin somewhere, with terms undefined forthe moment, though perhaps not permanently.All traditional pure mathematics, including analytical geometry, may be regarded as consistingwholly of propositions about the natural numbers.That is to say, the terms which occur can be definedby means of the natural numbers, and the propositions can be deduced from the properties of thenatural numbers—with the addition, in each case,of the ideas and propositions of pure logic.That all traditional pure mathematics can bederived 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 obstaclein the way of what is called the “arithmetising” ofmathematics. It was Pythagoras who discovered theexistence of incommensurables, and, in particular,the incommensurability of the side of a square andthe diagonal. If the length of the side is inch, thenumber of inches in the diagonal is the square rootof , which appeared not to be a number at all. The

problem thus raised was solved only in our ownday, and was only solved completely by the help ofthe reduction of arithmetic to logic, which will beexplained in following chapters. For the present, weshall take for granted the arithmetisation of mathematics, though this was a feat of the very greatestimportance. Having reduced all traditional pure mathemat- ics to the theory of the natural numbers, the nextstep in logical analysis was to reduce this theoryitself to the smallest set of premisses and undefinedterms from which it could be derived. This workwas accomplished by Peano. He showed that the entire theory of the natural numbers could be derivedfrom three primitive ideas and five primitive propositions in addition to those of pure logic. These threeideas and five propositions thus became, as it were,hostages for the whole of traditional pure mathematics. If they could be defined and proved in termsof 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 thathave been deduced from the theory of the naturalnumbers; the truth of this whole series is assured ifthe truth of the five primitive propositions is guaranteed, provided, of course, that there is nothing

erroneous in the purely logical apparatus which isalso involved. The work of analysing mathematics isextraordinarily facilitated by this work of Peano’s.The three primitive ideas in Peano’s arithmeticare: , number, successor.By “successor” he means the next number in thenatural order. That is to say, the successor of is ,the successor of is , and so on. By “number” hemeans, in this connection, the class of the naturalnumbers. He is not assuming that we know allthe members of this class, but only that we knowwhat we mean when we say that this or that is anumber, just as we know what we mean when wesay “Jones is a man,” though we do not know allmen individually.The five primitive propositions which Peano assumes are:( ) is a number.( ) The successor of any number is a number.( ) No two numbers have the same successor. ( ) is not the successor of any number. We shall use “number” in this sense in the present chapter.Afterwards the word will be used in a more general sense.

( ) Any property which belongs to , and also tothe successor of every number which has theproperty, belongs to all numbers.The last of these is the principle of mathematicalinduction. We shall have much to say concerning mathematical induction in the sequel; for thepresent, we are concerned with it only as it occursin Peano’s analysis of arithmetic.Let us consider briefly the kind of way in whichthe theory of the natural numbers results from thesethree ideas and five propositions. To begin with, wedefine as “the successor of ,” as “the successorof ,” and so on. We can obviously go on as longas we like with these definitions, since, in virtue of( ), every number that we reach will have a successor, and, in virtue of ( ), this cannot be any of thenumbers already defined, because, if it were, twodifferent numbers would have the same successor;and in virtue of ( ) none of the numbers we reach inthe series of successors can be . Thus the series ofsuccessors gives us an endless series of continuallynew numbers. In virtue of ( ) all numbers comein this series, which begins with and travels onthrough successive successors: for (a) belongs tothis series, and (b) if a number n belongs to it, sodoes its successor, whence, by mathematical induc-

tion, every number belongs to the series.Suppose we wish to define the sum of two numbers. Taking any number m, we define m as m,and m (n ) as the successor of m n. In virtueof ( ) this gives a definition of the sum of m andn, whatever number n may be. Similarly we candefine the product of any two numbers. The readercan easily convince himself that any ordinary elementary proposition of arithmetic can be provedby means of our five premisses, and if he has anydifficulty he can find the proof in Peano.It is time now to turn to the considerations whichmake it necessary to advance beyond the standpoint of Peano, who represents the last perfection of the “arithmetisation” of mathematics, to that ofFrege, who first succeeded in “logicising” mathematics, i.e. in reducing to logic the arithmeticalnotions which his predecessors had shown to besufficient for mathematics. We shall not, in thischapter, actually give Frege’s definition of numberand of particular numbers, but we shall give someof the reasons why Peano’s treatment is less finalthan it appears to be.In the first place, Peano’s three primitive ideas—namely, “ ,” “number,” and “successor”—are capable of an infinite number of different interpre-

tations, all of which will satisfy the five primitivepropositions. We will give some examples.( ) Let “ ” be taken to mean , and let “number” be taken to mean the numbers from onward in the series of natural numbers. Then all ourprimitive propositions are satisfied, even the fourth,for, though is the successor of , is not a“number” in the sense which we are now giving tothe word “number.” It is obvious that any numbermay be substituted for in this example.( ) Let “ ” have its usual meaning, but let “number” mean what we usually call “even numbers,”and let the “successor” of a number be what resultsfrom adding two to it. Then “ ” will stand for thenumber two, “ ” will stand for the number four,and so on; the series of “numbers” now will be , two, four, six, eight . . .All Peano’s five premisses are satisfied still.( ) Let “ ” mean the number one, let “number”mean the set , , , , ,.and let “successor” mean “half.” Then all Peano’sfive axioms will be true of this set.

It is clear that such examples might be multiplied indefinitely. In fact, given any seriesx , x , x , x , . . . xn , . . . which is endless, contains no repetitions, has a be- ginning, and has no terms that cannot be reachedfrom the beginning in a finite number of steps, wehave a set of terms verifying Peano’s axioms. Thisis easily seen, though the formal proof is somewhatlong. Let “ ” mean x , let “number” mean the wholeset of terms, and let the “successor” of xn mean xn .Then( ) “ is a number,” i.e. x is a member of theset.( ) “The successor of any number is a number,”i.e. taking any term xn in the set, xn is also in theset.( ) “No two numbers have the same successor,”i.e. if xm and xn are two different members of theset, xm and xn are different; this results from thefact that (by hypothesis) there are no repetitions inthe set.( ) “ is not the successor of any number,” i.e. noterm in the set comes before x .( ) This becomes: Any property which belongsto x , and belongs to xn provided it belongs to xn ,

belongs to all the x’s.This follows from the corresponding propertyfor numbers.A series of the formx , x , x , . . . xn , . . .in which there is a first term, a successor to eachterm (so that there is no last term), no repetitions,and every term can be reached from the start in afinite number of steps, is called a progression. Progressions are of great importance in the principles ofmathematics. As we have just seen, every progression verifies Peano’s five axioms. It can be proved,conversely, that every series which verifies Peano’sfive axioms is a progression. Hence these five axioms may be used to define the class of progressions: “progressions” are “those series which verifythese five axioms.” Any progression may be takenas the basis of pure mathematics: we may give thename “ ” to its first term, the name “number” to thewhole set of its terms, and the name “successor” tothe next in the progression. The progression neednot be composed of numbers: it may be composed of points in space, or moments of time, or any otherterms of which there is an infinite supply. Eachdifferent progression will give rise to a different in-

terpretation of all the propositions of traditionalpure mathematics; all these possible interpretationswill be equally true.In Peano’s system there is nothing to enable us todistinguish between these different interpretationsof his primitive ideas. It is assumed that we knowwhat is meant by “ ,” and that we shall not supposethat this symbol means or Cleopatra’s Needleor any of the other things that it might mean.This point, that “ ” and “number” and “successor” cannot be defined by means of Peano’s fiveaxioms, but must be independently understood, isimportant. We want our numbers not merely to verify mathematical formulæ, but to apply in the rightway to common objects. We want to have ten fingersand two eyes and one nose. A system in which “ ”meant , and “ ” meant , and so on, might beall right for pure mathematics, but would not suitdaily life. We want “ ” and “number” and “successor” to have meanings which will give us the rightallowance of fingers and eyes and noses. We havealready some knowledge (though not sufficientlyarticulate or analytic) of what we mean by “ ” and“ ” and so on, and our use of numbers in arithmeticmust conform to this knowledge. We cannot securethat this shall be the case by Peano’s method; all

that we can do, if we adopt his method, is to say “weknow what we mean by ‘ ’ and ‘number’ and ‘successor,’ though we cannot explain what we mean interms of other simpler concepts.” It is quite legitimate to say this when we must, and at some pointwe all must; but it is the object of mathematical philosophy to put off saying it as long as possible. Bythe logical theory of arithmetic we are able to put itoff for a very long time.It might be suggested that, instead of settingup “ ” and “number” and “successor” as terms ofwhich we know the meaning although we cannotdefine them, we might let them stand for any three terms that verify Peano’s five axioms. They will thenno longer be terms which have a meaning that isdefinite though undefined: they will be “variables,”terms concerning which we make certain hypotheses, namely, those stated in the five axioms, butwhich are otherwise undetermined. If we adopt thisplan, our theorems will not be proved concerningan ascertained set of terms called “the natural numbers,” but concerning all sets of terms having certainproperties. Such a procedure is not fallacious; indeed for certain purposes it represents a valuablegeneralisation. But from two points of view it failsto give an adequate basis for arithmetic. In the first

place, it does not enable us to know whether thereare any sets of terms verifying Peano’s axioms; itdoes not even give the faintest suggestion of anyway of discovering whether there are such sets. Inthe second place, as already observed, we want ournumbers to be such as can be used for countingcommon objects, and this requires that our numbers should have a definite meaning, not merely thatthey should have certain formal properties. Thisdefinite meaning is defined by the logical theory ofarithmetic.

Chapter IIDefinition of NumberThe question “What is a number?” is one which has been often asked, but has only been correctlyanswered in our own time. The answer was givenby Frege in , in his Grundlagen der Arithmetik. Although this book is quite short, not difficult, andof the very highest importance, it attracted almostno attention, and the definition of number which itcontains remained practically unknown until it wasrediscovered by the present author in .In seeking a definition of number, the first thingto be clear about is what we may call the grammar ofour inquiry. Many philosophers, when attemptingto define number, are really setting to work to defineplurality, which is quite a different thing. Number is The same answer is given more fully and with more development in his Grundgesetze der Arithmetik, vol. i., .

what is characteristic of numbers, as man is what ischaracteristic of men. A plurality is not an instanceof number, but of some particular number. A trio ofmen, for example, is an instance of the number ,and the number is an instance of number; but thetrio is not an instance of number. This point mayseem elementary and scarcely worth mentioning;yet it has proved too subtle for the philosophers,with few exceptions.A particular number is not identical with anycollection of terms having that number: the number is not identical with the trio consisting of Brown, Jones, and Robinson. The number is somethingwhich all trios have in common, and which distinguishes them from other collections. A numberis something that characterises certain collections,namely, those that have that number.Instead of speaking of a “collection,” we shall asa rule speak of a “class,” or sometimes a “set.” Otherwords used in mathematics for the same thing are“aggregate” and “manifold.” We shall have much tosay later on about classes. For the present, we willsay as little as possible. But there are some remarksthat must be made immediately.A class or collection may be defined in two waysthat at first sight seem quite distinct. We may enu-

merate its members, as when we say, “The collectionI mean is Brown, Jones, and Robinson.” Or we maymention a defining property, as when we speak of“mankind” or “the inhabitants of London.” The definition which enumerates is called a definition by“extension,” and the one which mentions a definingproperty is called a definition by “intension.” Ofthese two kinds of definition, the one by intension islogically more fundamental. This is shown by twoconsiderations: ( ) that the extensional definitioncan always be reduced to an intensional one; ( ) thatthe intensional one often cannot even theoreticallybe reduced to the extensional one. Each of thesepoints needs a word of explanation.( ) Brown, Jones, and Robinson all of them possess a certain property which is possessed by nothing else in the whole universe, namely, the propertyof being either Brown or Jones or Robinson. Thisproperty can be used to give a definition by intension of the class consisting of Brown and Jones andRobinson. Consider such a formula as “x is Brownor x is Jones or x is Robinson.” This formula willbe true for just three x’s, namely, Brown and Jonesand Robinson. In this respect it resembles a cubicequation with its three roots. It may be taken asassigning a property common to the members of the

class consisting of these three men, and peculiar to them. A similar treatment can obviously be appliedto any other class given in extension.( ) It is obvious that in practice we can oftenknow a great deal about a class without being ableto enumerate its members. No one man could actually enumerate all men, or even all the inhabitantsof London, yet a great deal is known about each ofthese classes. This is enough to show that definitionby extension is not necessary to knowledge about aclass. But when we come to consider infinite classes,we find that enumeration is not even theoreticallypossible for beings who only live for a finite time.We cannot enumerate all the natural numbers: theyare , , , , and so on. At some point we mustcontent ourselves with “and so on.” We cannot enumerate all fractions or all irrational numbers, or allof any other infinite collection. Thus our knowledgein regard to all such collections can only be derivedfrom a definition by intension.These remarks are relevant, when we are seekingthe definition of number, in three different ways.In the first place, numbers themselves form an infinite collection, and cannot therefore be defined byenumeration. In the second place, the collectionshaving a given number of terms themselves presum-

ably form an infinite collection: it is to be presumed,for example, that there are an infinite collection oftrios in the world, for if this were not the case thetotal number of things in the world would be finite, which, though possible, seems unlikely. In thethird place, we wish to define “number” in such away that infinite numbers may be possible; thus wemust be able to speak of the number of terms inan infinite collection, and such a collection must bedefined by intension, i.e. by a property common toall its members and

Introduction to Mathematical Philosophy by Bertrand Russell Originally published by George Allen & Unwin, Ltd., London. May 1919. Online Corrected Edition version 1:0 (February 5, 2010),