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VIEWS OF LIMITS AND NUMBERS 81 

can be put in such a relation to one another that to every 
object of each one belongs one and only one object of the other. 
We must not think that this implies that we have already the 
idea of the number one. It is true that " one and only one " 
seems to use this idea. But : " the class a has one and only 
one member,'' is simply a short way of expressing : " x is a 
member of a, and, if y is also a member of a, then y is iden- 
tical with x." It is true, also, that we use the idea of the 
unity or the individuality of the things considered. But this 
unity is a property of each individual, while the number 1 is 
a property of a class. If a class of pages of a book is itself, 
under the name of a " volume," a member of a class of books, 
the same class of pages has a number (say 360), and a unity 
as being itself a member of a class. 

The relation spoken of above in which two classes possessing 
the same number stand to one another does not involve 
counting. Think of the fingers on your hands. If to every 
finger of each hand belongs, by some process of corre- 
spondence, one and only one remember the above meaning of 
this phrase of the other, they are said to have "the same 
number." This is a definition of what " the same number " is 
to mean for us ; the word " number " by itself is to have, as 
yet, no meaning for us ; and, to avoid confusion, we had better 
replace the phrase " the same number " by the word " similar." 
Any other word would, of course, do, but this word happens to 
be fairly suggestive and customary. Now, if the variable u is 
any class, " the number of u " is defined as short for the 
phrase : " the class whose members are classes which are 
similar to u." Thus the number of u is an entity which is 
purely logical in its nature. Some people might urge that by 
" number " they mean something different from this, and that 
is quite possible. All that is maintained by those who agree 
to the process sketched above is: (1) Classes of the kind 
described are identical in all known arithmetical properties 
with the undefined things people call "integer numbers"; 
(2) It is futile to say : "These classes are not numbers" if it 
is not also said what numbers are, that is to say, if " the 
number of" is not defined in some more satisfactory way. 
There may be more satisfactory definitions, but this one is a 
perfectly sound foundation for all mathematics, including the