History of Mathematics

required sum and product of the cardinal numbers in question. With these definitions it is now possible to prove the following six premises applying to finite cardinal numbers, from which Peanol has shown that all arithmetic can be deduced : i. Cardinal numbers form a class. ii. Zero is a cardinal number. iii. If a is a cardinal number, a+1 is a cardinal number. iv. If s is any class and zero is a member of it, also if when x is a cardinal number and a member of s, also x-F i is a member of s, then the whole class of cardinal numbers is contained in s. v. If a and b are cardinal numbers, and a+ = 1, then a=b. vi. If a is a cardinal number, then a+ I = o.

It may be noticed that (iv.) is the familiar principle of mathe matical induction. Peano in an historical note refers its first explicit employment, although without a general enunciation, to Maurolycus in his work, Arithmeticorum libri duo (Venice, 1575).

But now the difficulty of confining mathematics to being the science of number and quantity is immediately apparent. For there is no self-contained science of cardinal numbers. The proof of the six premises requires an elaborate investigation into the general properties of classes and relations which can be de duced by the strictest reasoning from our ultimate logical prin ciples. Also it is purely arbitrary to erect. the consequences of these six principles into a separate science. They are excellent principles of the highest value, but they are in no sense the neces sary premises which must be proved before any other proposi tions of cardinal numbers can be established. On the contrary, the premises of arithmetic can be put in other forms, and, further more, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, the introduction of cardinal numbers makes no great break in this general science. It is merely a subdivision in a general theory.

Nature of Ordinal Numbers.

We must first understand what is meant by "order," that is, by "serial arrangement." An order of a set of things is to be sought in that relation holding hetween members of the set which constitutes that order. The set viewed as a class has many orders. Thus the telegraph posts along a certain road have a space-order very obvious to our senses; but they have also a time-order according to dates of erection, perhaps more important to the postal authorities who replace them after fixed intervals. A set of cardinal numbers has an order of mag nitude, often called the order of the set because of its insistent obviousness to us ; but, if they are the numbers drawn in a lot tery, their time-order of occurrence in that drawing also ranges them in an order of some importance. Thus the order is defined by the "serial" relation. A relation (R) is serial' when ( 1) it implies diversity, so that, if x has the relation R to y, x is diverse from y; (2) it is transitive, so that if x has the relation R to y, and y to z, then x has the relation R to z; (3) it has the property of connexity, so that if x and y are things to which any things bear the relation R, or which bear the relation R to any things, then either x is identical with y, or x has the relation R to y, or y has the relation R to x. These conditions are necessary and suffi

cient to secure that our ordinary ideas of "preceding" and "suc ceeding" hold in respect to the relation R. The "field" of the rela tion R is the class of things ranged in order by it. Two relations R and R' are said to be ordinally similar, if a one-one relation holds between the members of the two fields of R and R', such that if x and y are any two members of the field of R, such that x has the relation R to y, and if x' and y' are the correlates in the field of R' of x and y, then in all such cases x' has the relation R' to y', and conversely, interchanging the dashes on the letters, i.e., R and R', x and x', etc. It is evident that the ordinal similarity of two relations implies the cardinal similarity of their fields, but not conversely. Also, two relations need not be serial in order to be ordinally similar; but if one is serial, so is the other. The 'Cf. Formulaire mathematique (Turin, ed. of 1903) ; earlier formu lations of the bases of arithmetic are given by him in the editions of 1898 and of 1901. The variations are only trivial.

f. Russell, loc. cit., pp. 199-256.

relation-number of a relation is the class whose members are all those relations which are ordinally similar to it. This class will include the original relation itself. The relation-number of a rela tion should be compared with the cardinal number of a class. When a relation is serial its relation-number is often called its serial type. The addition and multiplication of two relation-num bers is defined by taking two relations R and S, such that (1 ) their fields have no terms in common ; (2) their relation-numbers are the two relation-numbers in question, and then by defining by reference to R and S two other suitable relations whose relation numbers are defined to be respectively the sum and product of the relation-numbers in question. We need not consider the details of this process. Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number. This relation number is the ordinal number corresponding to n; let it be sym bolized by it. Thus, corresponding to the cardinal numbers 2, 3, 4 • • • there are the ordinal numbers 4. . . . The definition of the ordinal number I requires some little ingenuity owing to the fact that no serial relation can have a field whose cardinal number is I ; but we must omit here the explanation of the process. The ordinal number o is the class whose sole member is the null rela tion—that is, the relation which never holds between any pair of entities. The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals. Here also it can be seen that the science of the finite ordinals is merely a subdivision of the general theory of classes and relations.