MATHEMATICS AND METAPHYSICIANS 87 



series, and its result tells us what type of series results 

 from this arrangement. In other words, it is impossible 

 to count things without counting some first and others 

 afterwards, so that counting always has to do with order. 

 Now when there are only a finite number of terms, we 

 can count them in any order we like ; but when there are 

 an infinite number, what corresponds to counting will 

 give us quite different results according to the way in 

 which we carry out the operation. Thus the ordinal 

 number, which results from what, in a general sense 

 may be called counting, depends not only upon how many 

 terms we have, but also (where the number of terms is 

 infinite) upon the way in which the terms are arranged. 



The fundamental infinite numbers are not ordinal, but 

 are what is called cardinal. They are not obtained by 

 putting our terms in order and counting them, but by a 

 different method, which tells us, to begin with, whether two 

 collections have the same number of terms, or, if not, 

 which is the greater. 1 It does not tell us, in the way in 

 which counting does, what number of terms a collection 

 has ; but if we define a number as the number of terms 

 in such and such a collection, then this method enables 

 us to discover whether some other collection that may be 

 mentioned has more or fewer terms. An illustration will 

 show how this is done. If there existed some country in 

 which, for one reason or another, it was impossible to 

 take a census, but in which it was known that every man 

 had a wife and every woman a husband, then (provided 

 polygamy was not a national institution) we should know, 

 without counting, that there were exactly as many men 

 as there were women in that country, neither more nor 



1 [Note added in 1917.] Although some infinite numbers are 

 greater than some others, it cannot be proved that of any two infinite 

 numbers one must be the greater. 



